1. TDG Leadership Seminar 2009 1 Pivotal Problems: Knowing When to Use Them Pivotal Solutions: Knowing When to Reveal Them George W. Bright gbright45@comcast.net Professor Emeritus University of North Carolina at Greensboro

2. TDG Leadership Seminar 2009 2 Introductions What grades do you teach? Where do you teach? What is your experience in delivering professional development for teachers? What is something others might not know about you?

3. TDG Leadership Seminar 2009 3 Rating Personal Understanding On a scale of 1 (low) to 10 (high), rate yourself in terms of: how much you know about proportional reasoning how confident you are about helping students learn about proportional reasoning how confident you are about helping teachers learn about proportional reasoning

4. TDG Leadership Seminar 2009 4 Goals for the Session Solve proportional reasoning problems and explore their solutions Identify why some problems and some solutions might be “pivotal” in helping learners understand proportional reasoning Reflect on when pivotal problems and pivotal solutions might be most effectively presented

5. TDG Leadership Seminar 2009 5 Pivotal Problems and Pivotal Solutions What does pivotal mean? When might a problem be pivotal in developing understanding? When might a solution to a problem be pivotal in developing understanding?

6. TDG Leadership Seminar 2009 6 National Mathematics Advisory Panel The Final Report of the National Mathematics Advisory Panel recommends that understanding of fractions is one of the Critical Foundations of Algebra. Their definition of “fractions” includes ratios and proportions, so they really are recommending “proportional reasoning” as a critical foundation of algebra.

7. TDG Leadership Seminar 2009 7 The “3/5’s Problem” What do you see? Thompson, P. W. (2002). 3/5s problem. In B. Litwiller (Ed.). Making sense of fractions, ratios, and proportions: 2002 yearbook(pp. 103-104). Reston, VA: National Council of Teachers of Mathematics.

8. TDG Leadership Seminar 2009 8 Reorganizing Thinking What were the greatest challenges for you in the questions about the diagram? How did the questions encourage you to reorganize your thinking about fractions and operations on fractions?

9. TDG Leadership Seminar 2009 9 Is this pivotal? When might this problem be pivotal? What might you want learners to understand before the problem is posed? What would you want learners to understand after solving this problem?

10. TDG Leadership Seminar 2009 10 Equal or Equivalent? Are 1/3 and 2/6 equal fractions or equivalent fractions or both or neither? Does it matter what language we use? How might the language we use influence what students learn?

11. TDG Leadership Seminar 2009 11 Fractions In Between, Part 1 Find three fractions between 4/7 and 5/7.

12. TDG Leadership Seminar 2009 12 Fractions In Between, Part 2 4.5 is halfway between 4 and 5. Is 4.5/7 halfway between 4/7 and 5/7? Why or why not?

13. TDG Leadership Seminar 2009 13 Fractions In Between, Part 3 Find three fractions equally spaced between a/b and (a+1)/b. Would you ask students to solve the general case?

14. TDG Leadership Seminar 2009 14 Fractions In Between, Part 4 Find three fractions equally spaced between a/b and (a+N)/b. Would you ask students to solve the even more general case?

15. TDG Leadership Seminar 2009 15 Is this pivotal? When might this problem be pivotal? What would this problem help learners learn that more traditional problems might not help them learn?

16. TDG Leadership Seminar 2009 16 Fractions In Between, Part 5 Does the same strategy work for this problem? Find three fractions between 5/7 and 5/6.

17. TDG Leadership Seminar 2009 17 Fractions In Between, Part 6 6.5 is halfway between 7 and 6. Is 5/6.5 between 5/7 and 5/6? If so, is it halfway between 5/7 and 5/6? Why or why not?

18. TDG Leadership Seminar 2009 18 Fractions In Between, Part 7 Find three fractions equally spaced between a/(b+1) and a/b. Would you ask students to solve the general case?

19. TDG Leadership Seminar 2009 19 Fractions In Between, Part 8 Find three fractions equally spaced between a/(b+N) and a/b. Would you ask students to solve the even more general case?

20. TDG Leadership Seminar 2009 20 Is this pivotal? Could this problem be a pivotal problem? What grade level would this problem be most appropriate for? What mathematics does this highlight that other kinds of problems might not highlight?

21. TDG Leadership Seminar 2009 21 What is the Point? What big mathematics ideas are embedded in these “in between” problems? How might the solutions to these problems help move learners’ thinking forward?

22. TDG Leadership Seminar 2009 22 What is the label? True or false: 6 ÷ 2 = 3. 3 what? True or false: 6 ÷ 3 = 2. 2 what? Why are the labels different for the quotients?

23. TDG Leadership Seminar 2009 23 What is the label? True or false: 6 ÷ 2/3 = 9 9 what? Is it important for students to understand what label is attached to a quotient? Is this a pivotal idea?

24. TDG Leadership Seminar 2009 24 Reversibility Asking learners to “reverse” their thinking helps them create connections among ideas. Suppose a rectangular prism has a volume of 40 cm 3 and height of 5 cm. What else can you tell me about the rectangular prism?

25. TDG Leadership Seminar 2009 25 Thinking Differently about Familiar Ideas Imagine what an inch looks like. Imagine what a centimeter looks like. What is the area of a rectangle that is 5 inches long and 3 centimeters wide?

26. TDG Leadership Seminar 2009 26 Changing Views How might understanding of a familiar idea change by solving a problem that presents the idea in an unfamiliar way? How might you decide when to pose such unfamiliar problems?

27. TDG Leadership Seminar 2009 27 Seeing the Big Picture Solve this problem: If 2(x - 3) = 8, then what is the value of (x - 3) 2 + 5(x - 3) - 2? How flexible is your thinking? Can you see the “big picture” in a problem or do you focus on the details?

28. TDG Leadership Seminar 2009 28 Interference For any linear measurement, let Y = number of yards for that measurement, let F = number of feet for that measurement. Write an equation showing the relationship of these two variables.

29. TDG Leadership Seminar 2009 29 Variables How did these problems expand your understanding of variable? Why is it important for learners to understand what a variable is?

30. TDG Leadership Seminar 2009 30 Numbers with a Simple Relationship Melissa bought 0.43 of a pound of wheat flour for which she paid $0.86. How many pounds of flour could she buy for one dollar? Post, T. R., Harel, G., Behr, M., & Lesh, R. (1991). Intermediate teachers’ knowledge of rational number concepts. In E. Fennema, T. P. Carpenter, & S. J. Lamon (Eds.), Integration research on teaching and learning mathematics (pp. 177-198). Ithaca, NY: SUNY Press.

31. TDG Leadership Seminar 2009 31 Numbers with a Not-so-simple Relationship Melissa bought 0.46 of a pound of wheat flour for which she paid $0.86. How many pounds of flour could she buy for one dollar?

32. TDG Leadership Seminar 2009 32 Reflection on Problem 1 modified Which problem was more difficult, the “simple” problem or the “not-so-simple” problem? What made that problem difficult? How does the choice of numbers in a problem affect the way you (or students) might think about the problem?

33. TDG Leadership Seminar 2009 33 Is this pivotal? Could these problems be pivotal? What would you want learners to take away from engagement with these problems?

34. TDG Leadership Seminar 2009 34 A Different Kind of “Adult” Problem In an adult condominium complex, 2/3 of the men are married to 3/5 of the women. What part of the residents are married? Lester, F. (2002). Condo problem. In B. Litwiller (Ed.). Making sense of fractions, ratios, and proportions: 2002 yearbook (pp. 191-192). Reston, VA: National Council of Teachers of Mathematics.

35. TDG Leadership Seminar 2009 35 Identifying Difficulties Why do people struggle with this problem? What makes it difficult? How do the different solutions reveal different aspects of the underlying mathematics ideas?

36. TDG Leadership Seminar 2009 36 Is this pivotal? Could this problem be pivotal? Could the solutions be pivotal? What mathematics might this problem or the solutions to this problem help learners internalize?

37. TDG Leadership Seminar 2009 37 Graphs How are the two graphs below alike? How are they different?

38. TDG Leadership Seminar 2009 38 Graphing Speed Joe walks down a straight path and then turns around a walks back to the starting point. The graph below displays how far away he was from the starting point. Sketch the graph of his walking speed(s).

39. TDG Leadership Seminar 2009 39 Graphing Speed Joe walks down a straight path and then turns around a walks back to the starting point. The graph below displays how far away he was from the starting point. Sketch the graph of his walking velocities. (How is this graph different from the previous graph?)

40. TDG Leadership Seminar 2009 40 Connecting to the Real World Speed and velocity are complex ideas. It takes considerable time and experience to understand them fully. When might you use these problems? For what purpose? How might discussion of the solutions help students understand mathematics more deeply? Is this problem pivotal?

41. TDG Leadership Seminar 2009 41 Rating Personal Understanding (reprise) On a scale of 1 (low) to 10 (high), rate yourself in terms of: how much you know about proportional reasoning how confident you are about helping students learn about proportional reasoning how confident you are about helping teachers learn about proportional reasoning