Kindergarten Grade Level Overview - Georgia Standards · Georgia Department of Education Richard...
Transcript of Kindergarten Grade Level Overview - Georgia Standards · Georgia Department of Education Richard...
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Georgia Standards of Excellence
Grade Level
Curriculum Overview
GSE Kindergarten
Mathematics
Georgia Department of Education
Georgia Department of Education
July 2018 Page 2 of 47
All Rights Reserved
Grade Level Overview
TABLE OF CONTENTS
Curriculum Maphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip4
Unpacking the Standards
bull Standards for Mathematical Practicehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip5
bull Content Standardshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip8
Mindset and Mathematics helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip18
Vertical Understanding of the Mathematic Learning Trajectory helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip19
Research of Interest to Teachers helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip20
GloSS and IKAN helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip20
Fluencyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip21
Arc of LessonMath Instructional Frameworkhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip22
Unpacking a Taskhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip23
Routines and Rituals
bull Teaching Math in Context and Through Problemshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip24
bull Use of Manipulativeshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip24
bull Use of Strategies and Effective Questioninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip25
bull 0-99 or 1-100 Charthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip26
bull Number Lines helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip28
bull Math Maintenance Activities helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip29
o Number CornerCalendar Time helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip30
o Number Talks helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip32
o EstimationEstimation 180 helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip34
Georgia Department of Education
Georgia Department of Education
July 2018 Page 3 of 47
All Rights Reserved
bull Mathematize the World through Daily Routineshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip37
bull Workstations and Learning Centershelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip38
bull Gameshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip38
bull Journalinghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip39
General Questions for Teacher Usehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip40
Questions for Teacher Reflectionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip41
Depth of Knowledgehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip42
Depth and Rigor Statementhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip44
Additional Resources Available
bull K-2 Problem Solving Rubric (creation of Richmond County Schools)helliphelliphelliphelliphelliphellip45
bull Literature Resourceshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip46
bull Technology Linkshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip46
Resources Consulted helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip47
Georgia Department of Education
Georgia Department of Education
July 2018 Page 4 of 47
All Rights Reserved
Georgia Standards of Excellence
Elementary School Mathematics
Kindergarten
NEW Click on the link in the table to view a video that shows instructional strategies for teaching the specified standard
NOTE Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among
mathematical topics
Grades K-2 Key CC = Counting and Cardinality G= Geometry MD=Measurement and Data NBT= Number and Operations in Base Ten OA = Operations and Algebraic Thinking
GSE Kindergarten Curriculum Map
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7
Comparing
Numbers
Counting With
Friends
Sophisticated
Shapes
Measuring and
Analyzing Data
Investigating
Addition and
Subtraction
Further
Investigation of
Addition and
Subtraction
Show What We
Know
MGSEKCC1
MGSEKCC2
MGSEKCC3
MGSEKCC4
MGSEKMD3
MGSEKNBT1
MGSEKCC3
MGSEKCC4a
MGSEKCC5
MGSEKCC6
MGSEKCC7
MGSEKMD3
MGSEKG1
MGSEKG2
MGSEKG3
MGSEKG4
MGSEKG5
MGSEKG6
MGSEKMD3
MGSEKMD1
MGSEKMD2
MGSEKMD3
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
ALL
These units were written to build upon concepts from prior units so later units contain tasks that depend upon the concepts and standards addressed in earlier units
All units include the Mathematical Practices and indicate skills to maintain
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Georgia Department of Education
July 2018 Page 5 of 47
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe the varieties of expertise that mathematics
educators at all levels should seek to develop in their students These practices rest on important
ldquoprocesses and proficienciesrdquo with longstanding importance in mathematics education
The first of these are the NCTM process standards of problem solving reasoning and proof
communication representation and connections
The second are the strands of mathematical proficiency specified in the National Research
Councilrsquos report Adding It Up adaptive reasoning strategic competence conceptual
understanding (comprehension of mathematical concepts operations and relations) procedural
fluency (skill in carrying out procedures flexibly accurately efficiently and appropriately) and
productive disposition (habitual inclination to see mathematics as sensible useful and
worthwhile coupled with a belief in diligence and onersquos own efficacy)
Students are expected to
1 Make sense of problems and persevere in solving them
Mathematically proficient students in Kindergarten begin to develop effective dispositions
toward problem solving In rich settings in which informal and formal possibilities for solving
problems are numerous young children develop the ability to focus attention test hypotheses
take reasonable risks remain flexible try alternatives exhibit self-regulation and persevere
(Copley 2010) Using both verbal and nonverbal means kindergarten students begin to explain
to themselves and others the meaning of a problem look for ways to solve it and determine if
their thinking makes sense or if another strategy is needed As the teacher uses thoughtful
questioning and provides opportunities for students to share thinking kindergarten students begin
to reason as they become more conscious of what they know and how they solve problems
2 Reason abstractly and quantitatively
Mathematically proficient students in Kindergarten begin to use numerals to represent a specific
amount (quantity) For example a student may write the numeral ldquo11rdquo to represent an amount of
objects counted select the correct number card ldquo17rdquo to follow ldquo16rdquo on the calendar or build a
pile of counters depending on the number drawn In addition kindergarten students begin to
draw pictures manipulate objects use diagrams or charts etc to express quantitative ideas such
as a joining situation (Mary has 3 bears Juanita gave her 1 more bear How many bears does
Mary have altogether) or a separating situation (Mary had 5 bears She gave some to Juanita
Now she has 3 bears How many bears did Mary give Juanita) Using the language developed
through numerous joining and separating scenarios kindergarten students begin to understand
how symbols (+ - =) are used to represent quantitative ideas in a written format
3 Construct viable arguments and critique the reasoning of others
In Kindergarten mathematically proficient students begin to clearly express explain organize
and consolidate their math thinking using both verbal and written representations Through
opportunities that encourage exploration discovery and discussion kindergarten students begin
to learn how to express opinions become skillful at listening to others describe their reasoning
and respond to othersrsquo thinking and reasoning They begin to develop the ability to reason and
Georgia Department of Education
Georgia Department of Education
July 2018 Page 6 of 47
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analyze situations as they consider questions such as ldquoAre you surehelliprdquo ldquoDo you think that
would happen all the timehelliprdquo and ldquoI wonder whyhelliprdquo
4 Model with mathematics
Mathematically proficient students in Kindergarten begin to experiment with representing real-
life problem situations in multiple ways such as with numbers words (mathematical language)
drawings objects acting out charts lists and number sentences For example when making
toothpick designs to represent the various combinations of the number ldquo5rdquo the student writes the
numerals for the various parts (such as ldquo4rdquo and ldquo1rdquo) or selects a number sentence that represents
that particular situation (such as 5 = 4 + 1)
Kindergarten students should see addition and subtraction equations and student writing of
equations in kindergarten is encouraged but it is not required However please note that it is
not until First Grade when ldquoUnderstand the meaning of the equal signrdquo is an expectation
(1OA7)
5 Use appropriate tools strategically
In Kindergarten mathematically proficient students begin to explore various tools and use them
to investigate mathematical concepts Through multiple opportunities to examine materials they
experiment and use both concrete materials (eg 3- dimensional solids connecting cubes ten
frames number balances) and technological materials (eg virtual manipulatives calculators
interactive websites) to explore mathematical concepts Based on these experiences they
become able to decide which tools may be helpful to use depending on the problem or task For
example when solving the problem ldquoThere are 4 dogs in the park 3 more dogs show up in the
park How many dogs are in the parkrdquo students may decide to act it out using counters and a
story mat draw a picture or use a handful of cubes
6 Attend to precision
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning
using words As their mathematical vocabulary increases due to exposure modeling and
practice kindergarteners become more precise in their communication calculations and
measurements In all types of mathematical tasks students begin to describe their actions and
strategies more clearly understand and use grade-level appropriate vocabulary accurately and
begin to give precise explanations and reasoning regarding their process of finding solutions For
example a student may use color words (such as blue green light blue) and descriptive words
(such as small big rough smooth) to accurately describe how a collection of buttons is sorted
7 Look for and make use of structure
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the
number system and other areas of mathematics For example when searching for triangles
around the room kindergarteners begin to notice that some triangles are larger than others or
come in different colors- yet they are all triangles While exploring the part-whole relationships
of a number using a number balance students begin to realize that 5 can be broken down into
sub-parts such as 4 and 1 or 3 and 2 and still remain a total of 5
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Georgia Department of Education
July 2018 Page 7 of 47
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8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
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July 2018 Page 8 of 47
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CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
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July 2018 Page 10 of 47
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b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
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Georgia Department of Education
July 2018 Page 11 of 47
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OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
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July 2018 Page 18 of 47
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MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
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Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
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FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
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ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
Georgia Department of Education
July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
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reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
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All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
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MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 2 of 47
All Rights Reserved
Grade Level Overview
TABLE OF CONTENTS
Curriculum Maphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip4
Unpacking the Standards
bull Standards for Mathematical Practicehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip5
bull Content Standardshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip8
Mindset and Mathematics helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip18
Vertical Understanding of the Mathematic Learning Trajectory helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip19
Research of Interest to Teachers helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip20
GloSS and IKAN helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip20
Fluencyhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip21
Arc of LessonMath Instructional Frameworkhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip22
Unpacking a Taskhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip23
Routines and Rituals
bull Teaching Math in Context and Through Problemshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip24
bull Use of Manipulativeshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip24
bull Use of Strategies and Effective Questioninghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip25
bull 0-99 or 1-100 Charthelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip26
bull Number Lines helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip28
bull Math Maintenance Activities helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip29
o Number CornerCalendar Time helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip30
o Number Talks helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip32
o EstimationEstimation 180 helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip34
Georgia Department of Education
Georgia Department of Education
July 2018 Page 3 of 47
All Rights Reserved
bull Mathematize the World through Daily Routineshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip37
bull Workstations and Learning Centershelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip38
bull Gameshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip38
bull Journalinghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip39
General Questions for Teacher Usehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip40
Questions for Teacher Reflectionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip41
Depth of Knowledgehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip42
Depth and Rigor Statementhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip44
Additional Resources Available
bull K-2 Problem Solving Rubric (creation of Richmond County Schools)helliphelliphelliphelliphelliphellip45
bull Literature Resourceshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip46
bull Technology Linkshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip46
Resources Consulted helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip47
Georgia Department of Education
Georgia Department of Education
July 2018 Page 4 of 47
All Rights Reserved
Georgia Standards of Excellence
Elementary School Mathematics
Kindergarten
NEW Click on the link in the table to view a video that shows instructional strategies for teaching the specified standard
NOTE Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among
mathematical topics
Grades K-2 Key CC = Counting and Cardinality G= Geometry MD=Measurement and Data NBT= Number and Operations in Base Ten OA = Operations and Algebraic Thinking
GSE Kindergarten Curriculum Map
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7
Comparing
Numbers
Counting With
Friends
Sophisticated
Shapes
Measuring and
Analyzing Data
Investigating
Addition and
Subtraction
Further
Investigation of
Addition and
Subtraction
Show What We
Know
MGSEKCC1
MGSEKCC2
MGSEKCC3
MGSEKCC4
MGSEKMD3
MGSEKNBT1
MGSEKCC3
MGSEKCC4a
MGSEKCC5
MGSEKCC6
MGSEKCC7
MGSEKMD3
MGSEKG1
MGSEKG2
MGSEKG3
MGSEKG4
MGSEKG5
MGSEKG6
MGSEKMD3
MGSEKMD1
MGSEKMD2
MGSEKMD3
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
ALL
These units were written to build upon concepts from prior units so later units contain tasks that depend upon the concepts and standards addressed in earlier units
All units include the Mathematical Practices and indicate skills to maintain
Georgia Department of Education
Georgia Department of Education
July 2018 Page 5 of 47
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe the varieties of expertise that mathematics
educators at all levels should seek to develop in their students These practices rest on important
ldquoprocesses and proficienciesrdquo with longstanding importance in mathematics education
The first of these are the NCTM process standards of problem solving reasoning and proof
communication representation and connections
The second are the strands of mathematical proficiency specified in the National Research
Councilrsquos report Adding It Up adaptive reasoning strategic competence conceptual
understanding (comprehension of mathematical concepts operations and relations) procedural
fluency (skill in carrying out procedures flexibly accurately efficiently and appropriately) and
productive disposition (habitual inclination to see mathematics as sensible useful and
worthwhile coupled with a belief in diligence and onersquos own efficacy)
Students are expected to
1 Make sense of problems and persevere in solving them
Mathematically proficient students in Kindergarten begin to develop effective dispositions
toward problem solving In rich settings in which informal and formal possibilities for solving
problems are numerous young children develop the ability to focus attention test hypotheses
take reasonable risks remain flexible try alternatives exhibit self-regulation and persevere
(Copley 2010) Using both verbal and nonverbal means kindergarten students begin to explain
to themselves and others the meaning of a problem look for ways to solve it and determine if
their thinking makes sense or if another strategy is needed As the teacher uses thoughtful
questioning and provides opportunities for students to share thinking kindergarten students begin
to reason as they become more conscious of what they know and how they solve problems
2 Reason abstractly and quantitatively
Mathematically proficient students in Kindergarten begin to use numerals to represent a specific
amount (quantity) For example a student may write the numeral ldquo11rdquo to represent an amount of
objects counted select the correct number card ldquo17rdquo to follow ldquo16rdquo on the calendar or build a
pile of counters depending on the number drawn In addition kindergarten students begin to
draw pictures manipulate objects use diagrams or charts etc to express quantitative ideas such
as a joining situation (Mary has 3 bears Juanita gave her 1 more bear How many bears does
Mary have altogether) or a separating situation (Mary had 5 bears She gave some to Juanita
Now she has 3 bears How many bears did Mary give Juanita) Using the language developed
through numerous joining and separating scenarios kindergarten students begin to understand
how symbols (+ - =) are used to represent quantitative ideas in a written format
3 Construct viable arguments and critique the reasoning of others
In Kindergarten mathematically proficient students begin to clearly express explain organize
and consolidate their math thinking using both verbal and written representations Through
opportunities that encourage exploration discovery and discussion kindergarten students begin
to learn how to express opinions become skillful at listening to others describe their reasoning
and respond to othersrsquo thinking and reasoning They begin to develop the ability to reason and
Georgia Department of Education
Georgia Department of Education
July 2018 Page 6 of 47
All Rights Reserved
analyze situations as they consider questions such as ldquoAre you surehelliprdquo ldquoDo you think that
would happen all the timehelliprdquo and ldquoI wonder whyhelliprdquo
4 Model with mathematics
Mathematically proficient students in Kindergarten begin to experiment with representing real-
life problem situations in multiple ways such as with numbers words (mathematical language)
drawings objects acting out charts lists and number sentences For example when making
toothpick designs to represent the various combinations of the number ldquo5rdquo the student writes the
numerals for the various parts (such as ldquo4rdquo and ldquo1rdquo) or selects a number sentence that represents
that particular situation (such as 5 = 4 + 1)
Kindergarten students should see addition and subtraction equations and student writing of
equations in kindergarten is encouraged but it is not required However please note that it is
not until First Grade when ldquoUnderstand the meaning of the equal signrdquo is an expectation
(1OA7)
5 Use appropriate tools strategically
In Kindergarten mathematically proficient students begin to explore various tools and use them
to investigate mathematical concepts Through multiple opportunities to examine materials they
experiment and use both concrete materials (eg 3- dimensional solids connecting cubes ten
frames number balances) and technological materials (eg virtual manipulatives calculators
interactive websites) to explore mathematical concepts Based on these experiences they
become able to decide which tools may be helpful to use depending on the problem or task For
example when solving the problem ldquoThere are 4 dogs in the park 3 more dogs show up in the
park How many dogs are in the parkrdquo students may decide to act it out using counters and a
story mat draw a picture or use a handful of cubes
6 Attend to precision
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning
using words As their mathematical vocabulary increases due to exposure modeling and
practice kindergarteners become more precise in their communication calculations and
measurements In all types of mathematical tasks students begin to describe their actions and
strategies more clearly understand and use grade-level appropriate vocabulary accurately and
begin to give precise explanations and reasoning regarding their process of finding solutions For
example a student may use color words (such as blue green light blue) and descriptive words
(such as small big rough smooth) to accurately describe how a collection of buttons is sorted
7 Look for and make use of structure
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the
number system and other areas of mathematics For example when searching for triangles
around the room kindergarteners begin to notice that some triangles are larger than others or
come in different colors- yet they are all triangles While exploring the part-whole relationships
of a number using a number balance students begin to realize that 5 can be broken down into
sub-parts such as 4 and 1 or 3 and 2 and still remain a total of 5
Georgia Department of Education
Georgia Department of Education
July 2018 Page 7 of 47
All Rights Reserved
8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
Georgia Department of Education
Georgia Department of Education
July 2018 Page 8 of 47
All Rights Reserved
CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
Georgia Department of Education
July 2018 Page 11 of 47
All Rights Reserved
OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
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July 2018 Page 13 of 47
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
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July 2018 Page 14 of 47
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are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
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July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
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July 2018 Page 18 of 47
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MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
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reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
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July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
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July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 3 of 47
All Rights Reserved
bull Mathematize the World through Daily Routineshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip37
bull Workstations and Learning Centershelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip38
bull Gameshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip38
bull Journalinghelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip39
General Questions for Teacher Usehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip40
Questions for Teacher Reflectionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip41
Depth of Knowledgehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip42
Depth and Rigor Statementhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip44
Additional Resources Available
bull K-2 Problem Solving Rubric (creation of Richmond County Schools)helliphelliphelliphelliphelliphellip45
bull Literature Resourceshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip46
bull Technology Linkshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip46
Resources Consulted helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip47
Georgia Department of Education
Georgia Department of Education
July 2018 Page 4 of 47
All Rights Reserved
Georgia Standards of Excellence
Elementary School Mathematics
Kindergarten
NEW Click on the link in the table to view a video that shows instructional strategies for teaching the specified standard
NOTE Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among
mathematical topics
Grades K-2 Key CC = Counting and Cardinality G= Geometry MD=Measurement and Data NBT= Number and Operations in Base Ten OA = Operations and Algebraic Thinking
GSE Kindergarten Curriculum Map
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7
Comparing
Numbers
Counting With
Friends
Sophisticated
Shapes
Measuring and
Analyzing Data
Investigating
Addition and
Subtraction
Further
Investigation of
Addition and
Subtraction
Show What We
Know
MGSEKCC1
MGSEKCC2
MGSEKCC3
MGSEKCC4
MGSEKMD3
MGSEKNBT1
MGSEKCC3
MGSEKCC4a
MGSEKCC5
MGSEKCC6
MGSEKCC7
MGSEKMD3
MGSEKG1
MGSEKG2
MGSEKG3
MGSEKG4
MGSEKG5
MGSEKG6
MGSEKMD3
MGSEKMD1
MGSEKMD2
MGSEKMD3
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
ALL
These units were written to build upon concepts from prior units so later units contain tasks that depend upon the concepts and standards addressed in earlier units
All units include the Mathematical Practices and indicate skills to maintain
Georgia Department of Education
Georgia Department of Education
July 2018 Page 5 of 47
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe the varieties of expertise that mathematics
educators at all levels should seek to develop in their students These practices rest on important
ldquoprocesses and proficienciesrdquo with longstanding importance in mathematics education
The first of these are the NCTM process standards of problem solving reasoning and proof
communication representation and connections
The second are the strands of mathematical proficiency specified in the National Research
Councilrsquos report Adding It Up adaptive reasoning strategic competence conceptual
understanding (comprehension of mathematical concepts operations and relations) procedural
fluency (skill in carrying out procedures flexibly accurately efficiently and appropriately) and
productive disposition (habitual inclination to see mathematics as sensible useful and
worthwhile coupled with a belief in diligence and onersquos own efficacy)
Students are expected to
1 Make sense of problems and persevere in solving them
Mathematically proficient students in Kindergarten begin to develop effective dispositions
toward problem solving In rich settings in which informal and formal possibilities for solving
problems are numerous young children develop the ability to focus attention test hypotheses
take reasonable risks remain flexible try alternatives exhibit self-regulation and persevere
(Copley 2010) Using both verbal and nonverbal means kindergarten students begin to explain
to themselves and others the meaning of a problem look for ways to solve it and determine if
their thinking makes sense or if another strategy is needed As the teacher uses thoughtful
questioning and provides opportunities for students to share thinking kindergarten students begin
to reason as they become more conscious of what they know and how they solve problems
2 Reason abstractly and quantitatively
Mathematically proficient students in Kindergarten begin to use numerals to represent a specific
amount (quantity) For example a student may write the numeral ldquo11rdquo to represent an amount of
objects counted select the correct number card ldquo17rdquo to follow ldquo16rdquo on the calendar or build a
pile of counters depending on the number drawn In addition kindergarten students begin to
draw pictures manipulate objects use diagrams or charts etc to express quantitative ideas such
as a joining situation (Mary has 3 bears Juanita gave her 1 more bear How many bears does
Mary have altogether) or a separating situation (Mary had 5 bears She gave some to Juanita
Now she has 3 bears How many bears did Mary give Juanita) Using the language developed
through numerous joining and separating scenarios kindergarten students begin to understand
how symbols (+ - =) are used to represent quantitative ideas in a written format
3 Construct viable arguments and critique the reasoning of others
In Kindergarten mathematically proficient students begin to clearly express explain organize
and consolidate their math thinking using both verbal and written representations Through
opportunities that encourage exploration discovery and discussion kindergarten students begin
to learn how to express opinions become skillful at listening to others describe their reasoning
and respond to othersrsquo thinking and reasoning They begin to develop the ability to reason and
Georgia Department of Education
Georgia Department of Education
July 2018 Page 6 of 47
All Rights Reserved
analyze situations as they consider questions such as ldquoAre you surehelliprdquo ldquoDo you think that
would happen all the timehelliprdquo and ldquoI wonder whyhelliprdquo
4 Model with mathematics
Mathematically proficient students in Kindergarten begin to experiment with representing real-
life problem situations in multiple ways such as with numbers words (mathematical language)
drawings objects acting out charts lists and number sentences For example when making
toothpick designs to represent the various combinations of the number ldquo5rdquo the student writes the
numerals for the various parts (such as ldquo4rdquo and ldquo1rdquo) or selects a number sentence that represents
that particular situation (such as 5 = 4 + 1)
Kindergarten students should see addition and subtraction equations and student writing of
equations in kindergarten is encouraged but it is not required However please note that it is
not until First Grade when ldquoUnderstand the meaning of the equal signrdquo is an expectation
(1OA7)
5 Use appropriate tools strategically
In Kindergarten mathematically proficient students begin to explore various tools and use them
to investigate mathematical concepts Through multiple opportunities to examine materials they
experiment and use both concrete materials (eg 3- dimensional solids connecting cubes ten
frames number balances) and technological materials (eg virtual manipulatives calculators
interactive websites) to explore mathematical concepts Based on these experiences they
become able to decide which tools may be helpful to use depending on the problem or task For
example when solving the problem ldquoThere are 4 dogs in the park 3 more dogs show up in the
park How many dogs are in the parkrdquo students may decide to act it out using counters and a
story mat draw a picture or use a handful of cubes
6 Attend to precision
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning
using words As their mathematical vocabulary increases due to exposure modeling and
practice kindergarteners become more precise in their communication calculations and
measurements In all types of mathematical tasks students begin to describe their actions and
strategies more clearly understand and use grade-level appropriate vocabulary accurately and
begin to give precise explanations and reasoning regarding their process of finding solutions For
example a student may use color words (such as blue green light blue) and descriptive words
(such as small big rough smooth) to accurately describe how a collection of buttons is sorted
7 Look for and make use of structure
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the
number system and other areas of mathematics For example when searching for triangles
around the room kindergarteners begin to notice that some triangles are larger than others or
come in different colors- yet they are all triangles While exploring the part-whole relationships
of a number using a number balance students begin to realize that 5 can be broken down into
sub-parts such as 4 and 1 or 3 and 2 and still remain a total of 5
Georgia Department of Education
Georgia Department of Education
July 2018 Page 7 of 47
All Rights Reserved
8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
Georgia Department of Education
Georgia Department of Education
July 2018 Page 8 of 47
All Rights Reserved
CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
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July 2018 Page 11 of 47
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OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
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are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 4 of 47
All Rights Reserved
Georgia Standards of Excellence
Elementary School Mathematics
Kindergarten
NEW Click on the link in the table to view a video that shows instructional strategies for teaching the specified standard
NOTE Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among
mathematical topics
Grades K-2 Key CC = Counting and Cardinality G= Geometry MD=Measurement and Data NBT= Number and Operations in Base Ten OA = Operations and Algebraic Thinking
GSE Kindergarten Curriculum Map
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7
Comparing
Numbers
Counting With
Friends
Sophisticated
Shapes
Measuring and
Analyzing Data
Investigating
Addition and
Subtraction
Further
Investigation of
Addition and
Subtraction
Show What We
Know
MGSEKCC1
MGSEKCC2
MGSEKCC3
MGSEKCC4
MGSEKMD3
MGSEKNBT1
MGSEKCC3
MGSEKCC4a
MGSEKCC5
MGSEKCC6
MGSEKCC7
MGSEKMD3
MGSEKG1
MGSEKG2
MGSEKG3
MGSEKG4
MGSEKG5
MGSEKG6
MGSEKMD3
MGSEKMD1
MGSEKMD2
MGSEKMD3
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
MGSEKOA1
MGSEKOA2
MGSEKOA3
MGSEKOA4
MGSEKOA5
ALL
These units were written to build upon concepts from prior units so later units contain tasks that depend upon the concepts and standards addressed in earlier units
All units include the Mathematical Practices and indicate skills to maintain
Georgia Department of Education
Georgia Department of Education
July 2018 Page 5 of 47
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe the varieties of expertise that mathematics
educators at all levels should seek to develop in their students These practices rest on important
ldquoprocesses and proficienciesrdquo with longstanding importance in mathematics education
The first of these are the NCTM process standards of problem solving reasoning and proof
communication representation and connections
The second are the strands of mathematical proficiency specified in the National Research
Councilrsquos report Adding It Up adaptive reasoning strategic competence conceptual
understanding (comprehension of mathematical concepts operations and relations) procedural
fluency (skill in carrying out procedures flexibly accurately efficiently and appropriately) and
productive disposition (habitual inclination to see mathematics as sensible useful and
worthwhile coupled with a belief in diligence and onersquos own efficacy)
Students are expected to
1 Make sense of problems and persevere in solving them
Mathematically proficient students in Kindergarten begin to develop effective dispositions
toward problem solving In rich settings in which informal and formal possibilities for solving
problems are numerous young children develop the ability to focus attention test hypotheses
take reasonable risks remain flexible try alternatives exhibit self-regulation and persevere
(Copley 2010) Using both verbal and nonverbal means kindergarten students begin to explain
to themselves and others the meaning of a problem look for ways to solve it and determine if
their thinking makes sense or if another strategy is needed As the teacher uses thoughtful
questioning and provides opportunities for students to share thinking kindergarten students begin
to reason as they become more conscious of what they know and how they solve problems
2 Reason abstractly and quantitatively
Mathematically proficient students in Kindergarten begin to use numerals to represent a specific
amount (quantity) For example a student may write the numeral ldquo11rdquo to represent an amount of
objects counted select the correct number card ldquo17rdquo to follow ldquo16rdquo on the calendar or build a
pile of counters depending on the number drawn In addition kindergarten students begin to
draw pictures manipulate objects use diagrams or charts etc to express quantitative ideas such
as a joining situation (Mary has 3 bears Juanita gave her 1 more bear How many bears does
Mary have altogether) or a separating situation (Mary had 5 bears She gave some to Juanita
Now she has 3 bears How many bears did Mary give Juanita) Using the language developed
through numerous joining and separating scenarios kindergarten students begin to understand
how symbols (+ - =) are used to represent quantitative ideas in a written format
3 Construct viable arguments and critique the reasoning of others
In Kindergarten mathematically proficient students begin to clearly express explain organize
and consolidate their math thinking using both verbal and written representations Through
opportunities that encourage exploration discovery and discussion kindergarten students begin
to learn how to express opinions become skillful at listening to others describe their reasoning
and respond to othersrsquo thinking and reasoning They begin to develop the ability to reason and
Georgia Department of Education
Georgia Department of Education
July 2018 Page 6 of 47
All Rights Reserved
analyze situations as they consider questions such as ldquoAre you surehelliprdquo ldquoDo you think that
would happen all the timehelliprdquo and ldquoI wonder whyhelliprdquo
4 Model with mathematics
Mathematically proficient students in Kindergarten begin to experiment with representing real-
life problem situations in multiple ways such as with numbers words (mathematical language)
drawings objects acting out charts lists and number sentences For example when making
toothpick designs to represent the various combinations of the number ldquo5rdquo the student writes the
numerals for the various parts (such as ldquo4rdquo and ldquo1rdquo) or selects a number sentence that represents
that particular situation (such as 5 = 4 + 1)
Kindergarten students should see addition and subtraction equations and student writing of
equations in kindergarten is encouraged but it is not required However please note that it is
not until First Grade when ldquoUnderstand the meaning of the equal signrdquo is an expectation
(1OA7)
5 Use appropriate tools strategically
In Kindergarten mathematically proficient students begin to explore various tools and use them
to investigate mathematical concepts Through multiple opportunities to examine materials they
experiment and use both concrete materials (eg 3- dimensional solids connecting cubes ten
frames number balances) and technological materials (eg virtual manipulatives calculators
interactive websites) to explore mathematical concepts Based on these experiences they
become able to decide which tools may be helpful to use depending on the problem or task For
example when solving the problem ldquoThere are 4 dogs in the park 3 more dogs show up in the
park How many dogs are in the parkrdquo students may decide to act it out using counters and a
story mat draw a picture or use a handful of cubes
6 Attend to precision
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning
using words As their mathematical vocabulary increases due to exposure modeling and
practice kindergarteners become more precise in their communication calculations and
measurements In all types of mathematical tasks students begin to describe their actions and
strategies more clearly understand and use grade-level appropriate vocabulary accurately and
begin to give precise explanations and reasoning regarding their process of finding solutions For
example a student may use color words (such as blue green light blue) and descriptive words
(such as small big rough smooth) to accurately describe how a collection of buttons is sorted
7 Look for and make use of structure
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the
number system and other areas of mathematics For example when searching for triangles
around the room kindergarteners begin to notice that some triangles are larger than others or
come in different colors- yet they are all triangles While exploring the part-whole relationships
of a number using a number balance students begin to realize that 5 can be broken down into
sub-parts such as 4 and 1 or 3 and 2 and still remain a total of 5
Georgia Department of Education
Georgia Department of Education
July 2018 Page 7 of 47
All Rights Reserved
8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
Georgia Department of Education
Georgia Department of Education
July 2018 Page 8 of 47
All Rights Reserved
CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
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July 2018 Page 11 of 47
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OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
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are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 5 of 47
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
The Standards for Mathematical Practice describe the varieties of expertise that mathematics
educators at all levels should seek to develop in their students These practices rest on important
ldquoprocesses and proficienciesrdquo with longstanding importance in mathematics education
The first of these are the NCTM process standards of problem solving reasoning and proof
communication representation and connections
The second are the strands of mathematical proficiency specified in the National Research
Councilrsquos report Adding It Up adaptive reasoning strategic competence conceptual
understanding (comprehension of mathematical concepts operations and relations) procedural
fluency (skill in carrying out procedures flexibly accurately efficiently and appropriately) and
productive disposition (habitual inclination to see mathematics as sensible useful and
worthwhile coupled with a belief in diligence and onersquos own efficacy)
Students are expected to
1 Make sense of problems and persevere in solving them
Mathematically proficient students in Kindergarten begin to develop effective dispositions
toward problem solving In rich settings in which informal and formal possibilities for solving
problems are numerous young children develop the ability to focus attention test hypotheses
take reasonable risks remain flexible try alternatives exhibit self-regulation and persevere
(Copley 2010) Using both verbal and nonverbal means kindergarten students begin to explain
to themselves and others the meaning of a problem look for ways to solve it and determine if
their thinking makes sense or if another strategy is needed As the teacher uses thoughtful
questioning and provides opportunities for students to share thinking kindergarten students begin
to reason as they become more conscious of what they know and how they solve problems
2 Reason abstractly and quantitatively
Mathematically proficient students in Kindergarten begin to use numerals to represent a specific
amount (quantity) For example a student may write the numeral ldquo11rdquo to represent an amount of
objects counted select the correct number card ldquo17rdquo to follow ldquo16rdquo on the calendar or build a
pile of counters depending on the number drawn In addition kindergarten students begin to
draw pictures manipulate objects use diagrams or charts etc to express quantitative ideas such
as a joining situation (Mary has 3 bears Juanita gave her 1 more bear How many bears does
Mary have altogether) or a separating situation (Mary had 5 bears She gave some to Juanita
Now she has 3 bears How many bears did Mary give Juanita) Using the language developed
through numerous joining and separating scenarios kindergarten students begin to understand
how symbols (+ - =) are used to represent quantitative ideas in a written format
3 Construct viable arguments and critique the reasoning of others
In Kindergarten mathematically proficient students begin to clearly express explain organize
and consolidate their math thinking using both verbal and written representations Through
opportunities that encourage exploration discovery and discussion kindergarten students begin
to learn how to express opinions become skillful at listening to others describe their reasoning
and respond to othersrsquo thinking and reasoning They begin to develop the ability to reason and
Georgia Department of Education
Georgia Department of Education
July 2018 Page 6 of 47
All Rights Reserved
analyze situations as they consider questions such as ldquoAre you surehelliprdquo ldquoDo you think that
would happen all the timehelliprdquo and ldquoI wonder whyhelliprdquo
4 Model with mathematics
Mathematically proficient students in Kindergarten begin to experiment with representing real-
life problem situations in multiple ways such as with numbers words (mathematical language)
drawings objects acting out charts lists and number sentences For example when making
toothpick designs to represent the various combinations of the number ldquo5rdquo the student writes the
numerals for the various parts (such as ldquo4rdquo and ldquo1rdquo) or selects a number sentence that represents
that particular situation (such as 5 = 4 + 1)
Kindergarten students should see addition and subtraction equations and student writing of
equations in kindergarten is encouraged but it is not required However please note that it is
not until First Grade when ldquoUnderstand the meaning of the equal signrdquo is an expectation
(1OA7)
5 Use appropriate tools strategically
In Kindergarten mathematically proficient students begin to explore various tools and use them
to investigate mathematical concepts Through multiple opportunities to examine materials they
experiment and use both concrete materials (eg 3- dimensional solids connecting cubes ten
frames number balances) and technological materials (eg virtual manipulatives calculators
interactive websites) to explore mathematical concepts Based on these experiences they
become able to decide which tools may be helpful to use depending on the problem or task For
example when solving the problem ldquoThere are 4 dogs in the park 3 more dogs show up in the
park How many dogs are in the parkrdquo students may decide to act it out using counters and a
story mat draw a picture or use a handful of cubes
6 Attend to precision
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning
using words As their mathematical vocabulary increases due to exposure modeling and
practice kindergarteners become more precise in their communication calculations and
measurements In all types of mathematical tasks students begin to describe their actions and
strategies more clearly understand and use grade-level appropriate vocabulary accurately and
begin to give precise explanations and reasoning regarding their process of finding solutions For
example a student may use color words (such as blue green light blue) and descriptive words
(such as small big rough smooth) to accurately describe how a collection of buttons is sorted
7 Look for and make use of structure
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the
number system and other areas of mathematics For example when searching for triangles
around the room kindergarteners begin to notice that some triangles are larger than others or
come in different colors- yet they are all triangles While exploring the part-whole relationships
of a number using a number balance students begin to realize that 5 can be broken down into
sub-parts such as 4 and 1 or 3 and 2 and still remain a total of 5
Georgia Department of Education
Georgia Department of Education
July 2018 Page 7 of 47
All Rights Reserved
8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
Georgia Department of Education
Georgia Department of Education
July 2018 Page 8 of 47
All Rights Reserved
CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
Georgia Department of Education
July 2018 Page 11 of 47
All Rights Reserved
OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
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July 2018 Page 14 of 47
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are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
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July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
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July 2018 Page 18 of 47
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MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 6 of 47
All Rights Reserved
analyze situations as they consider questions such as ldquoAre you surehelliprdquo ldquoDo you think that
would happen all the timehelliprdquo and ldquoI wonder whyhelliprdquo
4 Model with mathematics
Mathematically proficient students in Kindergarten begin to experiment with representing real-
life problem situations in multiple ways such as with numbers words (mathematical language)
drawings objects acting out charts lists and number sentences For example when making
toothpick designs to represent the various combinations of the number ldquo5rdquo the student writes the
numerals for the various parts (such as ldquo4rdquo and ldquo1rdquo) or selects a number sentence that represents
that particular situation (such as 5 = 4 + 1)
Kindergarten students should see addition and subtraction equations and student writing of
equations in kindergarten is encouraged but it is not required However please note that it is
not until First Grade when ldquoUnderstand the meaning of the equal signrdquo is an expectation
(1OA7)
5 Use appropriate tools strategically
In Kindergarten mathematically proficient students begin to explore various tools and use them
to investigate mathematical concepts Through multiple opportunities to examine materials they
experiment and use both concrete materials (eg 3- dimensional solids connecting cubes ten
frames number balances) and technological materials (eg virtual manipulatives calculators
interactive websites) to explore mathematical concepts Based on these experiences they
become able to decide which tools may be helpful to use depending on the problem or task For
example when solving the problem ldquoThere are 4 dogs in the park 3 more dogs show up in the
park How many dogs are in the parkrdquo students may decide to act it out using counters and a
story mat draw a picture or use a handful of cubes
6 Attend to precision
Mathematically proficient students in Kindergarten begin to express their ideas and reasoning
using words As their mathematical vocabulary increases due to exposure modeling and
practice kindergarteners become more precise in their communication calculations and
measurements In all types of mathematical tasks students begin to describe their actions and
strategies more clearly understand and use grade-level appropriate vocabulary accurately and
begin to give precise explanations and reasoning regarding their process of finding solutions For
example a student may use color words (such as blue green light blue) and descriptive words
(such as small big rough smooth) to accurately describe how a collection of buttons is sorted
7 Look for and make use of structure
Mathematically proficient students in Kindergarten begin to look for patterns and structures in the
number system and other areas of mathematics For example when searching for triangles
around the room kindergarteners begin to notice that some triangles are larger than others or
come in different colors- yet they are all triangles While exploring the part-whole relationships
of a number using a number balance students begin to realize that 5 can be broken down into
sub-parts such as 4 and 1 or 3 and 2 and still remain a total of 5
Georgia Department of Education
Georgia Department of Education
July 2018 Page 7 of 47
All Rights Reserved
8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
Georgia Department of Education
Georgia Department of Education
July 2018 Page 8 of 47
All Rights Reserved
CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
Georgia Department of Education
July 2018 Page 11 of 47
All Rights Reserved
OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
All Rights Reserved
MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
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July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
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into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
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July 2018 Page 18 of 47
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MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
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FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
All Rights Reserved
Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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July 2018 Page 29 of 47
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
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reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
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MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 7 of 47
All Rights Reserved
8 Look for and express regularity in repeated reasoning
In Kindergarten mathematically proficient students begin to notice repetitive actions in
geometry counting comparing etc For example a kindergartener may notice that as the number
of sides increase on a shape a new shape is created (triangle has 3 sides a rectangle has 4 sides a
pentagon has 5 sides a hexagon has 6 sides) When counting out loud to 100 kindergartners may
recognize the pattern 1-9 being repeated for each decade (eg Seventy-ONE Seventy-TWO
Seventy- THREEhellip Eighty-ONE Eighty-TWO Eighty-THREEhellip) When joining one more
cube to a pile the child may realize that the new amount is the next number in the count
sequence
Mathematical Practices 1 and 6 should be evident in EVERY lesson
Georgia Department of Education
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July 2018 Page 8 of 47
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CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
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July 2018 Page 11 of 47
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OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
All Rights Reserved
MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
All Rights Reserved
CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
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FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
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ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
Georgia Department of Education
July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
Georgia Department of Education
July 2018 Page 30 of 47
All Rights Reserved
Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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July 2018 Page 32 of 47
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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July 2018 Page 35 of 47
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Georgia Department of Education
July 2018 Page 36 of 47
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 8 of 47
All Rights Reserved
CONTENT STANDARDS
COUNTING AND CARDINALITY (CC)
CLUSTER 1 KNOW NUMBER NAMES AND THE COUNT SEQUENCE
MGSEKCC1 Count to 100 by ones and by tens
This standard calls for students to rote count by starting at one and count to 100 When
students count by tens they are only expected to master counting on the decade (0 10 20
30 40 hellip) This objective does not require recognition of numerals It is focused on the
rote number sequence
MGSEKCC2 Count forward beginning from a given number within the known sequence
(instead of having to begin at 1)
This standard includes numbers 0 to 100 This asks for students to begin a rote forward
counting sequence from a number other than 1 Thus given the number 4 the student
would count ldquo4 5 6 helliprdquo This objective does not require recognition of numerals It is
focused on the rote number sequence
MGSEKCC3 Write numbers from 0 to 20 Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects)
This standard addresses the writing of numbers and using the written numerals (0-20) to
describe the amount of a set of objects Due to varied development of fine motor and
visual development a reversal of numerals is anticipated for a majority of the students
While reversals should be pointed out to students the emphasis is on the use of numerals
to represent quantities rather than the correct handwriting formation of the actual numeral
itself
In addition the standard asks for students to represent a set of objects with a written
numeral The number of objects being recorded should not be greater than 20 Students
can record the quantity of a set by selecting a number card or tile (numeral recognition)
or writing the numeral Students can also create a set of objects based on the numeral
presented
CLUSTER 2 COUNT TO TELL THE NUMBER OF OBJECTS
Students use numbers including written numerals to represent quantities and to solve
quantitative problems such as counting objects in a set counting out a given number of objects
and comparing sets or numerals
MGSEKCC4 Understand the relationship between numbers and quantities connect
counting to cardinality
This standard asks students to count a set of objects and see sets and numerals in
relationship to one another rather than as isolated numbers or sets These connections
are higher-level skills that require students to analyze to reason about and to explain
relationships between numbers and sets of objects This standard should first be
addressed using numbers 1-5 with teachers building to the numbers 1-10 later in the year
Georgia Department of Education
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July 2018 Page 9 of 47
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The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
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July 2018 Page 10 of 47
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b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
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July 2018 Page 11 of 47
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OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
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July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
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July 2018 Page 13 of 47
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
All Rights Reserved
CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
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July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
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July 2018 Page 25 of 47
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introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
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July 2018 Page 26 of 47
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
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July 2018 Page 27 of 47
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
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July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 9 of 47
All Rights Reserved
The expectation is that students are comfortable with these skills with the numbers 1-10
by the end of Kindergarten
a When counting objects say the number names in the standard order pairing
each object with one and only one number name and each number name with one
and only one object (one-to-one correspondence)
This standard reflects the ideas that students implement correct counting procedures by
pointing to one object at a time (one-to-one correspondence) using one counting word for
every object (one-to-one taggingsynchrony) while keeping track of objects that have and
have not been counted This is the foundation of counting
b Understand that the last number name said tells the number of objects counted
The number of objects is the same regardless of their arrangement or the order in
which they were counted (cardinality)
This standard calls for students to answer the question ldquoHow many are thererdquo by
counting objects in a set and understanding that the last number stated when counting a
set (hellip8 9 10) represents the total amount of objects ldquoThere are 10 bears in this pilerdquo
(cardinality) It also requires students to understand that the same set counted three
different times will end up being the same amount each time Thus a purpose of keeping
track of objects is developed Therefore a student who moves each object as it is
counted recognizes that there is a need to keep track in order to figure out the amount of
objects present While it appears that this standard calls for students to have conservation
of number (regardless of the arrangement of objects the quantity remains the same)
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered The goal of this objective is for students to be able to
count a set of objects regardless of the formation those objects are placed
c Understand that each successive number name refers to a quantity that is one
larger
This standard represents the concept of ldquoone morerdquo while counting a set of objects
Students are to make the connection that if a set of objects was increased by one more
object then the number name for that set is to be increased by one as well Students are
asked to understand this concept with and without objects For example after counting a
set of 8 objects students should be able to answer the question ldquoHow many would there
be if we added one more objectrdquo and answer a similar question when not using objects
by asking hypothetically ldquoWhat if we have 5 cubes and added one more How many
cubes would there be thenrdquo This concept should be first taught with numbers 1-5 before
building to numbers 1-10 Students are expected to be comfortable with this skill with
numbers to 10 by the end of Kindergarten
MGSEKCC5 Count to answer lsquohow manyrdquo questions
a Count to answer ldquohow manyrdquo questions about as many as 20 things arranged in
a variety of ways (a line a rectangular array or a circle) or as many as 10 things in
a scattered configuration
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
Georgia Department of Education
July 2018 Page 11 of 47
All Rights Reserved
OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
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July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
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July 2018 Page 18 of 47
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MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 10 of 47
All Rights Reserved
b Given a number from 1-20 count out that many objects
c Identify and be able to count pennies within 20 (Use pennies as manipulatives in
multiple mathematical contexts)
This standard addresses various counting strategies Based on early childhood
mathematics experts such as Kathy Richardson students go through a progression of
four general ways to count These counting strategies progress from least difficult to
most difficult First students move objects and count them as they move them The
second strategy is that students line up the objects and count them Third students have a
scattered arrangement and they touch each object as they count Lastly students have a
scattered arrangement and count them by visually scanning without touching them Since
the scattered arrangements are the most challenging for students MGSEKCC5 calls for
students to only count 10 objects in a scattered arrangement and count up to 20 objects in
a line rectangular array or circle Out of these 3 representations a line is the easiest type
of arrangement to count
MGSEKCC6 Identify whether the number of objects in one group is greater than less
than or equal to the number of objects in another group eg by using matching and
counting strategies
This standard expects mastery of up to ten objects Students can use matching strategies
(Student 1) counting strategies or equal shares (Student 3) to determine whether one
group is greater than less than or equal to the number of objects in another group
(Student 2)
Student 1
I lined up one square and one
triangle Since there is one
extra triangle there are more
triangles than squares
Student 2
I counted the squares and I got
8 Then I counted the
triangles and got 9 Since 9 is
bigger than 8 there are more
triangles than squares
Student 3
I put them in a pile I then
took away objects Every time
I took a square I also took a
triangle When I had taken
almost all of the shapes away
there was still a triangle left
That means there are more
triangles than squares
MGSEKCC7 Compare two numbers between 1 and 10 presented as written numerals
This standard calls for students to apply their understanding of numerals 1-10 to compare
one from another Thus looking at the numerals 8 and 10 a student must be able to
recognize that the numeral 10 represents a larger amount than the numeral 8 Students
should begin this standard by having ample experiences with sets of objects
(MGSEKKCC3 and MGSEKKCC6) before completing this standard with just
numerals Based on early childhood research students should not be expected to be
comfortable with this skill until the end of kindergarten
Georgia Department of Education
Georgia Department of Education
July 2018 Page 11 of 47
All Rights Reserved
OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
All Rights Reserved
MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
All Rights Reserved
CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
All Rights Reserved
Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
Georgia Department of Education
July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
Georgia Department of Education
July 2018 Page 30 of 47
All Rights Reserved
Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
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July 2018 Page 32 of 47
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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July 2018 Page 35 of 47
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
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July 2018 Page 11 of 47
All Rights Reserved
OPERATIONS AND ALGEBRAIC THINKING (OA)
CLUSTER 1 UNDERSTAND ADDITION AS PUTTING TOGETHER AND ADDING
TO AND UNDERSTAND SUBTRACTION AS TAKING APART AND TAKING FROM
All standards in this cluster should only include numbers through 10 Students will model simple
joining and separating situations with sets of objects or eventually with equations such as 5 + 2
= 7 and 7 ndash 2 = 5 (Kindergarten students should see addition and subtraction equations and
student writing of equations in kindergarten is encouraged but it is not required) Students
choose combine and apply effective strategies for answering quantitative questions including
quickly recognizing the cardinalities of small sets of objects counting and producing sets of
given sizes counting the number of objects in combined sets or counting the number of objects
that remain in a set after some are taken away
MGSEKOA1 Represent addition and subtraction with objects fingers mental images
drawings sounds (eg claps) acting out situations verbal explanations expressions or
equations
This standard asks students to demonstrate the understanding of how objects can be
joined (addition) and separated (subtraction) by representing addition and subtraction
situations in various ways This objective is primarily focused on understanding the
concept of addition and subtraction rather than merely reading and solving addition and
subtraction number sentences (equations)
MGSEKOA2 Solve addition and subtraction word problems and add and subtract
within 10 eg by using objects or drawings to represent the problem
This standard asks students to solve problems presented in a story format (context) with a
specific emphasis on using objects or drawings to determine the solution This objective
builds upon their understanding of addition and subtraction from KOA1 to solve
problems Once again numbers should not exceed 10
Teachers should be cognizant of the three types of problems There are three types of
addition and subtraction problems Result Unknown Change Unknown and Start
Unknown These types of problems become increasingly difficult for students Research
has found that Result Unknown problems are easier than Change and Start Unknown
problems Kindergarten students should have experiences with all three types of
problems The level of difficulty can be decreased by using smaller numbers (up to 5) or
increased by using larger numbers (up to 10) Please see Appendix Table 1 for
additional examples
MGSEKOA3 Decompose numbers less than or equal to 10 into pairs in more than one
way eg by using objects or drawings and record each decomposition by a drawing or
equation (drawings need not include an equation)
This standard asks students to understand that a set of (5) objects can be broken into two
sets (3 and 2) and still be the same total amount (5) In addition this objective asks
students to realize that a set of objects (5) can be broken in multiple ways (3 and 2 4 and
1) Thus when breaking apart a set (decomposing) students develop the understanding
that a smaller set of objects exists within that larger set (inclusion) This should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
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MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 12 of 47
All Rights Reserved
developed in context before moving into how to represent decomposition with symbols
(+ - =)
Example
ldquoBobby Bear is missing 5 buttons on his jacket How many ways can you use blue and
red buttons to finish his jacket Draw a picture of all your ideas Students could draw
pictures of
bull4 blue and 1 red button bull3 blue and 2 red buttons
bull2 blue and 3 red buttons bull1 blue and 4 red buttons
After the students have had numerous experiences with decomposing sets of objects and
recording with pictures and numbers the teacher eventually makes connections between
the drawings and symbols 5=4+1 5=3+2 5=2+3 and 5=1+4
The number sentence only comes after pictures or work with manipulatives and students
should never give the number sentence without a mathematical representation
MGSEKOA4 For any number from 1 to 9 find the number that makes 10 when added to
the given number eg by using objects or drawings and record the answer with a drawing
or equation
This standard builds upon the understanding that a number can be decomposed into parts
(KOA3) Once students have had experiences breaking apart ten into various
combinations this asks students to find a missing part of 10
Example
ldquoA full case of juice boxes has 10 boxes There are only 6 boxes in this case How many
juice boxes are missing
Student 1
Using a Ten-Frame
I used 6 counters for the 6 boxes
of juice still in the case There
are 4 blank spaces so 4 boxes
have been removed This makes
sense since 6 and 4 more equals
10
Student 2
Think Addition
I counted out 10 cubes because I
knew there needed to be ten I
pushed these 6 over here because
there were in the container
These are left over So therersquos 4
missing
Student 3
Basic Fact
I know that itrsquos 4 because 6 and 4
is the same amount as 10
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
All Rights Reserved
MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
All Rights Reserved
CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 13 of 47
All Rights Reserved
MGSEKOA5 Fluently add and subtract within 5
Students are fluent when they display accuracy (correct answer) efficiency (a reasonable
amount of steps without resorting to counting) and flexibility (using strategies such as
the distributive property) Students develop fluency by understanding and internalizing
the relationships that exist between and among numbers Oftentimes when children think
of each ldquofactrdquo as an individual item that does not relate to any other ldquofactrdquo they are
attempting to memorize separate bits of information that can be easily forgotten Instead
in order to fluently add and subtract children must first be able to see sub-parts within a
number (inclusion KCC4c) Once they have reached this milestone children need
repeated experiences with many different types of concrete materials (such as cubes
chips and buttons) over an extended amount of time in order to recognize that there are
only particular sub-parts for each number Therefore children will realize that if 3 and 2
is a combination of 5 then 3 and 2 cannot be a combination of 6 For example after
making various arrangements with toothpicks students learn that only a certain number
of sub-parts exist within the number 4
Then after numerous opportunities to explore represent and discuss ldquo4rdquo a student
becomes able to fluently answer problems such as ldquoOne bird was on the tree Three more
birds came How many are on the tree nowrdquo and ldquoThere was one bird on the tree Some
more came There are now 4 birds on the tree How many birds camerdquo Traditional flash
cards or timed tests have not been proven as effective instructional strategies for
developing fluency Rather numerous experiences with breaking apart actual sets of
objects help children internalize parts of number
NUMBERS AND OPERATIONS IN BASE TEN (NBT)
CLUSTER 1 WORK WITH NUMBERS 11ndash19 TO GAIN FOUNDATIONS FOR
PLACE VALUE
MGSEKNBT1 Compose and decompose numbers from 11 to 19 into ten ones and some
further ones to understand that these numbers are composed of ten ones and one two
three four five six seven eight or nine ones eg by using objects or drawings and
record each composition or decomposition by a drawing or equation (eg 18 = 10 + 8)
This standard is the first time that students move beyond the number 10 with
representations such as objects (manipulatives) or drawings The spirit of this standard
is that students separate out a set of 11-19 objects into a group of ten objects with
leftovers This ability is a pre-cursor to later grades when they need to understand the
complex concept that a group of 10 objects is also one ten (unitizing) Ample
experiences with ten frames will help solidify this concept Research states that students
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July 2018 Page 14 of 47
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are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
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July 2018 Page 15 of 47
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CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
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them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
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into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
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MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
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FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
All Rights Reserved
Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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July 2018 Page 29 of 47
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
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reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
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MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 14 of 47
All Rights Reserved
are not ready to unitize until the end of first grade Therefore this work in Kindergarten
lays the foundation of composing tens and recognizing leftovers
MEASUREMENT AND DATA (MD)
CLUSTER 1 DESCRIBE AND COMPARE MEASURABLE ATTRIBUTES
MGSEKMD1 Describe several measurable attributes of an object such as length or
weight For example a student may describe a shoe as ldquoThis shoe is heavy It is also really
longrdquo
This standard calls for students to describe measurable attributes of objects such as
length weight and size For example a student may describe a shoe as ldquoThis shoe is
heavy Itrsquos also really longrdquo This standard focuses on using descriptive words and does
not mean that students should sort objects based on attributes Sorting appears later in the
Kindergarten standards
MGSEKMD2 Directly compare two objects with a measurable attribute in common to
see which object has ldquomore ofrdquo or ldquoless ofrdquo the attribute and describe the difference For
example directly compare the heights of two children and describe one child as
tallershorter
This standard asks for direct comparisons of objects Direct comparisons are made when
objects are put next to each other such as two children two books two pencils For
example a student may line up two blocks and say ldquoThis block is a lot longer than this
onerdquo Students are not comparing objects that cannot be moved and lined up next to each
other
Through ample experiences with comparing different objects children should recognize
that objects should be matched up at the end of objects to get accurate measurements
Since this understanding requires conservation of length a developmental milestone for
young children children need multiple experiences to move beyond the idea that hellip
ldquoSometimes this block is longer than this one and sometimes itrsquos shorter (depending on
how I lay them side by side) and thatrsquos okayrdquo ldquoThis block is always longer than this
block (with each end lined up appropriately)rdquo
Before conservation of length The striped block is longer than the plain block when they
are lined up like this But when I move the blocks around sometimes the plain block is
longer than the striped block
After conservation of length I have to line up the blocks to measure them The plain
block is always longer than the striped block
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
All Rights Reserved
CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 15 of 47
All Rights Reserved
CLUSTER 2 CLASSIFY OBJECTS AND COUNT THE NUMBER OF OBJECTS IN
EACH CATEGORY
MGSEKMD3 Classify objects into given categories count the numbers of objects in each
category and sort the categories by count (Limit category counts to be less than or equal to
10)
This standard asks students to identify similarities and differences between objects (eg
size color shape) and use the identified attributes to sort a collection of objects Once
the objects are sorted the student counts the amount in each set Once each set is
counted then the student is asked to sort (or group) each of the sets by the amount in
each set
For example when given a collection of buttons the student separates the buttons into
different piles based on color (all the blue buttons are in one pile all the orange buttons
are in a different pile etc) Then the student counts the number of buttons in each pile
blue (5) green (4) orange (3) purple (4) Finally the student organizes the groups by
the quantity in each group (Orange buttons (3) Green buttons next (4) Purple buttons
with the green buttons because purple also had (4) Blue buttons last (5)
This objective helps to build a foundation for data collection in future grades In later
grades students will transfer these skills to creating and analyzing various graphical
representations
GEOMETRY (G)
CLUSTER 1 IDENTIFY AND DESCRIBE SHAPES (SQUARES CIRCLES
TRIANGLES RECTANGLES HEXAGONS CUBES CONES CYLINDERS AND
SPHERES)
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides number of angles etc) and other attributes do not (color size
orientation) Then using geometric attributes the student identifies and describes particular
shapes listed above Throughout the year Kindergarten students move from informal language
to describe what shapes look like (eg ldquoThat looks like an ice cream conerdquo) to more formal
mathematical language (eg ldquoThat is a triangle It has three sidesrdquo)
In Kindergarten students need ample experiences exploring various forms of the shapes (eg
size big and small types triangles equilateral isosceles scalene orientation rotated slightly
to the left ldquoupside down‟) using geometric vocabulary to describe the different shapes In
addition students need numerous experiences comparing one shape to another rather than
focusing on one shape at a time This type of experience solidifies the understanding of the
various attributes and how those attributes are different- or similar- from one shape to another
Students in Kindergarten typically recognize figures by appearance alone often by comparing
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
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July 2018 Page 25 of 47
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introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
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July 2018 Page 26 of 47
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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July 2018 Page 27 of 47
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
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July 2018 Page 31 of 47
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 16 of 47
All Rights Reserved
them to a known example of a shape such as the triangle on the left For example students in
Kindergarten typically recognize that the figure on the left is a triangle but claim that the figure
on the right is not a triangle since it does not have a flat bottom The properties of a figure are
not recognized or known Students make decisions on identifying and describing shapes based
on perception not reasoning
MGSEKG1 Describe objects in the environment using names of shapes and describe the
relative positions of these objects using terms such as above below beside in front of
behind and next to
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment Kindergarten students need to focus first on location
and position of two-and-three-dimensional objects in their classroom prior to describing
location and position of two-and-three-dimensional representations on paper
MGSEKG2 Correctly name shapes regardless of their orientations or overall size
This standard addresses students ‟identification of shapes based on known examples
Students at this level do not yet recognize triangles that are turned upside down as
triangles since they donrsquot ldquolook likerdquo triangles Students need ample experiences looking
at and manipulating shapes with various typical and atypical orientations Through these
experiences students will begin to move beyond what a shape ldquolooks likerdquo to identifying
particular geometric attributes that define a shape
MGSEKG3 Identify shapes as two-dimensional (lying in a plane ldquoflatrdquo) or three
dimensional (ldquosolidrdquo)
This standard asks students to identify flat objects (2 dimensional) and solid objects (3
dimensional) This standard can be taught by having students sort flat and solid objects
or by having students describe the appearance or thickness of shapes
CLUSTER 2 ANALYZE COMPARE CREATE AND COMPOSE SHAPES
MGSEKG4 Analyze and compare two- and three-dimensional shapes in different sizes
and orientations using informal language to describe their similarities differences parts
(eg number of sides and verticesldquocornersrdquo) and other attributes (eg having sides of
equal length)
This standard asks students to note similarities and differences between and among 2-D
and 3-D shapes using informal language These experiences help young students begin to
understand how 3-dimensional shapes are composed of 2-dimensional shapes (eg the
base and the top of a cylinder is a circle a circle is formed when tracing a sphere)
MGSEKG5 Model shapes in the world by building shapes from components (eg sticks
and clay balls) and drawing shapes
This standard asks students to apply their understanding of geometric attributes of shapes
in order to create given shapes For example a student may roll a clump of play-dough
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
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ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
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July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
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July 2018 Page 25 of 47
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introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
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July 2018 Page 26 of 47
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
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July 2018 Page 27 of 47
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
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July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 17 of 47
All Rights Reserved
into a sphere or use their finger to draw a triangle in the sand table recalling various
attributes in order to create that particular shape
MGSEKG6 Compose simple shapes to form larger shapes For example ldquoCan you join
these two triangles with full sides touching to make a rectanglerdquo
This standard moves beyond identifying and classifying simple shapes to manipulating
two or more shapes to create a new shape This concept begins to develop as students
first move rotate flip and arrange puzzle pieces Next students use their experiences
with puzzles to move given shapes to make a design (eg ldquoUse the 7 tangram pieces to
make a foxrdquo) Finally using these previous foundational experiences students
manipulate simple shapes to make a new shape
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 18 of 47
All Rights Reserved
MINDSET AND MATHEMATICS
Growth mindset was pioneered by Carol Dweck Lewis and Virginia Eaton Professor of
Psychology at Stanford University She and her colleagues were the first to identify a link
between growth mindset and achievement They found that students who believed that their
ability and intelligence could grow and change otherwise known as growth mindset
outperformed those who thought that their ability and intelligence were fixed Additionally
students who were taught that they could grow their intelligence actually did better over time
Dwecks research showed that an increased focus on the process of learning rather than the
outcome helped increase a students growth mindset and ability
(from WITH+MATH=I CAN)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 19 of 47
All Rights Reserved
Jo Boaler Professor of Mathematics Education at the Stanford Graduate School of Education
and author of Mathematical Mindsets Unleashing Students Potential through Creative Math
Inspiring Messages and Innovative Teaching was one of the first to apply growth mindset to
math achievement
You can learn how to use the power of growth mindset for yourself and your students here
httpswwwamazoncomgpwithmathican
httpswwwmindsetkitorgtopicsabout-growth-mindset
httpswwwyoucubedorg
Growth and Fixed Mindset images courtesy of Katherine Lynas (katherinelynascom) Thank
you Katherine
VERTICAL UNDERSTANDING OF THE MATHEMATICS LEARNING
TRAJECTORY
Why does it matter if you know what happens in mathematics in the grades before and after the
one you teach Isnrsquot it enough just to know and understand the expectations for your grade
There are many reasons to devote a bit of your time to the progression of standards
Yoursquoll
bull Deepen your understanding of how development of algebraic thinking has proven to be a
critical element of student mathematics success as they transition from elementary to
middle school Elementary and middle school teachers must understand how algebraic
thinking develops prior to their grade in their grade and beyond their grade in order to
support student algebraic thinking
bull Know what to expect when students show up in your grade because you know what they
should understand from the years before
bull Understand how conceptual understanding develops making it easier to help students
who have missing bits and pieces
bull Be able to help students to see the connections between ideas in mathematics in your
grade and beyond helping them to connect to what they already know and what is to
come
bull Assess understanding more completely and develop better assessments
bull Know what the teachers in the grades to come expect your students to know and
understand
bull Plan more effectively with same-grade and other-grade colleagues
bull Deepen your understanding of the mathematics of your grade
We arenrsquot asking you to take a month off to study up just asking that you reference the
following resources when you want to deepen your understanding of where students are in their
mathematics learning understand why they are learning what they are learning in your grade
and understand the mathematical ideas and connections within your grade and beyond
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
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July 2018 Page 25 of 47
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introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
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July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 20 of 47
All Rights Reserved
Resources
The Coherence Map
httpachievethecoreorgpage1118coherence-map This resource diagrams the connections
between standards provides explanations of standards provides example tasks for
many standards and links to the progressions document when further detail is required
A visual learning trajectory of
Addition and Subtraction - httpgfletchycom20160304the-progression-of-addition-and-
subtraction
Multiplication - httpgfletchycom20151218the-progression-of-multiplication
Division - httpgfletchycom20160131the-progression-of-division
Fractions - httpsgfletchycom20161208the-progression-of-fractions
(Many thanks to Graham Fletcher the genius behind these videos)
The Mathematics Progression Documents
httpmatharizonaedu~imeprogressions
Learning Trajectories in Mathematics
httpwwwcpreorgimagesstoriescpre_pdfslearning20trajectories20in20math_ccii20r
eportpdf
RESEARCH OF INTEREST TO MATHEMATICS TEACHERS
Social Emotional Learning and Math
httpcurryvirginiaeduuploadsresourceLibraryTeachers_support_for_SEL_contributes_to_im
proved_math_teaching_and_learning_et_alpdf
Why how you teach math is important httpswwwyoucubedorg
GloSS IKAN and the overall Numeracy Project
Information on the Numeracy Project which includes GloSS and IKAN can be found here
Georgia Numeracy Project Overview
The GloSS and IKAN professional learning video found here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesFOAFoundations-of-Algebra-Day-
1aspx provides an in-depth look at the GloSS and IKAN While it was created for teachers of
Foundations of Algebra the information is important for teachers of grades K- 12
The GloSS and IKAN prezi found on georgiastandardsorg here
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesGlobal-Strategy-Stage-GloSS-and-
Individual-Knowledge-Assessment-of-Number-IKANaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
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July 2018 Page 23 of 47
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BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
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July 2018 Page 24 of 47
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
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July 2018 Page 25 of 47
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introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
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July 2018 Page 26 of 47
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
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July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
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July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 21 of 47
All Rights Reserved
FLUENCY
Fluency Procedural fluency is defined as skill in carrying out procedures flexibly accurately
efficiently and appropriately Fluent problem solving does not necessarily mean solving
problems within a certain time limit Fluency is based on a deep understanding of quantity and
number
Deep Understanding Teachers teach more than simply ldquohow to get the answerrdquo and instead
support studentsrsquo ability to access concepts from a number of perspectives Therefore students
are able to see math as more than a set of mnemonics or discrete procedures Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations as well as writing and speaking about their understanding
Memorization Memorization leads to the rapid recall of arithmetic facts or mathematical
procedures without the necessity of understanding This type of learning is not the goal of
numeracy Memorization is often confused with fluency and automaticity Fluency implies a
much richer kind of mathematical knowledge and experience Automaticity is based on strategy
development and the ability to become automatic with part-whole computation strategies
developed
Number Sense Students consider the context of a problem look at the numbers in a problem
make a decision about which strategy would be most efficient in each particular problem
Number sense is not a deep understanding of a single strategy but rather the ability to think
flexibly between a variety of strategies in context
Fluent students
bull flexibly use a combination of deep understanding number sense and automaticity
bull are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them
bull are able to articulate their reasoning
bull find solutions through a number of different paths
For more about fluency see
httpwwwyoucubedorgwp-contentuploads201503FluencyWithoutFear-2015pdf and
httpsbhi61nm2cr3mkdgk1dtaov18-wpenginenetdna-sslcomwp-contentuploadsnctm-timed-
testspdf
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
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ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
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July 2018 Page 25 of 47
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introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
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July 2018 Page 26 of 47
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
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July 2018 Page 31 of 47
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
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July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 22 of 47
All Rights Reserved
ARC OF LESSON (OPENING WORK SESSION CLOSING)
ldquoWhen classrooms are workshops-when learners are inquiring investigating and constructing-
there is already a feeling of community In workshops learners talk to one another ask one
another questions collaborate prove and communicate their thinking to one another The heart
of math workshop is this investigations and inquiries are ongoing and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon Children have the opportunity to explore to pursue
inquiries and to model and solve problems on their own creative ways Searching for patterns
raising questions and constructing onersquos own models ideas and strategies are the primary
activities of math workshop The classroom becomes a community of learners engaged in
activity discourse and reflectionrdquo Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk
ldquoStudents must believe that the teacher does not have a predetermined method for solving the
problem If they suspect otherwise there is no reason for them to take risks with their own ideas
and methodsrdquo Teaching Student-Centered Mathematics K-3 by John Van de Walle and Lou
Ann Lovin
Opening Set the stage
Get students mentally ready to work on the task
Clarify expectations for products or behavior
How
bull Begin with a simpler version of the task to be presented
bull Solve problem strings related to the mathematical ideas being investigated
bull Leap headlong into the task and begin by brainstorming strategies for approaching the
task
bull Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning Have students restate the task in their
own words Every task should require more of the students than just the answer
Work session Let them Grapple
Students- Grapple with the mathematics through sense-making discussion concretizing their
mathematical ideas and the situation record thinking in journals
Teacher- Let go Listen Respect student thinking Encourage testing of ideas Ask questions to
clarify or provoke thinking Provide gentle hints Observe assess and repeat
Closing Best Learning Happens Here
Students- Share answers justify thinking clarify understanding explain thinking question each
other identify patterns and make generalizations make additional connections to real world
Teacher- Listen attentively to all ideas asks for explanations offer comments such as ldquoPlease
tell me how you figured that outrdquo ldquoI wonder what would happen if you triedhelliprdquo
Take away from lesson- anchor charts and journal reflections using number pictures and words
Read Van de Walle K-3 Chapter 1
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
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bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 23 of 47
All Rights Reserved
BREAKDOWN OF A TASK (UNPACKING TASKS)
How do I go about tackling a task or a unit
Read the unit in its entirety Discuss it with your grade level colleagues Which parts do you feel
comfortable with Which make you wonder Brainstorm ways to implement the tasks
Collaboratively complete the culminating task with your grade level colleagues As students
work through the tasks you will be able to facilitate their learning with this end in mind The
structure of the units and tasks is similar task to task and grade to grade This structure allows
you to converse in a vertical manner with your colleagues school-wide There is a great deal of
mathematical knowledge and teaching support within each grade level guide unit and task
1 Read the first task your students will be engaged in Discuss it with your grade level
colleagues Which parts do you feel comfortable with Which make you wonder
Brainstorm ways to implement the tasks
2 If not already established use the first few weeks of school to establish routines and
rituals and to assess student mathematical understanding You might use some of the
tasks found in the unit or in some of the following resources as beginning tasks centers
or math tubs which serve the dual purpose of allowing you to observe and assess
Additional Resources
Math Their Way httpwwwcentereduMathTheirWayshtml
NZ Maths- httpwwwnzmathsconznumeracy-development-projects-
booksparent_node=
K-5 Math Teaching Resources- httpwwwk-5mathteachingresourcescomindexhtml
(this is a for-profit site with several free resources)
Math Solutions- httpwwwmathsolutionscomindexcfmpage=wp9ampcrid=56
3 Points to remember
bull Each task begins with a list of the standards specifically addressed in that task
however that does not mean that these are the only standards addressed in the
task Remember standards build on one another and mathematical ideas are
connected
bull Tasks are made to be modified to match your learnerrsquos needs If the names need
changing change them If the specified materials are not available use what is
available If a task doesnrsquot go where the students need to go modify the task or
use a different resource
bull The units are not intended to be all encompassing Each teacher and team will
make the units their own and add to them to meet the needs of the learners
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
All Rights Reserved
Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
Georgia Department of Education
July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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July 2018 Page 31 of 47
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
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July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
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July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
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MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 24 of 47
All Rights Reserved
ROUTINES AND RITUALS
Teaching Math in Context and Through Problems
ldquoBy the time they begin school most children have already developed a sophisticated informal
understanding of basic mathematical concepts and problem-solving strategies Too often
however the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledgerdquo (Carpenter et al 1999) The 8 Standards of Mathematical
Practice (SMP) should be at the forefront of every mathematics lesson and be the driving factor
of HOW students learn
One way to help ensure that students are engaged in the 8 SMP is to construct lessons built on
context or through story problems Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer probably in a way that the teacher wants ldquoContext
problems on the other hand are connected as closely as possible to childrenrsquos lives rather than
to lsquoschool mathematicsrsquo They are designed to anticipate and to develop childrenrsquos mathematical
modeling of the real worldrdquo
Traditionally mathematics instruction has been centered around a lot of problems in a single
math lesson focusing on rote procedures and algorithms which do not promote conceptual
understanding Teaching through word problems and in context is difficult however
ldquokindergarten students should be expected to solve word problemsrdquo (Van de Walle K-3)
A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods nor is there a perception by students that there is a specific correct
solution method A problem for learning mathematics also has these features
bull The problem must begin where the students are which makes it accessible to all learners
bull The problematic or engaging aspect of the problem must be due to the mathematics that
the students are to learn
bull The problem must require justifications and explanations for answers and methods
It is important to understand that mathematics is to be taught through problem solving That is
problem-based tasks or activities are the vehicle through which the standards are taught Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice (Van de Walle and Lovin Teaching
Student-Centered Mathematics K-3 page 11)
Use of Manipulatives
ldquoIt would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives or a ldquohands-on approachrdquo is the recommended way to teach mathematics There
is no doubt that these materials can and should play a significant role in your classroom Used
correctly they can be a positive factor in childrenrsquos learning But they are not a cure-all that some
educators seem to believe them to be It is important that you have a good perspective on how
manipulatives can help or fail to help children construct ideas We canrsquot just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent When a new model or new use of a familiar model is
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
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July 2018 Page 28 of 47
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
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July 2018 Page 30 of 47
All Rights Reserved
Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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July 2018 Page 35 of 47
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
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MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 25 of 47
All Rights Reserved
introduced into the classroom it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use rdquo
(Van de Walle and Lovin Teaching Student-Centered Mathematics K-3 page 6)
Once you are comfortable that the models have been explained you should not force their use on
students Rather students should feel free to select and use models that make sense to them In
most instances not using a model at all should also be an option The choice a student makes can
provide you with valuable information about the level of sophistication of the studentrsquos
reasoning
Whereas the free choice of models should generally be the norm in the classroom you can often
ask students to model to show their thinking This will help you find out about a childrsquos
understanding of the idea and also his or her understanding of the models that have been used in
the classroom
The following are simple rules of thumb for using models
bull Introduce new models by showing how they can represent the ideas for which they are
intended
bull Allow students (in most instances) to select freely from available models to use in solving
problems
bull Encourage the use of a model when you believe it would be helpful to a student having
difficultyrdquo (Van de Walle and Lovin Teaching Student-Centered Mathematics K-3
page 8-9)
bull Modeling also includes the use of mathematical symbols to representmodel the concrete
mathematical ideathought processsituation This is a very important yet often neglected
step along the way Modeling can be concrete representational and abstract Each type
of model is important to student understanding Modeling also means to ldquomathematizerdquo a
situation or problem to take a situation which might at first glance not seem
mathematical and view it through the lens of mathematics For example students notice
that the cafeteria is always out of their favorite flavor of ice cream on ice cream days
They decide to survey their schoolmates to determine which flavors are most popular
and share their data with the cafeteria manager so that ice cream orders reflect their
findings The problem Running out of ice cream flavors The solution Use math to
change the flavor amounts ordered
Use of Strategies and Effective Questioning
Teachers ask questions all the time They serve a wide variety of purposes to keep learners
engaged during an explanation to assess their understanding to deepen their thinking or focus
their attention on something This process is often semi-automatic Unfortunately there are many
common pitfalls These include
bull asking questions with no apparent purpose
bull asking too many closed questions
bull asking several questions all at once
bull poor sequencing of questions
bull asking rhetorical questions
bull asking lsquoGuess what is in my headrsquo questions
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
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bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
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July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 26 of 47
All Rights Reserved
bull focusing on just a small number of learners
bull ignoring incorrect answers
bull not taking answers seriously
In contrast the research shows that effective questioning has the following characteristics
bull Questions are planned well ramped in difficulty
bull Open questions predominate
bull A climate is created where learners feel safe
bull A lsquono handsrsquo approach is used for example when all learners answer at once using mini-
whiteboards or when the teacher chooses who answers
bull Probing follow-up questions are prepared
bull There is a sufficient lsquowait timersquo between asking and answering a question
bull Learners are encouraged to collaborate before answering
bull Learners are encouraged to ask their own questions
0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A Kndash8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)
Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics
Most often these charts are used to reinforce counting skills Counting involves two separate
skills (1) ability to produce the standard list of counting words (ie one two three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle
2007) The counting sequence is a rote procedure The ability to attach meaning to counting is
ldquothe key conceptual idea on which all other number concepts are developedrdquo (Van de Walle p
122) Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence Although both charts can be useful the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary
A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value Listed below are some of the benefits of using
the 0-99 Chart in your classroom
bull A 0-99 Chart begins with zero where as a hundredrsquos chart begins with 1 It is important
to include zero because it is a digit and just as important as 1-9
bull A 1-100 chart puts the decade numerals (10 20 30 etc) on rows without the remaining
members of the same decade For instance on a hundredrsquos chart 20 appears at the end of
the teensrsquo row This causes a separation between the number 20 and the numbers 21-29
The number 20 is the beginning of the 20rsquos family therefore it should be in the
beginning of the 20rsquos row like in a 99rsquos chart to encourage students to associate the
quantities together
bull A 0-99 chart ends with the last two-digit number 99 this allows the students to
concentrate their understanding using numbers only within the onesrsquo and tensrsquo place
values A hundredrsquos chart ends in 100 introducing a new place value which may change
the focus of the places
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
All Rights Reserved
Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
Georgia Department of Education
July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
Georgia Department of Education
July 2018 Page 30 of 47
All Rights Reserved
Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 27 of 47
All Rights Reserved
bull The understanding that 9 units fit in each place value position is crucial to the
development of good number sense It is also very important that students recognize that
zero is a number not merely a placeholder This concept is poorly modeled by a typical
1-100 chart base ten manipulatives and even finger counting We have no zero finger
zero block or zero space on typical 1-100 number charts Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity a quantity of
none Zero is the answer to a question such as ldquoHow many elephants are in the roomrdquo
bull Including zero presents the opportunity to establish zero correctly as an even number
when discussing even and odd Children see that it fits the same pattern as all of the
other even numbers on the chart
While there are differences between the 0-99 Chart and the 1-100 Chart both number charts
are valuable resources for your students and should be readily available in several places
around the classroom Both charts can be used to recognize number patterns such as the
increase or decrease by multiples of ten Provide students the opportunity to explore the
charts and communicate the patterns they discover
The number charts should be placed in locations that are easily accessible to students and
promote conversation Having one back at your math calendar or bulletin board area
provides you the opportunity to use the chart to engage students in the following kinds of
discussions Ask students to find the numeral that represents
bull the day of the month
bull the month of the year
bull the number of students in the class
bull the number of students absent or any other amount relevant to the moment
Using the number is 21 give directions and ask questions similar to those below
bull Name a number greater than 21
bull Name a number less than 21
bull What number is 3 more than or less than 21
bull What number is 5 more than or less than 21
bull What number is 10 more than or less than 21
bull Is 21 even or odd
bull What numbers live right next door to 21
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
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level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
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several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
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July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
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DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
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Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 28 of 47
All Rights Reserved
Ask students to pick an even number and explain how they know the number is even Ask
students to pick an odd number and explain how they know it is odd Ask students to count
by 2rsquos 5rsquos or 10rsquos Tell them to describe any patterns that they see (Accept any patterns that
students are able to justify There are many right answers)
Number Lines
The use of number lines in elementary mathematics is crucial in studentsrsquo development of
number and mathematical proficiency While the MGSEK explicitly state use number lines in
grades 2-5 number lines should be used in all grade levels and in multiple settings
According to John Van de Walle
A number line is also a worthwhile model but can initially present conceptual difficulties
for children below second grade and students with disabilities (National Research
Council Committee 2009) This is partially due to their difficulty in seeing the unit
which is a challenge when it appears in a continuous line A number line is also a shift
from counting a number of individual objects in a collection to continuous length units
There are however ways to introduce and model number lines that support young
learners as they learn this representation Familiarity with a number line is essential
because third grade students will use number lines to locate fractions and add and
subtract time intervals fourth graders will locate decimals and use them for
measurement and fifth graders will use perpendicular number lines in coordinate grids
(CCSSO2010)
A number line measures distance from zero the same way a ruler does If you donrsquot
actually teach the use of the number line through emphasis on the unit (length) students
may focus on the hash marks or numerals instead of the spaces (a misunderstanding that
becomes apparent when their answers are consistently off by one) At first students can
build a number path by using a given length such as a set of Cuisenaire rods of the same
color to make a straight line of multiple single units (Van de Walle and Lovin Teaching
Student-Centered Mathematics 3-5 pg 106-107)
Open number lines are particularly useful for building studentsrsquo number sense They can also
form the basis for discussions that require the precise use of vocabulary and quantities and are
therefore a good way to engage students in the Standards for Mathematical Practice
While the possibilities for integrating number lines into the mathematics classroom are endless
the following are some suggestionsideas
On a bulletin board attach a string which will function as an open number line Each
morning (or dedicated time for math routines) put a new number on each studentrsquos desk
Using some type of adhesive (thumb tack tape etc) students will place the number in
the appropriate location on the string In the beginning of the year provide students with
numbers that are more familiar to them As the year progresses move through more
complex problems such as skip counting fractions decimals or other appropriate grade
Georgia Department of Education
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July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
Georgia Department of Education
July 2018 Page 30 of 47
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Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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July 2018 Page 33 of 47
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place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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July 2018 Page 35 of 47
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
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July 2018 Page 36 of 47
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 29 of 47
All Rights Reserved
level problems Through daily integration the number line becomes part of the routine
Following the number placement have a brief discussiondebriefing of the reasoning
used by students to place the numbers
In the 3-Act tasks placed throughout the units students will be provided opportunities to
use an open number line to place estimates that are too low too high and just right as
related to the posed problem Similar opportunities can also be used as part of a daily
routine
Math Maintenance Activities
In addition to instruction centered on the current unit of study the math instructional block
should include time devoted to reviewing mathematics that have already been taught previewing
upcoming mathematics and developing mental math and estimation skills There is a saying that
if you donrsquot use it yoursquoll lose it If students donrsquot have opportunities to continuously apply and
refine the math skills theyrsquove learned previously then they may forget how to apply what
theyrsquove learned Unlike vocabulary words for literacy math vocabulary words are not used
much outside math class so it becomes more important to use those words in discussions
regularly Math maintenance activities incorporate review and preview of math concepts and
vocabulary and help students make connections across domains Itrsquos recommended that 15 to 30
minutes of the math instructional block be used for these math maintenance activities each day
Itrsquos not necessary nor is it recommended that teachers do every activity every day Teachers
should strive for a balance of math maintenance activities so that over the course of a week
students are exposed to a variety of these activities Math maintenance time may occur before or
after instruction related to the current math unit or it can occur at a different time during the day
The goals of this maintenance time should include
Deepening number sense including subitizing flexible grouping of quantities counting
forward and backward using whole numbers fractions decimals and skip counting
starting at random numbers or fractional amounts
Developing mental math skills by practicing flexible and efficient numerical thinking
through the use of operations and the properties of operations
Practicing estimation skills with quantities and measurements such as length mass and
liquid volume depending on grade level
Practicing previously-taught skills so that students deepen and refine their understanding
Reviewing previously-taught concepts that students struggled with as indicated on their
assessments including gaps in math concepts taught in previous grade levels
Using a variety of math vocabulary terms especially those that are used infrequently
Practicing basic facts using strategies learned in previous grade levels or in previous units
to develop or maintain fluency
Previewing prerequisite skills for upcoming math units of study
Georgia Department of Education
Georgia Department of Education
July 2018 Page 30 of 47
All Rights Reserved
Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
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GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 30 of 47
All Rights Reserved
Participating in mathematical discussions with others that require students to construct
viable arguments and critique the reasoning of others
To accomplish these goals math maintenance activities can take many different forms Some
activities include
Number Corner or Calendar Time
Number Talks
Estimation ActivitiesEstimation 180
Problem of the Day or Spiraled Review Problems
In addition math discussions math journals and math games are appropriate not only for the
current unit of study but also for maintaining math skills that were previously taught
Although there are commercially-available materials to use for math maintenance activities
there are also many excellent websites and internet resources that are free for classroom use
Here is a partial list of some recommended resources A more detailed explanation of some of
these components follows below
Math Maintenance Activity Possible Resources
Number Corner or Calendar Time bull httpteachelemmathweeblycomcalendarhtml
bull Every Day Counts Calendar Math from Houghton Mifflin
Harcourt
bull Number Corner from The Math Learning Center
Number Talks Number Talks by Sherry Parrish
httpkentuckymathematicsorgnumber_talk_resourcesphp
Estimation ActivitiesEstimation 180 httpwwwestimation180com
Problem of the DaySpiraled Review
Problems
wwwinsidemathematicsorg
httpnzmathsconzteaching-material
httpwwwk-5mathteachingresourcescom
httpmathlearnncsharpschoolcomcmsOneaspxportalId=
4507283amppageId=5856325
Childrenrsquos Mathematics Cognitively Guided Instruction by
Thomas Carpentar et Al
Extending Childrenrsquos Mathematics Fractions and Decimals
by Epson and Levi
Number CornerCalendar Time
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can be
found in Social Studies) during the primary classroom day This should be an interesting and
motivating time for students A calendar board or corner can be set up and there should be
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
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July 2018 Page 32 of 47
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
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July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
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July 2018 Page 35 of 47
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
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Georgia Department of Education
July 2018 Page 36 of 47
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 31 of 47
All Rights Reserved
several elements that are put in place The following elements should be set in place for students
to succeed during Number Corner
1 a safe environment
2 concrete models or math tools
3 opportunities to think first and then discuss
4 student interaction
Number Corner should relate several mathematics conceptsskills to real life experiences This
time can be as simple as reviewing the months days of the week temperature outside and the
schedule for the day but some teachers choose to add other components that integrate more
standards Number Corner should be used as a time to engage students in a discussion about
events which can be mathematized or as a time to engage in Number Talks
bull Find the number ___
bull If I have a nickel and a dime how much money do I have (any money combination)
bull What is ___ more than ___
bull What is ___ less than ___
bull Mystery number Give clues and they have to guess what number you have
bull This number has ___tens and ___ ones What number am I
bull What is the difference between ___ and ____
bull What number comes after ___ before ___
bull Tell me everything you know about the number ____ (Anchor Chart)
Number Corner is also a chance to familiarize your students with Data Analysis This creates an
open conversation to compare quantities which is a vital process that must be explored before
students are introduced to addition and subtraction
At first choose questions that have only two mutually exclusive answers such as yes or
no (eg Are you a girl or a boy) rather than questions that can be answered yes no or
maybe (or sometimes) This sets up the part-whole relationship between the number of
responses in each category and the total number of students present and it provides the
easiest comparison situation (between two numbers eg Which is more How much
more is it) Keep in mind that the concept of less than (or fewer) is more difficult than
the concept of greater than (or more) Be sure to frequently include the concept of less in
your questions and discussions about comparisons
Later you can expand the questions so they have more than two responses Expected
responses may include maybe Irsquom not sure I donrsquot know or a short predictable list of
categorical responses (eg In which season were you born)
Once the question is determined decide how to collect and represent the data Use a
variety of approaches including asking students to add their response to a list of names or
tally marks using Unifix cubes of two colors to accumulate response sticks or posting 3
x 5 cards on the board in columns to form a bar chart
The question should be posted for students to answer For example ldquoDo you have an
older sisterrdquo Ask students to contribute their responses in a way that creates a simple
visual representation of the data such as a physical model table of responses bar graph
etc
Georgia Department of Education
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July 2018 Page 32 of 47
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Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
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6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
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July 2018 Page 35 of 47
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The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
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Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
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Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
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Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 32 of 47
All Rights Reserved
Each day ask students to describe compare and interpret the data by asking questions
such as these ldquoWhat do you notice about the data Which group has the most Which
group has the least How many more answered [this] compared to [that] Why do you
suppose more answered [this]rdquo Sometimes ask data gathering questions ldquoDo you think
we would get similar data on a different day Would we get similar data if we asked the
same question in another class Do you think these answers are typical for first graders
Why or why notrdquo
Ask students to share their thinking strategies that justify their answers to the questions
Encourage and reward attention to specific details Focus on relational thinking and
problem-solving strategies for making comparisons Also pay attention to identifying
part-whole relationships and reasoning that leads to interpretations
Ask students questions about the ideas communicated by the representation used What
does this graph represent How does this representation communicate this information
clearly Would a different representation communicate this idea better
The representation analysis and discussion of the data are the most important parts of
the routine (as opposed to the data gathering process or the particular question being
asked) These mathematical processes are supported by the computational aspects of
using operations on the category totals to solve part-whole or ldquocomparerdquo problems
Number Talks
In order to be mathematically proficient students must be able to compute accurately efficiently
and flexibly Daily classroom number talks provide a powerful avenue for developing ldquoefficient
flexible and accurate computation strategies that build upon the key foundational ideas of
mathematicsrdquo (Parrish 2010) Number talks involve classroom conversations and discussions
centered upon purposefully planned computation problems
In Sherry Parrishrsquos book Number Talks Helping Children Build Mental Math and Computation
Strategies teachers will find a wealth of information about Number Talks including
bull Key components of Number Talks
bull Establishing procedures
bull Setting expectations
bull Designing purposeful Number Talks
bull Developing specific strategies through Number Talks
There are four overarching goals upon which K-2 teachers should focus during Number Talks
These goals are
1 Developing number sense
2 Developing fluency with small numbers
3 Subitizing
4 Making Tens
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems A Number Talk is a short ongoing daily routine that provides students with
meaningful ongoing practice with computation Number Talks should be structured as short
sessions alongside (but not necessarily directly related to) the ongoing math curriculum A great
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 33 of 47
All Rights Reserved
place to introduce a Number Talk is during Number CornerCalendar Time It is important to
keep Number Talks short as they are not intended to replace current curriculum or take up
the majority of the time spent on mathematics In fact teachers only need to spend 5 to 15
minutes on Number Talks Number Talks are most effective when done every day The primary
goal of Number Talks is to build computational fluency Children develop computational fluency
while thinking and reasoning like mathematicians When they share their strategies with others
they learn to clarify and express their thinking thereby developing mathematical language This
in turn serves them well when they are asked to express their mathematical processes in writing
In order for children to become computationally fluent they need to know particular
mathematical concepts that go beyond what is required to memorize basic facts or procedures
Students will begin to understand major characteristics of numbers such as
bull Numbers are composed of smaller numbers
bull Numbers can be taken apart and combined with other numbers to make new numbers
bull What we know about one number can help us figure out other numbers
bull What we know about parts of smaller numbers can help us with parts of larger numbers
bull Numbers are organized into groups of tens and ones (and hundreds tens and ones and so
forth)
bull What we know about numbers to 10 helps us with numbers to 100 and beyond
All Number Talks follow a basic six-step format The format is always the same but the
problems and models used will differ for each number talk
1 Teacher presents the problem Problems are presented in many different ways as dot
cards ten frames sticks of cubes models shown on the overhead a word problem or a
numerical expression Strategies are not explicitly taught to students instead the
problems presented lead to various strategies
2 Students figure out the answer Students are given time to figure out the answer To
make sure students have the time they need the teacher asks them to give a ldquothumbs-uprdquo
when they have determined their answer The thumbs up signal is unobtrusive- a message
to the teacher not the other students
3 Students share their answers Four or five students volunteer to share their answers and
the teacher records them on the board
4 Students share their thinking Three or four students volunteer to share how they got
their answers (Occasionally students are asked to share with the person(s) sitting next to
them) The teacher records the students thinking
5 The class agrees on the real answer for the problem The answer that together the
class determines is the right answer is presented as one would the results of an
experiment The answer a student comes up with initially is considered a conjecture
Models andor the logic of the explanation may help a student see where their thinking
went wrong may help them identify a step they left out or clarify a point of confusion
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong A student who is still unconvinced of an
answer should be encouraged to keep thinking and to keep trying to understand For some
students it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time The mantra should be
If you are not sure or it doesnt make sense yet keep thinking
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 34 of 47
All Rights Reserved
6 The steps are repeated for additional problems Similar to other procedures in your
classroom there are several elements that must be in place to ensure students get the most from
their Number Talk experiences These elements are
1 A safe environment
2 Problems of various levels of difficulty that can be solved in a variety of ways
3 Concrete models
4 Opportunities to think first and then check
5 Interaction
6 Self-correction
Estimation 180
Estimation is a skill that has many applications such as checking computation answers quickly
Engaging in regular estimation activities will develop studentsrsquo reasoning skills number sense
and increase their repertoire of flexible and efficient strategies As students gain more
experiences with estimation their accuracy will improve
According to John Van de Walle there are three types of estimation that students should
practice
Measurement estimation ndash determining an approximate measurement such as weight
length or capacity
Quantity estimation ndash approximating the number of items in a collection
Computational estimation ndash determining a number that is an approximation of a
computation
One resource which provides contexts for all three types of estimation is Andrew Stadelrsquos
website httpwwwestimation180com In his website Mr Stadel has posted daily estimation
contexts Here are his directions for using his website
1 Click on a picture
2 Read the question
3 Look for context clues
4 Make an estimate
5 Tell us how confident you are
6 Share your reasoning (what context clues did you use)
7 See the answer
8 See the estimates of others
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 35 of 47
All Rights Reserved
The most important part is step 6 After you make an estimate feel free to give a brief
description Its so valuable to a classroom when students share their logic or use of context
clues when formulating an estimate
Andrew Stadel has collaborated with Michael Fenton to create a recording sheet for students to
use with the estimation contexts on the website The recording sheet can also be found at
httpwwwestimation180com Here are his directions for the recording sheet
Column use descriptions from Andrew Stadel
Day
In Estimation 180s first year I was just trying to keep up with creating these estimation
challenges in time for use in my own classroom There really wasnt a scope and
sequence involved That said now that there are over 160 estimation challenges
available teachers and students can use them at any time throughout the school year and
without completing them in sequential order Therefore use the Day column simply to
number your daily challenges according to the site Tell your students or write it up on
the board that youre doing the challenge from Day 135 even though you might be on the
fifth day of school
Description
In my opinion this column is more important than the Day column Dont go crazy
here Keep it short and sweet but as specific as possible For example theres a lot of
scattered height estimates on the site Dont write down How tall for Day 110 Instead
write Bus height because when you get to Day 111 Id write in Parking structure
height I believe the teacher has the ultimate say here but it can be fun to poll your
students for a short description in which you all can agree Give students some
ownership right If unit measurement is involved try and sneak it in here Take Day
125 for instance Id suggest entering Net Wt (oz) of lg Hersheys bar Keep in mind
that Day 126 asks the same question but Id suggest you encourage your class to use
pounds if they dont think of it
By the way sometimes unit measurement(s) are already included in the question Use
discretion
Too Low
Think of an estimate that is too low
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 36 of 47
All Rights Reserved
Dont accept one (1) thats just rubbish unless one (1) is actually applicable to the
context of the challenge Stretch your students Think of it more as an answer thats too
low but reasonably close After all this is a site of estimation challenges not gimmes
Too High
Refer to my notes in Too Low Just dont accept 1 billion unless its actually applicable
Discuss with students the importance of the Too Low and Too High sections we are
trying to eliminate wrong answers while creating a range of possible answers
My Estimate
This is the place for students to fill in their answer If the answer requires a unit of
measurement we better see one Not every estimation challenge is How
many marshmallows or Christmas lights or cheese balls
Even if a unit of measurement has already been established (see the Description notes)
Id still encourage your students to accompany their numerical estimate with a unit of
measurement
For example on Day 41 Whats the height of the Giant [Ferris] Wheel use what
makes sense to you your students and your countrys customary unit of measurement
Discuss the importance of unit measurements with students Dont accept 108 What does
that 108 represent Pancakes Oil spills Bird droppings NO It represents 108 feet
My Reasoning
The My Reasoning section is the most recent addition to the handout and Im extremely
thrilled about it This is a students chance to shine Encourage their reasoning to be short
and sweet When a student writes something down theyll be more inclined to share it or
remember it Accept bullet points or phrases due to the limited space We dont need
students to write paragraphs However we are looking for students to identify any
context clues they used personal experiences andor prior knowledge Hold students
accountable for their reasoning behind the estimate of the day
Dont let student reasoning go untapped If youre doing a sequence of themed
estimation challenges dont accept I just guessed after the first day in the sequence
For example if youre doing the flight distance themed estimate challenges starting
on Day 136 you will establish the distance across the USA on the first day Sure go
ahead and guess on Day 136 but make sure you hold students accountable for their
reasoning every day thereafter
Have students share their reasoning before and after revealing the answer Utilize Think-
Pair-Share This will help create some fun conversations before revealing the answer
After revealing the answer get those who were extremely close (or correct) to share their
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 37 of 47
All Rights Reserved
reasoning I bet youll have some great mathematical discussions Im also curious to hear
from those that are way off and how their reasoning could possibly be improved
Id say the My Reasoning section was born for Mathematical Practice 3 Construct viable
arguments and critique the reasoning of others Keep some of these thoughts in mind
regarding Mathematical Practice 3
Explain and defend your estimate
Construct a detailed explanation referencing context clues prior knowledge or
previous experiences
Invest some confidence in it
Try to initiate a playful and respectful argument in class
Ask Was anyone convinced by this explanation Why Why not or Are you
guys going to let [student name] off the hook with that explanation
Theres reasoning behind every estimate (not guess)
Find out what that reasoning is
DONT let student reasoning go untapped
Answer
Jot down the revealed answer Id also encourage students to write down the unit of
measurement used in the answer The answer might use a different unit of measurement
than what you and your class agreed upon Take the necessary time to discuss the most
relative unit of measurement I might be subjectively wrong on some of the answers
posted As for more thoughts on unit of measurement refer to the My Estimate notes
above Continue having mathematical discussion after revealing the answer Refer to my
notes regarding the use of Mathematical Practice 3 in the My Reasoning section
Error
Find the difference between My Estimate and Answer Have students circle either the +
or the - if they didnt get it exactly correct
+ Your estimate was greater than (above) the actual answer
- Your estimate was less than (below) the actual answer
Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated Daily
routines must include such obvious activities such as taking attendance doing a lunch count
determining how many items are needed for snack lining up in a variety of ways (by height age
type of shoe hair color eye color etc) daily questions 99 chart questions and calendar
activities They should also include less obvious routines such as how to select materials how to
use materials in a productive manner how to put materials away how to open and close a door
etc An additional routine is to allow plenty of time for children to explore new materials before
attempting any directed activity with these new materials The regular use of the routines are
important to the development of studentsrsquo number sense flexibility and fluency which will
support studentsrsquo performances on the tasks in this unit
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 38 of 47
All Rights Reserved
Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards-based classroom These workstations typically occur during
the ldquoexploringrdquo part of the lesson which follows the mini-lesson Your role is to introduce the
concept and allow students to identify the problem Once students understand what to do and you
see that groups are working towards a solution offer assistance to the next group
Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year Avoid grouping students by ability Students in the
lower group will not experience the thinking and language of the top group and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28 Van
de Walle and Lovin 2006)
In order for students to work efficiently and to maximize participation several guidelines must
be in place (Burns 2007)
1 You are responsible for your own work and behavior
2 You must be willing to help any group member who asks
3 You may ask the teacher for help only when everyone in your group has the same question
These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member Once these guidelines are established you
should be able to successfully lead small groups which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others
The types of activities students engage in within the small groups will not always be the same
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills Possible activities include math games related previous Framework tasks problems
and computer-based activities With all tasks regardless if they are problems games etc include
a recording sheet for accountability This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006)
Games
ldquoA game or other repeatable activity may not look like a problem but it can nonetheless be
problem based The determining factor is this Does the activity cause students to be reflective
about new or developing relationships If the activity merely has students repeating procedure
without wrestling with an emerging idea then it is not a problem-based experience However
the few examples just mentioned and many others do have children thinking through ideas that
are not easily developed in one or two lessons In this sense they fit the definition of a problem-
based task
Just as with any task some form of recording or writing should be included with stations
whenever possible Students solving a problem on a computer can write up what they did and
explain what they learned Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they usedrdquo (Van de Walle and Lovin
Teaching Student-Centered Mathematics K-3 page 26)
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 39 of 47
All Rights Reserved
Journaling
Students should be writing and talking about math topics every day Putting thoughts into words
helps to clarify and solidify thinking By sharing their mathematical understandings in written
and oral form with their classmates teachers and parents students develop confidence in
themselves as mathematical learners this practice also enables teachers to better monitor student
progress NJ DOE
Language whether used to express ideas or to receive them is a very powerful tool and should
be used to foster the learning of mathematics Communicating about mathematical ideas is a way
for students to articulate clarify organize and consolidate their thinking Students like adults
exchange thoughts and ideas in many waysmdashorally with gestures and with pictures objects
and symbols By listening carefully to others students can become aware of alternative
perspectives and strategies By writing and talking with others they learn to use more-precise
mathematical language and gradually conventional symbols to express their mathematical
ideas Communication makes mathematical thinking observable and therefore facilitates further
development of that thought It encourages students to reflect on their own knowledge and their
own ways of solving problems Throughout the early years students should have daily
opportunities to talk and write about mathematics NCTM
When beginning math journals the teacher should model the process initially showing
students how to find the front of the journal the top and bottom of the composition book how
to open to the next page in sequence (special bookmarks or ribbons) and how to date the
page Discuss the usefulness of the book and the way in which it will help students retrieve
their math thinking whenever they need it
When beginning a task you can ask What do we need to find out and then How do we
figure it out Then figure it out usually by drawing representations and eventually adding
words numbers and symbols During the closing of a task have students show their journals
with a document camera or overhead when they share their thinking This is an excellent
opportunity to discuss different ways to organize thinking and clarity of explanations
Use a composition notebook (the ones with graph paper are terrific for math) for recording or
drawing answers to problems The journal entries can be from Frameworks tasks but should
also include all mathematical thinking Journal entries should be simple to begin with and
become more detailed as the childrens problem-solving skills improve Children should
always be allowed to discuss their representations with classmates if they desire feedback
The childrens journal entries demonstrate their thinking processes Each entry could first be
shared with a buddy to encourage discussion and explanation then one or two children
could share their entries with the entire class Dont forget to praise children for their thinking
skills and their journal entries These journals are perfect for assessment and for parent
conferencing The studentrsquos thinking is made visible
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 40 of 47
All Rights Reserved
GENERAL QUESTIONS FOR TEACHER USE
Adapted from Growing Success and materials from Math GAINS and TIPS4RM
Reasoning and Proving
bull How can we show that this is true for all cases
bull In what cases might our conclusion not hold true
bull How can we verify this answer
bull Explain the reasoning behind your prediction
bull Why does this work
bull What do you think will happen if this pattern continues
bull Show how you know that this statement is true
bull Give an example of when this statement is false
bull Explain why you do not accept the argument as proof
bull How could we check that solution
bull What other situations need to be considered
Reflecting
bull Have you thought abouthellip
bull What do you notice abouthellip
bull What patterns do you see
bull Does this problemanswer make sense to you
bull How does this compare tohellip
bull What could you start with to help you explore the possibilities
bull How can you verify this answer
bull What evidence of your thinking can you share
bull Is this a reasonable answer given thathellip
Selecting Tools and Computational Strategies
bull How did the learning tool you chose contribute to your understandingsolving of the
problem How did they assist in your communication
bull In what ways would [name a tool] assist in your investigationsolving of this problem
bull What other tools did you consider using Explain why you chose not to use them
bull Can you think of a different way to do the calculation that may be more efficient
bull What estimation strategy did you use
Connections
bull What other math have you studied that has some of the same principles properties or
procedures as this
bull How do these different representations connect to one another
bull When could this mathematical concept or procedure be used in daily life
bull What connection do you see between a problem you did previously and todayrsquos problem
Representing
bull What would other representations of this problem demonstrate
bull Explain why you chose this representation
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 41 of 47
All Rights Reserved
bull How could you represent this idea algebraically Graphically
bull Does this graphical representation of the data bias the viewer Explain
bull What properties would you have to use to construct a dynamic representation of this
situation
bull In what way would a scale model help you solve this problem
QUESTIONS FOR TEACHER REFLECTION
bull How did I assess for student understanding
bull How did my students engage in the 8 mathematical practices today
bull How effective was I in creating an environment where meaningful learning could take
place
bull How effective was my questioning today Did I question too little or say too much
bull Were manipulatives made accessible for students to work through the task
bull Name at least one positive thing about todayrsquos lesson and one thing you will change
bull How will todayrsquos learning impact tomorrowrsquos instruction
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 42 of 47
All Rights Reserved
MATHEMATICS DEPTH-OF-KNOWLEDGE LEVELS
Level 1 (Recall) includes the recall of information such as a fact definition term or a simple
procedure as well as performing a simple algorithm or applying a formula That is in
mathematics a one-step well-defined and straight algorithmic procedure should be included at
this lowest level Other key words that signify a Level 1 include ldquoidentifyrdquo ldquorecallrdquo ldquorecognizerdquo
ldquouserdquo and ldquomeasurerdquo Verbs such as ldquodescriberdquo and ldquoexplainrdquo could be classified at different
levels depending on what is to be described and explained
Level 2 (SkillConcept) includes the engagement of some mental processing beyond a habitual
response A Level 2 assessment item requires students to make some decisions as to how to
approach the problem or activity whereas Level 1 requires students to demonstrate a rote
response perform a well-known algorithm follow a set procedure (like a recipe) or perform a
clearly defined series of steps Keywords that generally distinguish a Level 2 item include
ldquoclassifyrdquo ldquoorganizerdquo rdquoestimaterdquo ldquomake observationsrdquo ldquocollect and display datardquo and
ldquocompare datardquo These actions imply more than one step For example to compare data requires
first identifying characteristics of the objects or phenomenon and then grouping or ordering the
objects Some action verbs such as ldquoexplainrdquo ldquodescriberdquo or ldquointerpretrdquo could be classified at
different levels depending on the object of the action For example if an item required students
to explain how light affects mass by indicating there is a relationship between light and heat this
is considered a Level 2 Interpreting information from a simple graph requiring reading
information from the graph also is a Level 2 Interpreting information from a complex graph that
requires some decisions on what features of the graph need to be considered and how
information from the graph can be aggregated is a Level 3 Caution is warranted in interpreting
Level 2 as only skills because some reviewers will interpret skills very narrowly as primarily
numerical skills and such interpretation excludes from this level other skills such as
visualization skills and probability skills which may be more complex simply because they are
less common Other Level 2 activities include explaining the purpose and use of experimental
procedures carrying out experimental procedures making observations and collecting data
classifying organizing and comparing data and organizing and displaying data in tables
graphs and charts
Level 3 (Strategic Thinking) requires reasoning planning using evidence and a higher level of
thinking than the previous two levels In most instances requiring students to explain their
thinking is a Level 3 Activities that require students to make conjectures are also at this level
The cognitive demands at Level 3 are complex and abstract The complexity does not result from
the fact that there are multiple answers a possibility for both Levels 1 and 2 but because the task
requires more demanding reasoning An activity however that has more than one possible
answer and requires students to justify the response they give would most likely be a Level 3
Other Level 3 activities include drawing conclusions from observations citing evidence and
developing a logical argument for concepts explaining phenomena in terms of concepts and
using concepts to solve problems
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 43 of 47
All Rights Reserved
DOK contrsquodhellip
Level 4 (Extended Thinking) requires complex reasoning planning developing and thinking
most likely over an extended period of time The extended time period is not a distinguishing
factor if the required work is only repetitive and does not require applying significant conceptual
understanding and higher-order thinking For example if a student has to take the water
temperature from a river each day for a month and then construct a graph this would be
classified as a Level 2 However if the student is to conduct a river study that requires taking
into consideration a number of variables this would be a Level 4 At Level 4 the cognitive
demands of the task should be high and the work should be very complex Students should be
required to make several connectionsmdashrelate ideas within the content area or among content
areasmdashand have to select one approach among many alternatives on how the situation should be
solved in order to be at this highest level Level 4 activities include designing and conducting
experiments making connections between a finding and related concepts and phenomena
combining and synthesizing ideas into new concepts and critiquing experimental designs
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 44 of 47
All Rights Reserved
DEPTH AND RIGOR STATEMENT
By changing the way we teach we are not asking children to learn less we are asking them to
learn more We are asking them to mathematize to think like mathematicians to look at numbers
before they calculate to think rather than to perform rote procedures Children can and do
construct their own strategies and when they are allowed to make sense of calculations in their
own ways they understand better In the words of Blaise Pascal ldquoWe are usually convinced
more easily by reasons we have found ourselves than by those which have occurred to othersrdquo
By changing the way we teach we are asking teachers to think mathematically too We are
asking them to develop their own mental math strategies in order to develop them in their
students Catherine Twomey Fosnot and Maarten Dolk Young Mathematicians at Work
While you may be tempted to explain and show students how to do a task much of the learning
comes as a result of making sense of the task at hand Allow for the productive struggle the
grappling with the unfamiliar the contentious discourse for on the other side of frustration lies
understanding and the confidence that comes from ldquodoing it myselfrdquo
DIFFERENTIATION
Information and instructional strategies for gifted students English Language Learners and
students with disabilities is available at httpeducationohiogovTopicsAcademic-Content-
StandardsNew-Learning-Standards
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 45 of 47
All Rights Reserved
Problem Solving Rubric (K-2)
SMP 1-Emergent 2-Progressing 3- MeetsProficient 4-Exceeds
Make sense of
problems and
persevere in
solving them
The student was unable to
explain the problem and
showed minimal perseverance
when identifying the purpose
of the problem
The student explained the
problem and showed some
perseverance in identifying the
purpose of the problem AND
selected and applied an
appropriate problem-solving
strategy that led to a partially
complete andor partially accurate
solution
The student explained the
problem and showed
perseverance when identifying
the purpose of the problem
AND selected and applied an
appropriate problem-solving
strategy that led to a generally
complete and accurate solution
The student explained the problem
and showed perseverance by
identifying the purpose of the problem
and selected and applied an
appropriate problem-solving strategy
that led to a thorough and accurate
solution
Attends to
precision
The student was unclear in
their thinking and was unable
to communicate
mathematically
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using
appropriate vocabulary in their
process of finding solutions
The student was precise by
clearly describing their actions
and strategies while showing
understanding and using grade-
level appropriate vocabulary in
their process of finding
solutions
The student was precise by clearly
describing their actions and strategies
while showing understanding and
using above-grade-level appropriate
vocabulary in their process of finding
solutions
Reasoning and
explaining
The student was unable to
express or justify their opinion
quantitatively or abstractly
using numbers pictures charts
or words
The student expressed or justified
their opinion either quantitatively
OR abstractly using numbers
pictures charts OR words
The student expressed and
justified their opinion both
quantitatively and abstractly
using numbers pictures charts
andor words
The student expressed and justified
their opinion both quantitatively and
abstractly using a variety of numbers
pictures charts and words
Models and use
of tools
The student was unable to
select an appropriate tool draw
a representation to reason or
justify their thinking
The student selected an
appropriate tool or drew a correct
representation of the tools used to
reason and justify their response
The student selected an efficient
tool andor drew a correct
representation of the efficient
tool used to reason and justify
their response
The student selected multiple efficient
tools and correctly represented the
tools to reason and justify their
response In addition this student was
able to explain why their tool model
was efficient
Seeing
structure and
generalizing
The student was unable to
identify patterns structures or
connect to other areas of
mathematics andor real-life
The student identified a pattern or
structure in the number system
and noticed connections to other
areas of mathematics or real-life
The student identified patterns or
structures in the number system
and noticed connections to other
areas of mathematics and real-
life
The student identified various patterns
and structures in the number system
and noticed connections to multiple
areas of mathematics and real-life
Many thanks to Richmond County Schools for sharing this rubric
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 46 of 47
All Rights Reserved
SUGGESTED LITERATURE
bull The Greedy Triangle by Marilyn Burns
bull When a Line BeginshellipA Shape Begins by Rhonda Gowler Greene
bull Round Trip by Ann Jonas
bull Eight Hands Round by Ann Whitford Paul
bull Pattern by Henry Arthur Pluckrose
bull Shape by Henry Arthur Pluckrose
bull The Shape of Things by Dayle Ann Dobbs
bull Grandfather Tangrsquos Story by Ann Tompert and Robert Andrew Parker
bull Three Pigs One Wolf and Seven Magic Shapes by Maccarone and Neuhaus and
bull The Tangram Magician by Lisa Campbell Ernst
bull Measuring Penny by Loreen Leedy or
bull Me and the Measure of Things by Joan Sweeney
bull The Best Bug Parade by Stuart J Murphy
bull Chrysanthemum by Kevin Henkes
bull Mighty Maddie by Stuart Murphy
bull If You Give a Mouse a Cookie by Laura Numeroff
bull Ten Black Dots by Donald Crews
bull Annorsquos Counting House by Mitsumasa Anno
bull The Penny Pot by Barbara Derubertis
bull Ten Flashing Fireflies by Philemon Sturges
bull 12 Ways to Get to 11 by Eve Merriam
bull One Is a Snail Ten Is a Crab by April Pulley Sayre and Jeff Sayre
bull Two of Everything by Lily Toy Hong
Additional Resources
httpkentuckymathematicsorgpimser_printablesphp
TECHNOLOGY LINKS
httpcatalogmathlearningcenterorgapps
httpnzmathsconzdigital-learning-objects
httpwwwgadoeorgTechnology-ServicesSLDSPagesTeacher-Resource-Linkaspx
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea
Georgia Department of Education
Georgia Department of Education
July 2018 Page 47 of 47
All Rights Reserved
RESOURCES CONSULTED
Content
Mathematics Progressions Documents httpimematharizonaeduprogressions
Illustrative Mathematics httpswwwillustrativemathematicsorgcontent-standardsK
NZ Maths
httpnzmathsconz
TeacherStudent Sense-making
httpwwwyoutubecomusermitcccnyorgfeature=watch
httpswwwgeorgiastandardsorgGeorgia-StandardsPagesMathaspx or
httpseccsedlorgcommon_core_videos
Journaling
httpwwwkindergartenkindergartencom201007setting-up-math-problemsolving-notebookshtml
Community of Learners
httpwwwedutopiaorgmath-social-activity-cooperative-learning-video
httpwwwedutopiaorgmath-social-activity-sel
httpwwwyoutubecomuserresponsiveclassroomvideos
Tea