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Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu • Position paper by Al Cuoco, E. Paul Goldenberg, and June Mark.
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Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu
• 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.”
Traditional courses • mechanisms for communicating established results
Traditional courses • mechanisms for communicating established results • give students a bag of facts
Traditional courses • mechanisms for communicating established results • give students a bag of facts • reform means replacing one set of established results by another
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
Goals • Train large numbers of university mathematicians.
Goals • Train large numbers of university mathematicians.
Goals • Help students learn and adopt some of the ways that mathematicians think about problems.
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.
Curricula should encourage • false starts
Curricula should encourage • false starts • experiments
Curricula should encourage • false starts • experiments • calculations
Curricula should encourage • false starts • experiments • calculations • special cases
Curricula should encourage • false starts • experiments • calculations • special cases • using lemmas
Curricula should encourage • false starts • experiments • calculations • special cases • using lemmas • looking for logical connections
What students say about their mathematics courses • It’s about triangles. • It’s about solving equations. • It’s about doing percent.
What we want students to say about mathematics It’s about ways of solving problems.
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!)
Students should be tinkerers. “What is 35i?”
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?