Building Foundations for Mathematics Defining Numerical Fluency

Goals & Purposes  Increase teacher knowledge regarding the refinements of the TEKS relating to numerical fluency.  Develop a working definition of numerical fluency.  Increase teacher knowledge of composing and decomposing numbers.  Increase teacher knowledge of developmental stages of numerical fluency.  Increase teacher knowledge of strategies to develop numerical fluency.  Develop an understanding of the use of metacognition in 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 Skills Throughout mathematics in Kindergarten-Grade 2, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Kindergarten-Grade 2 use basic number sense to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 2, students know basic addition and subtraction facts and are using them to work flexibly, efficiently, and accurately with numbers during addition and subtraction computation.

Texas Essential Knowledge and Skills Throughout mathematics in Grades 3-5, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Grades 3-5 use knowledge of the base- ten place value system to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 5, students know basic addition, subtraction, multiplication, and division facts and are using them to work flexibly, efficiently, and accurately with numbers during addition, subtraction, 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 Decomposing  Building and taking apart numbers  Looking for patterns/relationships between numbers  Unitizing numbers  Using numbers as reference points

Definition of Number Sense  Number sense is a “…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 (p. 11).”  Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36(6), 6-11.

Problem Solving  The potential to model (use manipulatives, draw pictures, create tables, charts, or graphs) the situation must be a natural progression of the problem.  The problem needs to be well defined so that children can analyze the problem and comprehend what they are to accomplish.  The problem encourages children to delve deeper into the problem by asking questions and identifying patterns. Fosnot, C. & M. Dolk. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.

Reading Fluency Fluency is important because it provides a bridge between word recognition and comprehension. Because fluent readers do not have to concentrate on decoding the words, they can focus their attention on what the text means. They can make connections among the ideas in the text and between the text and their background knowledge. In other words, fluent readers recognize words and comprehend at the same time. Less fluent readers, however, must focus their attention on figuring out the words, leaving them little attention for understanding the text. Institute for Literacy. (2006, March). Put reading first - k-3 (fluency) online at

Numerical Fluency Fluency is important because it provides a bridge between number recognition and problem solving comprehension. Because people who are numerically fluent do not have to concentrate on operation facts, they can focus their attention on what the problem means. They can make connections among the ideas in the problem and their background knowledge. In other words, people who are numerically fluent recognize how to compose and decompose numbers based on patterns and comprehend how to use those numerical patterns to solve problems. People who are less fluent, however, must focus their attention on the operations, leaving them little attention for understanding the problem. Smith, K. H. and Schielack, J. (2006)

Development of Numerical Fluency  First the student MUST build an understanding of composing and decomposing number through meaningful problems.  Then through much meaningful practice, children build automaticity, which is the fast, effortless composing and decomposing of numbers. Fosnot, C. & M. Dolk. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann

Numerical Fluency Numerical Fluency is the ability to compose and decompose numbers flexibly, efficiently, and accurately within the context of meaningful situations. Smith, K. H., Lopez, A., Reid, G., & Sullivan, C. (2006)

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

Abby 8+8=8+9= How would you describe Abby’s numerical fluency?

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Developing Numerical Fluency

Goals & Purposes  Increase teacher knowledge regarding the refinements of the TEKS relating to the Development of Numerical Fluency.  Increase teacher knowledge of composing and decomposing numbers.  Increase teacher knowledge of developmental stages of numerical fluency.  Develop an understanding of the use of 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 find? Picture Table/Chart/List Number 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 share solutions and strategies with groups.

TEKS  Each group will be assigned a grade level.  Identify the TEKS in your grade level (K-5) that students must master in order to have success in solving this 5 th grade problem.  Are there any refinements that need to be identified? Add refined TEKS needed to teach this concept to the TEKS Refinement Wall.

Numerical Sequences Children struggle with patterns like the following:  Arithmetic sequences  Geometric sequences  Figurate Numbers Composing and Decomposing Numerical Patterns

Subitizing

Foundations of Numerical Fluency Group 1 - One-to-one Correspondence Group 2 - Inclusion of Set Group 3 - Counting On/Counting Down Group 4 - More Than/Less Than/Equal To Group 5 - Part/Part/Whole Group 6 - Unitizing

Reflection  What concepts did you struggle with?  What concepts do your students struggle with?  What actions will you take to help students develop the foundations of patterns?  Should these strategies be taught in a particular sequence? Explain why or why not.  If you are teaching older students who have not developed these strategies, how can you schedule your instruction to include remediation of these concepts?

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Strategies for Numerical Fluency

Goals & Purposes  Increase teacher knowledge regarding the refinements of the TEKS relating to numerical fluency.  Increase teacher knowledge of composing and decomposing numbers.  Increase teacher knowledge of strategies to develop numerical fluency.  Develop an understanding of the use of metacognition in problem solving.

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?  Share your strategies with your neighbor.  Share your strategies with the whole group.

TEKS  Each group will be assigned a grade level.  Identify the TEKS in your grade level (K-5) that students must master in order to have success in solving this 5 th grade problem.  Are there any refinements that need to be identified? Add refined TEKS needed to teach this concept to the TEKS Refinement Wall.

Models  Concrete (such as)  Counters  Double-sided counters  Thematic counters  Rods  Base Ten Blocks  Semi-Concrete/Pictorial (such as)  Ten-Frame Templates  Drawing Pictures  Number Lines

Bridging 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 Unitizing  Spotting Numbers  Let’s Frame It

Strategies for Addition and Subtraction of Whole Numbers A. Give Me Ten! B. Think Addition C. Seeing Doubles D. Half of Doubles E. Doubles Plus One F. Speedy Tens

Strategies of Compensation  = 29  29 – 12 = 17

Fact Families  Using subitizing to teach fact families.

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Operations and Numerical Fluency =   = 10, 20, 25, 35, 45, 50   = = 50   I just thought of it as 2 quarters and 2 quarters is 50 cents. So, = 50

Goals & Purposes  Increase teacher knowledge regarding the refinements of the TEKS relating to numerical fluency.  Increase teacher knowledge of composing and decomposing numbers.  Increase teacher knowledge of the use of strategies to teach numerical fluency for operations of whole numbers.  Develop an understanding of the use of metacognition in problem solving.

Defining Addition, Subtraction, Multiplication, and Division of Whole Numbers  At your table, develop a definition of addition, subtraction, multiplication, and division based on the TEKS for your particular grade level.  Small groups share their answer with the 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?

What am I supposed to find? Picture Table/Chart/List Number 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 share solutions and strategies with groups.

TEKS  Each group will be assigned a grade level.  Identify the TEKS in your grade level (K-5) that students must master in order to have success in solving this 5 th grade problem.  Are there any refinements that need to be identified? Add refined TEKS needed to teach this concept to the poster board.

Estimation  Measurement estimation  Quantity estimation  Computational estimation

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

Computational Estimation  Front-end Approach  Rounding Methods  Compatible Numbers

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Addition and Subtraction of Whole Numbers  Amy is 8 years old. She was assigned a school project regarding her family. She did not know the year that her grandmother was born, but did know that she just celebrated her 86 th birthday. How could Amy determine the year her grandmother was born?

Double Digit Addition and Subtraction Through the Use of Strategies  Reflect on how you have solved previous problems. Have you always used a traditional algorithm to solve the problem?  Think about how children use inventive strategies to solve problems.  How important is students’ metacognition of solving mathematical problems?

Relationships in Multiplication and Division 12 1x12 2x6 3x4 12x1 6x2 4x3

Relationships of Operations  Brainstorm at your table all the relationships between the operations of whole numbers.  Walk 7 steps from where you are now and share relationships with someone near you.  Take 7 more steps and repeat this procedure.  Share with the whole group relationships that were found.

When to Develop Automaticity  Once you have taught two strategies, drill based on those strategies.  Teach more strategies.  Automaticity is needed ONLY after students have developed a meaningful concept of addition, subtraction, multiplication, or division and they have also developed flexible and useful strategies for those operations.

How to Develop Automaticity  The competition is to be developed from within the child (intrinsic motivation), not against other children.  Using time as the goal.  Using number of problems as the goal.

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Fractions and Numerical Fluency

Goals & Purposes  Increase teacher knowledge regarding the refinements of the TEKS relating to numerical fluency.  Increase teacher knowledge of composing and decomposing numbers.  Increase teacher knowledge of rational numbers.  Develop an understanding of the use of metacognition in problem solving.

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?

What am I supposed to find? Picture Table/Chart/List Number Sentence Explain how you derived your answer(s)? Write down your thought processes as you solve the following problem.

How Did You Solve the Problem?  Share your strategies with your neighbor.  Share your strategies with the whole group.

TEKS  Each group will be assigned a grade level.  Identify the TEKS in your grade level (K-5) that students must master in order to have success in solving this 5 th grade problem.  Are there any refinements that need to be identified? Add refined TEKS needed to teach this concept to the poster board.

Development of Fractions Where and How Do We See Fractions? Importance of the Whole

Referents Reference points of 0, ½, and 1  Fraction Estimators  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.

What am I supposed to find? Picture Table/Chart/List Number Sentence Explain how you derived your answer(s)? Write down your thought processes as you solve the following problem.

TEKS  Each group will be assigned a grade level.  Identify the TEKS in your grade level (K-5) that students must master in order to have success in solving this 5 th grade problem.  Are there any refinements that need to be identified? Add refined TEKS needed to teach this concept to the TEKS Refinement Wall.

Equivalent Fractions  How Do I Compare? Let Me Count the Ways.  Equality for All  Mixing It Up

Building Foundations for Mathematics Numerical Fluency Defined!

“Mingle to Music”  Stand with mind map and a writing utensil.  When the music begins, walk around the room with your paper.  When music stops, freeze and turn to the person closest to you.  If you do not have a partner, raise your hand and walk to another person raising his/her hand.  Each partner shares his/her mind map. As you are listening, add ideas you did not have on your mind map.  Tell participants they will have 2 minutes each (total of 4 minutes) to share and add to mind map.  When the music begins again, thank your partner and walk around the room again.  Continue this process two more times.

Building Foundations for Mathematics Numerical Fluency Defined!

What Are the Next Steps?