Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes...
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![Page 1: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/1.jpg)
Habits of Mind: An organizing principle
for mathematics curriculum and instruction
Michelle Manes
Department of Mathematics
Brown University
![Page 2: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/2.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.comSearch for the title “Habits of Mind: An Organizing Principle for Mathematics Curricula.”
![Page 3: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/3.jpg)
Analytic geometry or fractal geometry?
![Page 4: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/4.jpg)
Analytic geometry or fractal geometry?
Modeling with algebra
or modeling with spreadsheets?
![Page 5: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/5.jpg)
Analytic geometry or fractal geometry?
Modeling with algebra
or modeling with spreadsheets?
Graph theory or solid geometry?
![Page 6: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/6.jpg)
Analytic geometry or fractal geometry?
Modeling with algebra
or modeling with spreadsheets?
Graph theory or solid geometry?
These are the wrong questions.
![Page 7: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/7.jpg)
Traditional courses
• mechanisms for communicating established results
![Page 8: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/8.jpg)
Traditional courses
• mechanisms for communicating established results
• give students a bag of facts
![Page 9: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/9.jpg)
Traditional courses
• mechanisms for communicating established results
• give students a bag of facts• reform means replacing one set of established results by another
![Page 10: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/10.jpg)
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 curriculumResults of mathematics research
Methods used to create mathematics
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New view of curriculumResults of mathematics research
Methods used to create mathematics
![Page 13: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/13.jpg)
New view of curriculumResults of mathematics research
Methods used to create mathematics
![Page 14: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/14.jpg)
Goals
• Train large numbers of university mathematicians.
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Goals
• Train large numbers of university mathematicians.
![Page 16: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/16.jpg)
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
![Page 21: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/21.jpg)
Curricula should encourage
• false starts• experiments• calculations• special cases
![Page 22: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/22.jpg)
Curricula should encourage
• false starts• experiments• calculations• special cases• using lemmas
![Page 23: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/23.jpg)
Curricula should encourage
• false starts• experiments• calculations• special cases• using lemmas• looking for logical connections
![Page 24: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/24.jpg)
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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Give precise directions
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Invent notation
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Invent notation
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Invent notation
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Argue
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Argue
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Argue
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Students should be tinkerers.
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Students should be tinkerers.
“What is 35i?”
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Students should be inventors.
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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.
![Page 51: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/51.jpg)
Students should be visualizers.
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
![Page 53: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/53.jpg)
Students should be conjecturers.
![Page 54: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/54.jpg)
Inspi
to inspi :side :angle :incrementforward :sideright :angleinspi :side (:angle + :increment) :incrementend
![Page 55: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/55.jpg)
Inspi
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Inspi
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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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Students should be guessers.
“Guess x.”
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Mathematicians talk big and think small.
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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.
![Page 65: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/65.jpg)
Mathematicians talk small and think big.
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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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.
![Page 71: Habits of Mind: An organizing principle for mathematics curriculum and instruction Michelle Manes Department of Mathematics Brown University mmanes@math.brown.edu.](https://reader030.fdocuments.in/reader030/viewer/2022032522/56649d6a5503460f94a4871c/html5/thumbnails/71.jpg)
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.