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
• mmanes_at_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 geometry or fractal geometry?


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Analytic geometry or fractal geometry?
Modeling with algebra or modeling with spreadsheets?

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Analytic geometry or fractal geometry?
Modeling with algebra or modeling with spreadsheets?
Graph theory or solid geometry?
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Analytic geometry or fractal geometry?
Modeling with algebra or modeling with spreadsheets?
Graph theory or solid 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.
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Students think about the same topics that mathematicians do.
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What students say about their mathematics courses

• Its about triangles.
• Its about solving equations.
• Its about doing percent.

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What we want students to say about mathematics
Its about ways of solving problems.
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What we want students to say about mathematics
Its about ways of solving problems.
(Not Its 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.
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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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

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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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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.
You cant 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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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?

Yes, but you probably wont 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.