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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
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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.”
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Analytic geometryorfractal geometry?
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Analytic geometryorfractal geometry? Modeling with algebra ormodeling with spreadsheets?
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Analytic geometryorfractal geometry? Modeling with algebra ormodeling with spreadsheets? Graph theoryorsolid geometry?
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Analytic geometryorfractal geometry? Modeling with algebra ormodeling with spreadsheets? Graph theoryorsolid geometry? These are the wrong questions.
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Traditional courses mechanisms for communicating established results
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Traditional courses mechanisms for communicating established results give students a bag of facts
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Traditional courses mechanisms for communicating established results give students a bag of facts reform means replacing one set of established results by another
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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
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New view of curriculum Results of mathematics research Methods used to create mathematics
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New view of curriculum Results of mathematics research Methods used to create mathematics
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New view of curriculum Results of mathematics research Methods used to create mathematics
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Goals Train large numbers of university mathematicians.
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Goals Train large numbers of university mathematicians.
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Goals Help students learn and adopt some of the ways that mathematicians think about problems.
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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.
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Curricula should encourage false starts
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Curricula should encourage false starts experiments
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Curricula should encourage false starts experiments calculations
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Curricula should encourage false starts experiments calculations special cases
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Curricula should encourage false starts experiments calculations special cases using lemmas
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Curricula should encourage false starts experiments calculations special cases using lemmas looking for logical connections
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A caveat Students think about mathematics the way mathematicians do.
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A caveat Students think about mathematics the way mathematicians do. NOT Students think about the same topics that mathematicians do.
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What students say about their mathematics courses It’s about triangles. It’s about solving equations. It’s about doing percent.
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What we want students to say about mathematics It’s about ways of solving problems.
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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!)
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Students should be pattern sniffers.
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Students should be experimenters.
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Students should be describers.
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Give precise directions
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Invent notation
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Argue
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Students should be tinkerers.
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“What is 3 5i ?”
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Students should be inventors.
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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?
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Students should look for isomorphic structures.
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Students should be visualizers.
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How many windows are in your house?
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Categories of visualization reasoning about subsets of 2D or 3D space visualizing data visualizing relationships visualizing processes reasoning by continuity visualizing calculations
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Students should be conjecturers.
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Inspi to inspi :side :angle :increment forward :side right :angle inspi :side (:angle + :increment) :increment end
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Inspi
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Two incorrect conjectures If :angle + :increment = 6, there are two pods
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Two incorrect conjectures If :angle + :increment = 6, there are two pods If :increment = 1, there are two pods
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Students should be guessers.
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“Guess x.”
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Mathematicians talk big and think small.
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“You can’t map three dimensions into two with a matrix unless things get scrunched.”
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Mathematicians talk small and think big.
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Ever notice that a sum of two squares times a sum of two squares is also a sum of two squares?
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Mathematicians talk small and think big. This can be explained with Gaussian integers.
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Mathematicians mix deduction and experiment. Experimental evidence is not enough.
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Are there integer solutions to this equation?
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Yes, but you probably won’t find them experimentally.
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Mathematicians mix deduction and experiment. Experimental evidence is not enough. The proof of a statement suggests new theorems.
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Mathematicians mix deduction and experiment. Experimental evidence is not enough. The proof of a statement suggests new theorems. Proof is what sets mathematics apart from other disciplines. In a sense, it is the mathematical habit of mind.
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