1. Early Number Sense The “Phonics” of Mathematics Presenters: Lisa Zapalac, Head of Lower School Kevin Moore, 4 th Grade Math Brooke Carmichael, Kindergarten November 19, 2010 10:15 – 11:30 a.m. www.austintrinity.org

2. 3,996 + 4,246 Simplify the following expression:

3. 1 1 1 1 3,996 +4,246 8,242 Is this how you simplified it?

4. 2 nd Grader Simplifying 3,996 + 4,246 Example of 2nd Grader Using Compensation

5. How did the 2 nd grader simplify the expression? He used an addition strategy called compensation, but there are many underlying concepts that are embedded in compensation 1)He noticed that 3,996 is 4 less than 4,000 2)He recognized 4,246 as being equivalent to 4,242+ 4 3)He then associated (3,996 + 4) + 2,242

6. Recognizes values in their various forms Number Sense Demonstrates proficiency with estimation and evaluation of quantities Recognizes unreasonable conclusions Possesses a repertoire of mental computation strategies Evidence of Number Sense

7. Let’s try another expression 50 x 48 4th Grade Video

8. 4 48 x50 2400 Is this how you simplified it? 4 48 x 50 00 +2400 2400

9. 4 th Graders Simplifying 48 (50) Example 1 Example 2 Example 3

10. Solve 76 x 89 One more… 4th Grader

11. Number Sense…. How do we build it? There are many effective strategies for building number sense. At Trinity, “strings” are one power practice used.

12. Using “Strings” to Develop Number Sense Strings are a set of arithmetic problems in which the children are developing very specific strategies. Strings are generally done mentally. Each string begins with a known expression and moves towards the unknown, scaffolding the development of key strategies. The following slides contain examples of strings at various grade levels.

13. 1 st Grade String 5 + 5 5 + 6 6 + 6 6 + 7 7 + 7 7 + 8 8 + 8 9 + 7 6 + 8

14. Building Number Sense through Facts Doubles plus or minus 1 – Ex. 6 + 7 = 6 + 6 + 1 (or 7 + 7 – 1) = 13 Doubles plus or minus 2 – Ex. 5 + 7 = 5 + 5 + 2 (or 7 + 7 - 2) Working with the structure of five – Ex. 6 + 7 = 5 + 1 + 5 + 2 = 10 + 3 = 13 Making tens – Ex. 8 + 4 = 8 + 2 + 2 Using tens to solve nines – Ex. 9 + 7 = 10 + 7 - 1 Using compensation – Ex. 6 + 8 = 7 + 7 (adding one to one addend, while subtracting one from the other addend) Possesses a repertoire of mental computation strategies 5 + 5 5 + 6 6 + 6 6 + 7 7 + 7 7 + 8 8 + 8 9 + 7 6 + 8

15. Using Tools and Models to Develop Number Sense Recognizes values in their various forms The rekenrek, or arithmetic rack, is a tool consisting of two rows of ten beads with two sets of five in each. The rekenrek was developed by Adri Treffers, a researcher at the Freudenthal Institute in the Netherlands, and it provides a powerful model for exploring the composing and decomposing of number (Treffers 1991)

16. Kindergarten String Example 5 on the top, 5 on the bottom 7 on the top, 3 on the bottom 4 on the top, 6 on the bottom 6 on the top, 4 on the bottom 8 on the top, 2 on the bottom Possesses a repertoire of mental computation strategies Recognizes values in their various forms Kindergarten String

17. 2 3840 80 Moving Beyond Facts Modeling 38 + 42 The open number line is a tool used to model students’ thinking. In this problem, 38 + 42, a student might solve it by moving to a landmark number first. Or, they might first make jumps of ten. 38 78 40 80 2

18. Example of 2 nd Grade String Big Idea: Keeping One Number Whole and Taking Leaps of 10 75 + 20 75 + 25 75 + 24 55 + 30 55 + 39 69 + 21 69 + 29 2nd Grade String

19. Building Number Sense with Multiplication Constructing facts through relationships and models 4(4) = 162[(4)2] or 2(8)

20. Multiplication (3 rd & 4 th Grade Strategies) Doubling ▪6 x 6 = 2 x 3 x 6 Halving and doubling ▪4 x 3 = 2 x 6 Using the distributive property ▪7 x 8 = (5 x 8) + (2 x 8), or ▪7 x 8 = (8 x 8) – (1 x 8) Using the commutative property ▪5 x 8 = 8 x 5 Possesses a repertoire of mental computation strategies

21. Example of 4 th Grade String 4 x 8 14 x 8 6 x 9 26 x 9 12 x 13 15 x 24 4th Grade Multiplication String

22. 4 th Grade String Revisited – Connecting to Algebra a (8) (a + b) 8 (3a)2 (2a + c) (5) (a + 3) (a + 2)

23. Number sense is the bridge between arithmetic and algebra ArithmeticAlgebra Number Sense

24. Resources Books Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Devlin, K. (2000). The Math Gene. Great Britain: Weidenfeld & Nicolson Stigler & Hiebert (1999). The Teaching Gap. New York, NY: The Free Press Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Portsmouth, NH: Heinemann Carpenter, T., Franke, M., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Early Addition and Subtraction. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Extending Addition and Subtraction. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Early Multiplication and Division. Portsmouth, NH: Heinemann Fosnot, C. & Uittenbogaard, W. (2007). Minilessons for Extending Multiplication and Division. Portsmouth, NH: Heinemann Articles Faulkner, V. (2009). The Components of Number Sense – An Instructional Model for Teachers. – Teaching Exceptional Children, Vol. 41, No. 5, 24-30 Gersten, R. & Chard, D. (2010). Validating a Number Sense Screening Tool for Use in Kindergarten and First Grade: Prediction of Mathematics Proficiency in Third Grade – School Psychology Review, Vol. 39, No. 2, 181-195 Harel, G. & Rabin, J. (2010). Teaching Practices Associated With the Authoritative Proof Scheme – Journal for Research in Mathematics Education, Vol. 41, No. 1, 14-19 Web Sites DreamBox Learning - www.dreambox.comwww.dreambox.com To order a rekenrek: www.eNasco.comwww.eNasco.com