Erma Anderson MS Parent Coffee 2013
Transcript of Erma Anderson MS Parent Coffee 2013
At The Crossroads—Meeting the Challenges of a Changing
World —
Parent Session SAS 2013
What Are Our Challenges?
Of the 20 fastest growing occupations, 15 need extensive math and science preparation
Parent Session SAS 2013
Parent Session SAS 2013
Mathematical thinking . . .
A gateway to higher mathematics?
OR
A wall blocking path for
students? Parent Session SAS 2013
5
The Future
75 %75 % of jobs will be in STEM
Not justjust STEM careers,
it is STEM in everyevery job
Technology Technology as a “global knowledge economy” is the futurefuture, and it requires different skills.
Business and industry wantwant employees with these skills!
21st Century Learning
“We are responsible for preparing students to address problems we cannot foresee with knowledge that has not yet been developed using technology not yet invented.” Ralph Wolf
Parent Session EARJ 2013
Jobs of the Future
The TOP 10 jobs in 2015 are not yet invented.
Parent Session EARJ 2013
Thinking and Learning Skills
• Critical Thinking & Problem Solving Skills• Creativity & Innovation Skills• Communication & Information Skills• Collaboration Skills
21st Century Skills Framework
Parent Session EARJ 2013
Problem Solving
Computational & Procedural Skills
DOING MATH
Conceptual Understanding
“WHERE” THEMATHEMATICSWORKS
“HOW” THE
MATHEMATICSWORKS
“WHY” THE
MATHEMATICSWORKS
Parent Session SAS 2013
What Does It Mean to Understand Mathematics?
Knowing Understanding
Understanding is the measure of quality and quantity of connections between new ideas and existing ideas
ASB MCI2 Problem Solving
“Understanding is the key to remembering what is learned and being able to use it flexibly.”
- Hiebert, in Lester & Charles,
Teaching Mathematics through
Problem Solving, 2004.
A Thought
“People who do not understand mathematics today are like those who could not read or write in the industrial age.”
Robert Moses
ASB MCI2 Problem Solving
The Bridge To Understanding
Representation
“SEEING” Stage
Concrete Abstract
“DOING” Stage “SYMBOLIC”Stage
Parent Session EARJ 2013
Building Mathematical Concepts
Concrete Manipulativ
es
Pictorial Representatio
n I I I I
I I I I
Abstract Symbols
4 + 4 = 8
2 x 4 = 8
*Significant time must be spentworking with concrete materials
and constructing pictorial representations
in order for abstract symbol and operational understanding to occur
PARENT 2013
Value Multiple Representations…
concrete or pictorial
tabular
verbal
symbolic
graphical Parent Session EARJ 2013
Conceptual vs. Procedural Knowledge
Conceptual (connected networks)Knowledge and understanding of logical relationships and representations with an ability to talk, write and give examples of these relationships.
Procedural (sequence of actions)Knowledge of rules and procedures used in carrying out routine mathematical tasks and the symbols used to represent mathematics.
-- David Allen The question of which kind of knowledge is most important is the wrong question to ask. Both kinds of knowledge are required for mathematical expertise...
Instead, we should focus on designing teaching environments that help students build internal representations of procedures that become part of larger conceptual networks.
James Heibert and Tom Carpenter, Learning and Teaching with Understanding, 1992
Phil Daro
GradePriorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding
K–2Addition and subtraction, measurement using whole number quantities
3–5Multiplication and division of whole numbers and fractions
6Ratios and proportional reasoning; early expressions and equations
7Ratios and proportional reasoning; arithmetic of rational numbers
8 Linear algebra
Priorities in Mathematics
5/29/12
Key FluenciesGrade Required Fluency
K Add/subtract within 5
1 Add/subtract within 10
2Add/subtract within 20
Add/subtract within 100 (pencil and paper)
3Multiply/divide within 100
Add/subtract within 1000
4 Add/subtract within 1,000,000
5 Multi-digit multiplication
6Multi-digit division
Multi-digit decimal operations
7 Solve px + q = r, p(x + q) = r
8Solve simple 22 systems by inspection 19
Number Sense …
Howden (1989) described it as “good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.”
.
Prior Understandings
2.G.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
www.JennyRay.net 21
Distributive Property & Area Models
22www.JennyRay.net
3
5 + 2
15 6+
3 x 7 =3 x (5 + 2) = (3 x 5) + (3 x 2)= 15 + 6 = 21
3 x 7 =__
Area Models25 x 38=
23www.JennyRay.net
20 +
530 + 8
150 40
600 160
750 + 200 = 950
950
Partitioning Strategies for Multiplication
27 x 4 27 x 4
4 x 20 = 80 27 + 27 + 27 + 27
4 x 7 = 28 108
54 54
108
-----------------------------------
267 x 7
7 x 200 = 1400 1820
7 x 60 = 420 1876
7 x 8 = 56www.JennyRay.net 24
14 x 25: An Area Model
*Sketch is not drawn to scale.
25www.JennyRay.net
20 + 5
80 20
200 50
Algebra 1: Multiplying Binomials
*Sketch is not drawn to scale.
26www.JennyRay.net
x + 5
4x 20
x2 5x
Partial Products
38 78
x 19 x 54
30 x 10 300 70 x 50 3500
30 x 9 270 70 x 4 280
8 x 10 80 8 x 50 400
8 x 9 + 72 8 x 4 + 32
722 4212
Partial Products (Area Model) 62
x 18
60 2 600
480
10 600 20 20
16
8 480 16 1116
1500+350+120+28 =1998
50 4
30
7
1500
350
120
28
54 x 37
Decimals
3 x 0.24
0.3 x 0.60.3 x 0.6
0.12 + 0.60 = 0.720.12 + 0.60 = 0.72
Draw a picture that shows
4
3
3
2
Array
2 of 3 rows
3 of 4 in each row
2
1
12
6
43
32
ofrows
ofrows
areatotal
areashaded
Mixed Numbers, too!
8 x 3 ¾
8 x 3 = 248 x 3 = 24
64
24
4
38 x
24 + 6 = 3024 + 6 = 30
Where’s the Math?
Models help students explore concepts and build understanding
Models provide a context for students to solve problems and explain reasoning
Models provide opportunities for students to generalize conceptual understanding
–œ ‡°± ·þ
þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþðþþþýþþþýþþþýþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ þþþþþþþþþþþþ þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ
Multiplication (rational numbers)
What does 4 x 3 mean?
4 x 2/3 = 4 groups of 2/3 each
=
4X2/3 =2/3+2/3 +2/3+2/3+2/3 = 8/3
Why is it not 8/12?
Multiplication – number line model Aoife earns €12 per hour. What would she earn in 2, 3, 4, 3/4 hours?
Notice “of “ becoming multiplication
3/4 x12 = 12 x ¾ =9
Multiplication – Area Model
Cara had 2/5 of her birthday cake left from her party. She ate ¾ of the leftover cake. How much of the original cake did she eat?
2/5 cakeDivide into quarters
¾ of 2/5
Area of 3x2 out of area of 4x5
3 2 6 3
4 5 20 10
http://www.learner.org/courses/learningmath/number/session9/part_a/try.html
Multiplication making smaller!
A muffin recipe requires 2/3 of a cup of milk. Each recipe makes 12 muffins. How many muffins can be made using 6 cups of milk?
Adapted from Multiplicative Thinking. Workshop 1. Properties of Multiplication and Division. http.nzmaths.com, 2010.
The Additive Thinker
A muffin recipe requires 2/3 of a cup of milk. Each recipe makes 12 muffins. How many muffins can be made using 6 cups of milk?
1
2
2
34
Each rectangle represents a third of a cup of milk.
A muffin recipe requires 2/3 of a cup of milk. Each recipe makes 12 muffins. How many muffins can be made using 6 cups of milk?
Content + Practices
“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.”
(CCSS, 2010)
Parent Session EARJ 2013
“Learning happens within students, not to them. Learning is a process of making meaning that happens one student at a time.”
Carol Ann Tomlinson and Jay McTighe
Integrating Differentiated Instruction and
Understanding by Design © 2006
Standards for Mathematical Practice
Make sense of problems and persevere in solving them
11
Reason abstractly and quantitatively
22 Construct viable arguments and critique the reasoning of others
33
Model with mathematics
44
Use appropriate tools strategically.
55
Look for and make use of structure
66
Attend to precision
77Look forand expressregularity in repeated reasoning
88
1. Make sense of problems and persevere in solving them.
6. Attend to precision.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Reasoning and explaining
Modeling and using tools.
Seeing structure and generalizing.
Grouping the Standards of Mathematical Practice
William McCallum University of Arizona- April 1, 2011
Overarching habits of mind of a productive mathematical thinker.
1. Make sense of problems and persevere in solving
Do students:
• EXPLAIN?
• Make CONJECTURES?
• PLAN a solution pathway?
• MULTIPLE representations?
• Use DIFFERENT METHODS to check?
• Check that it all makes sense?
• Understand other approaches?
• See connections among different approaches?
• ANALYZE?
2. Reason abstractly and quantitatively
Do students:
• Make sense of quantities & their relationships?
• Decontextualize?
• Contextualize?
• Create a coherent representation?
• Consider units involved?
• Deal with the meaning of the quantities?
3. Construct viable arguments and critique the reasoning of others.
Do students:Understand & use stated assumptions, definitions, and previous results?
Analyze situations, recognize & use counterexamples?
Justify conclusions, communicate to others & respond to arguments?
Compare the effectiveness of 2 plausible arguments?
Distinguish correct logical reasoning from flawed & articulate the flaw?
Look at an argument, decide if it makes sense,& ask useful questions to clarify or improve it?
Make conjectures& build a logical progression?
Use mathematical induction as technique for proof?
Write geometric proofs, including proofs of contradiction?
4. Model with mathematics
Do students:
• Analyze relationships mathematically to draw conclusions?
• Apply the mathematics they know everyday?
• Initially use what they know to simplify the problem?
• Identify important qualities in a practical situation?
• Interpret results In the context of the situation?
• Reflect on whether the results make sense?
5. Use appropriate tools strategically.Do students:
• Consider available tools?
• Know the tools appropriate for their grade or course?
• Make sound decisions about when tools are helpful?
• Identify & use relevant external math Sources?
• Use technology tools to explore & deepen understanding of concepts?
6. Attend to precision.
Do students:
• Communicate precisely with others?
• Use clear definitions?
• Use the equal sign consistently & appropriately?
• Calculate accurately & efficiently?
7. Look for and make use of structure.
Do students:
• Look closely to determine a pattern or structure?
• Use properties?
• Decompose & recombine numbers & expressions?
• Have the facility to shift perspectives?
8. Look for and express regularity in repeated reasoning.
Do students:
• Notice if calculations are repeated?
• Look for general methods & shortcuts?
• Maintain process while attending to details?
• Evaluate the reasonableness of intermediate results?
a% of b = b% of a
Shift in Mathematics #1Deeper Learning Fewer Concepts
•How Parents Can Help Students at Home
Students must … Parents can …Spend more time on fewer concepts
Know what the priority work is for the grade level
Represent math in multiple ways Ask, “Can you show me that in another way?”
Apply strategies, not just get answers
Focus on how the child is tackling the problem over what the answer is
Shift in Mathematics #2Focus on Strong Number Sense and Problem Solving
•How Parents Can Help Students at HomeStudents must … Parents can …Be able to apply strategies and use core math facts quickly
Ask the child’s teacher what core math facts should be practiced at home Ask students which strategies they are using
Compose and decompose numbers Help children break apart and put together numbers to make problem solving easier
Shift in Mathematics #3Focus on Communication of Thinking and Language Rich Classrooms
•How Parents Can Help Students at Home
Students must … Parents can …Understand why the math works—explain and justify
Ask questions to find out whether the child really knows why the answer is correct
Talk about why the math works—explain and justify
Ask children to explain how they solved the problem and why they chose the strategies they used
Prove that they know why and how the math works—explain and justify
Ask children to show how they know they have the correct solution Talk about alternative strategies
Use academic vocabulary to explain their reasoning and critique that of others
Expect children to use the language of mathTalk about math
Shift in Mathematics #4Perseverance and Grappling with Mathematics
•How Parents Can Help Students at Home
Students must … Parents can …See mistakes as learning opportunities Help their children use their mistakes
as windows into their thinking
Understand that there is usually more than one way to solve a problem
Celebrate and value alternative responses Ask, “Is there another way to solve this?”
Spend more time solving a single problem in a deep way
Expect fewer problems but more writing and explaining in homework
Making Sense of Mathematics?
• ?:??
PARENT 2013
• Which is more rigorous ?
• 1895? 1931? 2012?
Eighth Grade Test questions---1895 Arithmetic [Time, 1.25 hours]
• 1. Name and define the Fundamental Rules of Arithmetic.
•2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How many bushels of wheat will it hold?
•3. If a load of wheat weighs 3942 lbs., what is it worth at 50cts/bushel, deducting 1050 lbs. for tare?
•4. District No. 33 has a valuation of $35,000. What is the necessary levy to carry on a school seven months at $50 per month, and have $104 for incidentals?
•5. Find the cost of 6720 lbs. coal at $6.00 per ton.
Parent Session EARJ 2013
Eighth Grade Test
• 6. Find the interest of $512.60 for 8 months and 18 days at 7 percent.
•7. What is the cost of 40 boards 12 inches wide and 16 ft. long at $20 per metre?
•8. Find bank discount on $300 for 90 days (no grace) at 10 percent.
•9. What is the cost of a square farm at $15 per acre, the distance of which is 640 rods?
•10. Write a Bank Check, a Promissory Note, and a Receipt
Parent Session EARJ 2013
Parent Session EARJ 2013
Parent Session EARJ 2013
Parent Session EARJ 2013
Instruction Matters!
PARENT 2013
A thought…
• We can best close the achievement gap by eliminating the opportunity gap. If we,as mathematics teachers K-12, each make it our personal goal for every student to have the opportunity to learn mathematics in ways that promote the habits of mind espoused
• in the standards for mathematical practice,we will be successful in helping all students tobe successful in learning and doing mathematics.
Parent Session EARJ 2013
Together we make a difference!
PARENT 2013