Habits of Mind: An organizing principle for mathematics curriculum and instruction

1. Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu

2. • Position paper by Al Cuoco, E. Paul Goldenberg, and June Mark. • Published in the Journal of Mathematical Behavior, volume 15, (1996). • Full text available online at http://www.sciencedirect.com Search for the title “Habits of Mind: An Organizing Principle for Mathematics Curricula.”

6. These are the wrong questions.

7. Traditional courses • mechanisms for communicating established results

8. Traditional courses • mechanisms for communicating established results • give students a bag of facts

9. Traditional courses • mechanisms for communicating established results • give students a bag of facts • reform means replacing one set of established results by another

10. Traditional courses • mechanisms for communicating established results • give students a bag of facts • reform means replacing one set of established results by another • learn properties, apply the properties, move on

11. New view of curriculum

12. New view of curriculum

13. New view of curriculum

14. Goals • Train large numbers of university mathematicians.

15. Goals • Train large numbers of university mathematicians.

16. Goals • Help students learn and adopt some of the ways that mathematicians think about problems.

17. Goals • Help students learn and adopt some of the ways that mathematicians think about problems. • Let students in on the process of creating, inventing, conjecturing, and experimenting.

18. Curricula should encourage • false starts

19. Curricula should encourage • false starts • experiments

20. Curricula should encourage • false starts • experiments • calculations

21. Curricula should encourage • false starts • experiments • calculations • special cases

22. Curricula should encourage • false starts • experiments • calculations • special cases • using lemmas

23. Curricula should encourage • false starts • experiments • calculations • special cases • using lemmas • looking for logical connections

24. A caveat

25. A caveat

26. What students say about their mathematics courses • It’s about triangles. • It’s about solving equations. • It’s about doing percent.

27. What we want students to say about mathematics It’s about ways of solving problems.

28. What we want students to say about mathematics It’s about ways of solving problems. (Not: It’s about the five steps for solving a problem!)

29. Students should be pattern sniffers.

31. Students should be experimenters.

35. Students should be describers.

36. Give precise directions

37. Give precise directions

38. Invent notation

39. Invent notation

40. Invent notation

41. Argue

42. Argue

43. Argue

44. Students should be tinkerers.

45. Students should be tinkerers. “What is 35i?”

46. Students should be inventors.

47. Students should be inventors. Alice offers to sell Bob her iPod for $100. Bob offers $50. Alice comes down to $75, to which Bob offers $62.50. They continue haggling like this. How much will Bob pay for the iPod?

48. Students should look for isomorphic structures.

50. Students should be visualizers.