Habits of mind an organizing principle for mathematics curriculum and instruction
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Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University [email protected] • 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 l.jpg

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

Michelle Manes

Department of Mathematics

Brown University

[email protected]


Slide2 l.jpg

• 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 l.jpg
Traditional courses Mark.

  • mechanisms for communicating established results


Traditional courses8 l.jpg
Traditional courses Mark.

  • mechanisms for communicating established results

  • give students a bag of facts


Traditional courses9 l.jpg
Traditional courses Mark.

  • 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 Mark.

  • 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 l.jpg
Goals Mark.

  • Train large numbers of university mathematicians.


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Goals Mark.

  • Train large numbers of university mathematicians.


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Goals Mark.

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


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Goals Mark.

  • 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 Mark.

  • false starts


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Curricula should encourage Mark.

  • false starts

  • experiments


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Curricula should encourage Mark.

  • false starts

  • experiments

  • calculations


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Curricula should encourage Mark.

  • false starts

  • experiments

  • calculations

  • special cases


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Curricula should encourage Mark.

  • false starts

  • experiments

  • calculations

  • special cases

  • using lemmas


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Curricula should encourage Mark.

  • false starts

  • experiments

  • calculations

  • special cases

  • using lemmas

  • looking for logical connections


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A caveat Mark.


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A caveat Mark.


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What students say about their mathematics courses Mark.

  • It’s about triangles.

  • It’s about solving equations.

  • It’s about doing percent.


What we want students to say about mathematics l.jpg
What we want students to say about mathematics Mark.

It’s about ways of solving problems.


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What we want students to say about mathematics Mark.

It’s about ways of solving problems.

(Not: It’s about the five steps for

solving a problem!)










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Argue Mark.


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Argue Mark.


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Argue Mark.



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Students should be tinkerers. Mark.

“What is 35i?”



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Students should be inventors. Mark.

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 be visualizers. Mark.

How many windows are in your house?


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Categories of visualization Mark.

  • reasoning about subsets of 2D or 3D space

  • visualizing data

  • visualizing relationships

  • visualizing processes

  • reasoning by continuity

  • visualizing calculations



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Inspi Mark.

to inspi :side :angle :increment

forward :side

right :angle

inspi :side (:angle + :increment) :increment

end


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Inspi Mark.


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Inspi Mark.


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Inspi Mark.


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Two incorrect conjectures Mark.

  • If :angle + :increment = 6, there are two pods


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Two incorrect conjectures Mark.

  • If :angle + :increment = 6, there are two pods

  • If :increment = 1, there are two pods



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Students should be guessers. Mark.

“Guess x.”



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Mathematicians talk big and think small. Mark.

“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. Mark.

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. Mark.

This can be explained with Gaussian integers.


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Mathematicians mix deduction and experiment. Mark.

  • Experimental evidence is not enough.



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Are there integer solutions to this equation? Mark.

Yes, but you probably won’t find them experimentally.


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Mathematicians mix deduction and experiment. Mark.

  • Experimental evidence is not enough.

  • The proof of a statement suggests new theorems.


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Mathematicians mix deduction and experiment. Mark.

  • 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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