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Pivotal Problems: Knowing When to Use Them Pivotal Solutions: Knowing When to Reveal Them

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. Introductions. • What grades do you teach? • Where do you teach?

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Pivotal Problems: Knowing When to Use Them Pivotal Solutions: Knowing When to Reveal Them

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

  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? TDG Leadership Seminar 2009 2

  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 TDG Leadership Seminar 2009 3

  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 TDG Leadership Seminar 2009 4

  5. Pivotal Problems and Pivotal Solutions TDG Leadership Seminar 2009 5 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. National Mathematics Advisory Panel TDG Leadership Seminar 2009 6 • 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. 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. TDG Leadership Seminar 2009 7

  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? TDG Leadership Seminar 2009 8

  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? TDG Leadership Seminar 2009 9

  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? TDG Leadership Seminar 2009 10

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

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

  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? TDG Leadership Seminar 2009 13

  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? TDG Leadership Seminar 2009 14

  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? TDG Leadership Seminar 2009 15

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

  17. Fractions In Between, Part 6 TDG Leadership Seminar 2009 17 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. 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? TDG Leadership Seminar 2009 18

  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? TDG Leadership Seminar 2009 19

  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? TDG Leadership Seminar 2009 20

  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? TDG Leadership Seminar 2009 21

  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? TDG Leadership Seminar 2009 22

  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? TDG Leadership Seminar 2009 23

  24. Reversibility • Asking learners to “reverse” their thinking helps them create connections among ideas. • Suppose a rectangular prism has a volume of 40 cm3 and height of 5 cm. • What else can you tell me about the rectangular prism? TDG Leadership Seminar 2009 24

  25. Thinking Differently about Familiar Ideas TDG Leadership Seminar 2009 25 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. Changing Views TDG Leadership Seminar 2009 26 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. 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? TDG Leadership Seminar 2009 27

  28. Interference TDG Leadership Seminar 2009 28 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. Variables TDG Leadership Seminar 2009 29 How did these problems expand your understanding of variable? Why is it important for learners to understand what a variable is?

  30. Numbers with a Simple Relationship TDG Leadership Seminar 2009 30 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. Numbers with a Not-so-simple Relationship TDG Leadership Seminar 2009 31 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. 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? TDG Leadership Seminar 2009 32

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

  34. A Different Kind of “Adult” Problem TDG Leadership Seminar 2009 34 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. Identifying Difficulties TDG Leadership Seminar 2009 35 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. 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? TDG Leadership Seminar 2009 36

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

  38. Graphing Speed TDG Leadership Seminar 2009 38 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. Graphing Speed TDG Leadership Seminar 2009 39 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. 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? TDG Leadership Seminar 2009 40

  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 TDG Leadership Seminar 2009 41

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