Building Foundations for Mathematics Defining Numerical Fluency.

Building Foundations for Building Foundations for MathematicsMathematics

Defining Numerical FluencyDefining Numerical Fluency

Goals & PurposesGoals & Purposes

Increase teacher knowledge regarding the refinements Increase teacher knowledge regarding the refinements of the TEKS relating to numerical fluency.of the TEKS relating to numerical fluency.

Develop a working definition of numerical fluency.Develop a working definition of numerical fluency. Increase teacher knowledge of composing and Increase teacher knowledge of composing and

decomposing numbers.decomposing numbers. Increase teacher knowledge of developmental stages of Increase teacher knowledge of developmental stages of

numerical fluency.numerical fluency. Increase teacher knowledge of strategies to develop Increase teacher knowledge of strategies to develop

numerical fluency.numerical fluency. Develop an understanding of the use of metacognition in Develop an understanding of the use of metacognition in

problem solving.problem solving.

Math

Just patterns

Waiting to be found

From 50 Problem-Solving Lessons, Grades 1-6, by Marilyn Burns. Page 116. Copyright © 1996 by Math Solutions Publications. Reprinted by permission. All rights reserved.

Texas Essential Knowledge and Texas Essential Knowledge and SkillsSkills

Throughout mathematics in Kindergarten-Grade 2, Throughout mathematics in Kindergarten-Grade 2, students develop numerical fluency with conceptual students develop numerical fluency with conceptual understanding and computational accuracy. understanding and computational accuracy. Students in Kindergarten-Grade 2 use basic number Students in Kindergarten-Grade 2 use basic number sense to compose and decompose numbers in sense to compose and decompose numbers in order to solve problems requiring precision, order to solve problems requiring precision, estimation, and reasonableness. By the end of estimation, and reasonableness. By the end of Grade 2, Grade 2, students know basic addition and students know basic addition and subtraction facts and are using them to work subtraction facts and are using them to work flexibly, efficiently, and accurately with numbers flexibly, efficiently, and accurately with numbers during addition and subtraction computation.during addition and subtraction computation.

Texas Essential Knowledge and Texas Essential Knowledge and SkillsSkills

Throughout mathematics in Grades 3-5, students Throughout mathematics in Grades 3-5, students develop numerical fluency with conceptual develop numerical fluency with conceptual understanding and computational accuracy. understanding and computational accuracy. Students in Grades 3-5 use knowledge of the base-Students in Grades 3-5 use knowledge of the base-ten place value system to compose and ten place value system to compose and decompose numbers in order to solve problems decompose numbers in order to solve problems requiring precision, estimation, and requiring precision, estimation, and reasonableness. By the end of Grade 5, reasonableness. By the end of Grade 5, students students know basic addition, subtraction, multiplication, know basic addition, subtraction, multiplication, and division facts and are using them to work and division facts and are using them to work flexibly, efficiently, and accurately with flexibly, efficiently, and accurately with numbers during addition, subtraction, numbers during addition, subtraction, multiplication, and division computation.multiplication, and division computation.

How Do You Use Numerical Fluency? Solve the following problem mentally.

Ms. Hill wants to carpet her rectangular living room, which measures 14 feet by 11 feet. If the carpet she wants to purchase costs $1.50 per square foot, including tax, how much will it cost to carpet her living room?

Write down your thought processes of how you solved the problem.

Turn to someone next to you and share your problem solving strategies.

Composing and DecomposingComposing and Decomposing

Building and taking apart numbers Building and taking apart numbers Looking for patterns/relationships between Looking for patterns/relationships between

numbersnumbers Unitizing numbersUnitizing numbers Using numbers as reference pointsUsing numbers as reference points

Definition of Number SenseDefinition of Number Sense

Number sense is a “…good intuition about Number sense is a “…good intuition about numbers and their relationships. It develops numbers and their relationships. It develops gradually as a result of exploring numbers, gradually as a result of exploring numbers, visualizing them in a variety of contexts and visualizing them in a variety of contexts and relating them in ways that are not limited by relating them in ways that are not limited by traditional algorithms (p. 11).”traditional algorithms (p. 11).”

Howden, H. (1989). Teaching number sense. Howden, H. (1989). Teaching number sense. Arithmetic TeacherArithmetic Teacher, 36(6), 6-11., 36(6), 6-11.

Problem SolvingProblem Solving The potential to model (use manipulatives, draw The potential to model (use manipulatives, draw

pictures, create tables, charts, or graphs) the situation pictures, create tables, charts, or graphs) the situation must be a natural progression of the problem.must be a natural progression of the problem.

The problem needs to be well defined so that children The problem needs to be well defined so that children can analyze the problem and comprehend what they are can analyze the problem and comprehend what they are to accomplish.to accomplish.

The problem encourages children to delve deeper into The problem encourages children to delve deeper into the problem by asking questions and identifying patterns.the problem by asking questions and identifying patterns.

Fosnot, C. & M. Dolk. (2001). Fosnot, C. & M. Dolk. (2001). Young mathematicians at work: Constructing Young mathematicians at work: Constructing number sense, addition, and subtractionnumber sense, addition, and subtraction. Portsmouth, NH: Heinemann.. Portsmouth, NH: Heinemann.

Reading FluencyReading Fluency

Fluency is important because it provides a Fluency is important because it provides a bridge bridge between word recognition and comprehensionbetween word recognition and comprehension. . Because fluent readers do not have to concentrate on Because fluent readers do not have to concentrate on decoding the words, they can focus their attention on decoding the words, they can focus their attention on what the text means. They can make connections what the text means. They can make connections among the ideas in the text and between the text and among the ideas in the text and between the text and their background knowledge. In other words, their background knowledge. In other words, fluent fluent readers recognize words and comprehend at the same readers recognize words and comprehend at the same time. Less fluent readers, however, must focus their time. Less fluent readers, however, must focus their attention on figuring out the words, leaving them little attention on figuring out the words, leaving them little attention for understanding the text.attention for understanding the text.

Institute for Literacy. (2006, March). Institute for Literacy. (2006, March). Put reading first - k-3 (fluency)Put reading first - k-3 (fluency) online at online at http://www.nifl.gov/partnershipforreading/publications/reading_first1fluency.html http://www.nifl.gov/partnershipforreading/publications/reading_first1fluency.html

Numerical FluencyNumerical FluencyFluency is important because it provides a Fluency is important because it provides a bridge bridge between number recognition and problem solving between number recognition and problem solving comprehension.comprehension. Because people who are numerically Because people who are numerically fluent do not have to concentrate on operation facts, they fluent do not have to concentrate on operation facts, they can focus their attention on what the problem means. can focus their attention on what the problem means. They can make connections among the ideas in the They can make connections among the ideas in the problem and their background knowledge. In other problem and their background knowledge. In other words, words, people who are numerically fluent recognize how people who are numerically fluent recognize how to compose and decompose numbers based on patterns to compose and decompose numbers based on patterns and comprehend how to use those numerical patterns to and comprehend how to use those numerical patterns to solve problems.solve problems. People who are less fluent, however, People who are less fluent, however, must focus their attention on the operations, leaving must focus their attention on the operations, leaving them little attention for understanding the problem.them little attention for understanding the problem.

Smith, K. H. and Schielack, J. (2006)

Development of Numerical FluencyDevelopment of Numerical Fluency

First the student MUST build an First the student MUST build an understanding of composing and understanding of composing and decomposing number through meaningful decomposing number through meaningful problems.problems.

Then through much meaningful practice, Then through much meaningful practice, children build automaticity, which is the children build automaticity, which is the fast, effortless composing and fast, effortless composing and decomposing of numbers. decomposing of numbers.

Fosnot, C. & M. Dolk. (2001). Fosnot, C. & M. Dolk. (2001). Young mathematicians at work: Constructing Young mathematicians at work: Constructing number number sense, addition, and subtractionsense, addition, and subtraction. Portsmouth, NH: . Portsmouth, NH: HeinemannHeinemann

Numerical FluencyNumerical Fluency

Numerical Fluency is the ability to Numerical Fluency is the ability to compose and decompose numbers compose and decompose numbers flexibly, efficiently, and accurately within flexibly, efficiently, and accurately within the context of meaningful situations.the context of meaningful situations.

Smith, K. H., Lopez, A., Reid, G., & Sullivan, C. (2006)

Numerical FluencyNumerical Fluency

How does this definition of numerical How does this definition of numerical fluency relate to how you approached fluency relate to how you approached solving the problem?solving the problem?

AbbyAbby

8+8=8+8=

8+9=8+9=

How would you describe Abby’s How would you describe Abby’s numerical fluency? numerical fluency?

BREAKBREAK

Developing Numerical FluencyDeveloping Numerical Fluency


Increase teacher knowledge regarding the Increase teacher knowledge regarding the refinements of the TEKS relating to the refinements of the TEKS relating to the Development of Numerical Fluency.Development of Numerical Fluency.

Increase teacher knowledge of composing and Increase teacher knowledge of composing and decomposing numbers.decomposing numbers.

Increase teacher knowledge of developmental Increase teacher knowledge of developmental stages of numerical fluency.stages of numerical fluency.

Develop an understanding of the use of Develop an understanding of the use of metacognition in problem solving.metacognition in problem solving.

Tanya is building a staircase in the pattern shown. The blocks are 1-inch cubes. She wants the last step to be 10 inches tall. How many cubes does she need in order to build the staircase?

What am I supposed to What am I supposed to find?find?

PicturePicture

Table/Chart/ListTable/Chart/List Number SentenceNumber Sentence

Explain how you derived your answer(s)?

Write down your thought processes as you solve the following problem.

Let us come back together and Let us come back together and share solutions and strategies share solutions and strategies with groups.with groups.

TEKSTEKS

Each group will be assigned a grade level.Each group will be assigned a grade level. Identify the TEKS in your grade level (K-5) Identify the TEKS in your grade level (K-5)

that students must master in order to have that students must master in order to have success in solving this 5success in solving this 5thth grade problem. grade problem.

Are there any refinements that need to be Are there any refinements that need to be identified? Add refined TEKS needed to identified? Add refined TEKS needed to teach this concept to the TEKS teach this concept to the TEKS Refinement Wall.Refinement Wall.

Numerical SequencesNumerical Sequences

Children struggle with patterns like the Children struggle with patterns like the following:following: Arithmetic sequencesArithmetic sequences Geometric sequencesGeometric sequences Figurate NumbersFigurate Numbers

Composing and Decomposing Composing and Decomposing

Numerical PatternsNumerical Patterns

SubitizingSubitizing

Foundations of Numerical FluencyFoundations of Numerical Fluency

Group 1 - One-to-one CorrespondenceGroup 1 - One-to-one Correspondence

Group 2 - Inclusion of SetGroup 2 - Inclusion of Set

Group 3 - Counting On/Counting DownGroup 3 - Counting On/Counting Down

Group 4 - More Than/Less Than/Equal ToGroup 4 - More Than/Less Than/Equal To

Group 5 - Part/Part/WholeGroup 5 - Part/Part/Whole

Group 6 - UnitizingGroup 6 - Unitizing

ReflectionReflection

What concepts did you struggle with?What concepts did you struggle with? What concepts do your students struggle with?What concepts do your students struggle with? What actions will you take to help students What actions will you take to help students

develop the foundations of patterns?develop the foundations of patterns? Should these strategies be taught in a particular Should these strategies be taught in a particular

sequence? Explain why or why not. sequence? Explain why or why not. If you are teaching older students who have not If you are teaching older students who have not

developed these strategies, how can you schedule developed these strategies, how can you schedule your instruction to include remediation of these your instruction to include remediation of these concepts?concepts?

BREAKBREAK

Strategies for Numerical FluencyStrategies for Numerical Fluency


Increase teacher knowledge regarding the Increase teacher knowledge regarding the refinements of the TEKS relating to refinements of the TEKS relating to numerical fluency.numerical fluency.

Increase teacher knowledge of composing Increase teacher knowledge of composing and decomposing numbers.and decomposing numbers.

Increase teacher knowledge of strategies Increase teacher knowledge of strategies to develop numerical fluency.to develop numerical fluency.


Solve the Following Problem Solve the following problem mentally.

A group of teachers at a local school are involved in a walking contest. They are asked to wear a pedometer for eight weeks. The first week Janice walked 65,787 steps. The next three weeks she walked a total of 214,241 steps. On average how many steps did Janice walk per day during the four week period?

Write down the thought processes you used to solve the problem.

How Did You Solve the Problem?How Did You Solve the Problem?

Share your strategies with your neighbor.Share your strategies with your neighbor. Share your strategies with the whole Share your strategies with the whole

group. group.

TEKSTEKS




ModelsModels Concrete (such as)Concrete (such as)

CountersCounters Double-sided countersDouble-sided counters Thematic countersThematic counters

Rods Rods Base Ten BlocksBase Ten Blocks

Semi-Concrete/Pictorial (such as)Semi-Concrete/Pictorial (such as) Ten-Frame TemplatesTen-Frame Templates Drawing PicturesDrawing Pictures Number LinesNumber Lines

BridgingBridging•Begin with stated problems that require children to

think.

•Have children use manipulatives to develop a visualization of the problem.

•Have students record about their work.

•As the teacher, lead the students to abstraction.

Building on UnitizingBuilding on Unitizing

Spotting NumbersSpotting Numbers Let’s Frame ItLet’s Frame It

Strategies for Addition and Strategies for Addition and Subtraction of Whole NumbersSubtraction of Whole Numbers

A.A. Give Me Ten!Give Me Ten!

B.B. Think AdditionThink Addition

C.C. Seeing DoublesSeeing Doubles

D.D. Half of Doubles Half of Doubles

E.E. Doubles Plus OneDoubles Plus One

F.F. Speedy TensSpeedy Tens

Strategies of CompensationStrategies of Compensation

17 +12 = 2917 +12 = 29

29 – 12 = 1729 – 12 = 17

Fact FamiliesFact Families

Using subitizing to teach fact families.Using subitizing to teach fact families.

BREAKBREAK

Operations and Numerical FluencyOperations and Numerical Fluency

25 + 25 =10 + 10 + 5 + 10 + 10 + 5 =

10, 20, 25, 35, 45, 5020 + 20 + 5 + 5 = 40 + 10 =

50 I just thought of it as 2 quarters

and 2 quarters is 50 cents. So, 25 + 25 = 50


Increase teacher knowledge regarding the Increase teacher knowledge regarding the refinements of the TEKS relating to numerical refinements of the TEKS relating to numerical fluency.fluency.

Increase teacher knowledge of composing and Increase teacher knowledge of composing and decomposing numbers.decomposing numbers.

Increase teacher knowledge of the use of Increase teacher knowledge of the use of strategies to teach numerical fluency for strategies to teach numerical fluency for operations of whole numbers.operations of whole numbers.


Defining Addition, Subtraction, Defining Addition, Subtraction, Multiplication, and Division of Whole Multiplication, and Division of Whole

NumbersNumbers

At your table, develop a definition of At your table, develop a definition of addition, subtraction, multiplication, and addition, subtraction, multiplication, and division based on the TEKS for your division based on the TEKS for your particular grade level.particular grade level.

Small groups share their answer with the Small groups share their answer with the large group.large group.

Solve the Following Problem

Mrs. Parks is buying ice-cream bars for the 17 dozen students at her school. The ice cream bars are packaged 10 to a box. What is an estimate of the number of boxes she has to buy so that each student gets at least 1 ice cream bar?


PicturePicture




Let us come back together and Let us come back together and share solutions and strategies share solutions and strategies with groups.with groups.

TEKSTEKS



Are there any refinements that need to be Are there any refinements that need to be identified? Add refined TEKS needed to identified? Add refined TEKS needed to teach this concept to the poster board.teach this concept to the poster board.

EstimationEstimation

Measurement estimationMeasurement estimation Quantity estimationQuantity estimation Computational estimationComputational estimation

Computational EstimationComputational Estimation

Computational estimation is the ability to Computational estimation is the ability to quickly produce an approximate result for quickly produce an approximate result for a computation that will be adequate for the a computation that will be adequate for the situation.situation.

Computational EstimationComputational Estimation

Front-end ApproachFront-end Approach Rounding MethodsRounding Methods Compatible NumbersCompatible Numbers

BREAKBREAK

Addition and Subtraction of Addition and Subtraction of Whole NumbersWhole Numbers

Amy is 8 years old. She was assigned a Amy is 8 years old. She was assigned a school project regarding her family. She school project regarding her family. She did not know the year that her did not know the year that her grandmother was born, but did know that grandmother was born, but did know that she just celebrated her 86she just celebrated her 86thth birthday. How birthday. How could Amy determine the year her could Amy determine the year her grandmother was born? grandmother was born?

Double Digit Addition and Double Digit Addition and Subtraction Through the Use of Subtraction Through the Use of

StrategiesStrategies Reflect on how you have solved previous Reflect on how you have solved previous

problems. Have you always used a problems. Have you always used a traditional algorithm to solve the problem?traditional algorithm to solve the problem?

Think about how children use inventive Think about how children use inventive strategies to solve problems.strategies to solve problems.

How important is students’ metacognition How important is students’ metacognition of solving mathematical problems?of solving mathematical problems?

Relationships in Multiplication and Relationships in Multiplication and DivisionDivision

12

1x12

2x6

3x412x1

6x24x3

Relationships of OperationsRelationships of Operations Brainstorm at your table all the relationships Brainstorm at your table all the relationships

between the operations of whole numbers.between the operations of whole numbers.

Walk 7 steps from where you are now and share Walk 7 steps from where you are now and share relationships with someone near you.relationships with someone near you.

Take 7 more steps and repeat this procedure.Take 7 more steps and repeat this procedure.

Share with the whole group relationships that Share with the whole group relationships that were found. were found.

When to Develop AutomaticityWhen to Develop Automaticity

Once you have taught two strategies, drill Once you have taught two strategies, drill based on those strategies.based on those strategies.

Teach more strategies.Teach more strategies. Automaticity is needed ONLY after Automaticity is needed ONLY after

students have developed a meaningful students have developed a meaningful concept of addition, subtraction, concept of addition, subtraction, multiplication, or division and they have multiplication, or division and they have also developed flexible and useful also developed flexible and useful strategies for those operations. strategies for those operations.

How to Develop AutomaticityHow to Develop Automaticity

The competition is to be developed from The competition is to be developed from within the child (intrinsic motivation), not within the child (intrinsic motivation), not against other children.against other children.

Using time as the goal.Using time as the goal. Using number of problems as the goal.Using number of problems as the goal.

BREAKBREAK

Fractions and Numerical Fractions and Numerical FluencyFluency


Increase teacher knowledge regarding the Increase teacher knowledge regarding the refinements of the TEKS relating to refinements of the TEKS relating to numerical fluency.numerical fluency.

Increase teacher knowledge of composing Increase teacher knowledge of composing and decomposing numbers.and decomposing numbers.

Increase teacher knowledge of rational Increase teacher knowledge of rational numbers.numbers.


Solve the Following Problem Think about the strategy/strategies you used

to solve the following problem.

Yesterday I baked my family a 9x13 pan of brownies. I cut the brownies in individual servings. Russell took ½ of the brownies, Chris took 1/3 of what was left, Natalie took ¼ of what was then left in the pan. An hour later Chris came back and took two more brownies, leaving one brownie for me. How many individual pieces did I cut the brownies into to begin with?


PicturePicture




How Did You Solve the How Did You Solve the Problem?Problem?

Share your strategies with your neighbor.Share your strategies with your neighbor. Share your strategies with the whole Share your strategies with the whole

group. group.

TEKSTEKS



Are there any refinements that need to be Are there any refinements that need to be identified? Add refined TEKS needed to identified? Add refined TEKS needed to teach this concept to the poster board.teach this concept to the poster board.

Development of FractionsDevelopment of Fractions

Where and How Do We See Where and How Do We See Fractions?Fractions?

Importance of the WholeImportance of the Whole

Referents Referents

Reference points of 0, ½, and 1Reference points of 0, ½, and 1 Fraction EstimatorsFraction Estimators How Big Am I?How Big Am I?

Solve the Following Problem Think about the strategy/strategies you

used to solve the following problem.

2/4, 3/6, 4/8, and 5/10 are all equivalent to 1/2. What is the relationship between the numerator and denominator in each fraction? Explain why they are equivalent to 1/2.


PicturePicture




TEKSTEKS




Equivalent FractionsEquivalent Fractions

How Do I Compare? Let Me Count the How Do I Compare? Let Me Count the Ways.Ways.

Equality for AllEquality for All Mixing It UpMixing It Up


Numerical Fluency Defined!Numerical Fluency Defined!

““Mingle to Music”Mingle to Music”

Stand with mind map and a writing utensil.Stand with mind map and a writing utensil. When the music begins, walk around the room with your When the music begins, walk around the room with your

paper.paper. When music stops, freeze and turn to the person closest When music stops, freeze and turn to the person closest

to you.to you. If you do not have a partner, raise your hand and walk to If you do not have a partner, raise your hand and walk to

another person raising his/her hand.another person raising his/her hand. Each partner shares his/her mind map. As you are Each partner shares his/her mind map. As you are

listening, add ideas you did not have on your mind map.listening, add ideas you did not have on your mind map. Tell participants they will have 2 minutes each (total of 4 Tell participants they will have 2 minutes each (total of 4

minutes) to share and add to mind map.minutes) to share and add to mind map. When the music begins again, thank your partner and When the music begins again, thank your partner and

walk around the room again.walk around the room again. Continue this process two more times. Continue this process two more times.


Numerical Fluency Defined!Numerical Fluency Defined!

What Are the Next Steps?What Are the Next Steps?

Building Foundations for Mathematics Defining Numerical Fluency.

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Transcript of Building Foundations for Mathematics Defining Numerical Fluency.