1. Making Sense of Division National Council of Teachers of Mathematics Salt Lake City, Utah April 12, 2008 Beth Schefelker Milwaukee Public Schools Melissa Hedges Milwaukee Public Schools

2. Session Overview Use connections to contextual situations to explore the meaning of division Explore several number oriented strategies for multi-digit division of whole numbers Analyze student work samples

3. Our journey… What mathematical knowledge and skills are needed to successfully navigate division of large numbers? In what ways might the role of contextual situations for division impact students' understanding? How might context be used for construction of division concepts, not for application at the end of the unit? (Fosnot & Dolk, 2001)‏

4. Consider the following: Context #1 Evan has 24 apples. He put them into bags containing 6 apples each. How many bags did Evan use? Context #2 Evan has 24 apples. He shares them equally among 6 friends. How many apples with each friend get? What is the answer to each? How might a young child model each problem?

5. You might say the answer is “the same.” But is it?

6. Two Contexts for Division Context #1 – Measurement Evan has 24 apples. He put them into bags containing 6 apples each. How many bags did Evan use?  Known: Size of group  Unknown: Number of groups Context #2 – Partitive Evan has 24 apples. He shares them equally among 6 friends. How many apples with each friend get?  Known: Number of groups  Unknown: Size of groups

7. Let's do some math Pick a problem: Solve the problem – no standard algorithm Share your strategy 125 /13 or 95/4

8. Fundamental Understandings… What are some mathematical understandings you need to compute these problems?

9. Solving Contextual Problems – Are Computation Skills Enough? Poor problem solvers fail to spend adequate time thinking about the problem and what it is about. They rush in and begin doing calculations, believing that “number crunching” is what solves problems. That is simply not the case. (Van de Walle, J. 2004. p. 151)‏

10. Developing Meaningful Contexts  Pick a problem: 125 /13 or 95/4  Write a context.

11. Discussing Meaningful Contexts In triads: #1 Shares their story #2 Retells the story and acts it out #3 Label the problem as partitive or measurement Switch roles !

12. The strategies… Repeated subtraction (measurement division)‏ Dealing out in groups (partitive division)‏ Using multiplication Partitioning the dividend Direct modeling with measurement

13. Contextual Situations Lead the Way Different contexts; same numbers. Identify as partitive or measurement Problem A  Tyrone has a collection of 169 Yu-Gi-Oh cards. He shares them equally among his 14 friends. How many cards will each friend get? Problem B  Tyrone has a collection of 169 Yu-Gi-Oh cards. He wants to give each friend 14 cards. How many friends can he share his cards with?

14. Shoulder Partner Activity Individually solve either A or B Try “repeated subtraction” and “dealing out” for both Problem A and Problem B Reflect: In what way do these strategies model the context of the problems?

15. The strategies… Repeated subtraction (measurement division)‏ Dealing out in groups (partitive division)‏ Using multiplication Partitioning the dividend Direct modeling with measurement

16. Viewing student work 259/24 Read the story Reflect on the context  Partitive or Measurement What do the students understand about division? In what ways do their strategies connect to the context?

17. Reflections on this experience The critical role context plays when supporting children's understanding of division. The insights gained while applying strategies to different contexts.

18. Contact information Milwaukee Mathematics Partnership www.mmp.uwm.edu Beth Schefelkerschefeba@milwaukee.k12.wi.usschefeba@milwaukee Melissa Hedges melissahedges@sbcglobal.net