From: Michelle Manes
Subject: Habits of Mind: Paper Summary
Organization: Education Development Center, Inc.
Date: Fri, 14 Apr 1995 20:29:04 GMT

Thinking about the future is risky business. Past experience tells us that today's first graders will graduate from high school most likely facing problems that do not yet exist. Given the uncertain needs of the next generation of high school graduates, how do we decide what mathematics to teach? Should it be graph theory or solid geometry? Analytic geometry or fractal geometry? Modeling with algebra or modeling with spreadsheets? These are the wrong questions, and designing the new curriculum around answers to them is a bad idea.

For generations, high school students have studied something in school that has been called mathematics, but which has very little to do with the way mathematics is created or applied outside of school. One reason for this has been a view of curriculum in which mathematics courses are seen as mechanisms for communicating established results and methods -- for preparing students for life after school by giving them a bag of facts ... Given this view of mathematics, curriculum reform simply means replacing one set of established results by another one (perhaps newer or more fashionable) ...

There is another way to think about it, and it involves turning the priorities around. Much more important than the specific mathematical results are the habits of mind used by the people who create those results, and we envision a curriculum that elevates the methods by which mathematics is created, the techniques used by researchers, to a status equal to that enjoyed by the results of that research. The goal is not to train large numbers of high school students to be university mathematicians, but rather to allow high school students to become comfortable with ill-posed and fuzzy problems, to see the benefit of systematizing and abstraction, and to look for and develop new ways of describing situations. While it is necessary to infuse courses and curricula with modern content, what's even more important is to give students the tools they'll need to use, understand, and even make mathematics that doesn't yet exist.

Mathematical Habits of Mind:

• Pattern sniffing
• Experimenting
• Describing
• Tinkering
  

  (Tinkering always needs more explanation. If you take a situation apart and put it back together, what happens if the pieces are put back slightly differently? For example, if you experiment with rotating a figure, followed by translating the figure, a natural next question is what happens if you translate and then rotate? Since we know every integer is the product of primes, we should wonder if every integer can be expressed as a sum of primes. If so, is it a unique summation, as the factorization is unique?)

• Investigating
• Visualizing
• Conjecturing
• Guessing

Mathematical Approaches to Things:

• Mathematicians talk big and think small.
  (Talk in generalizations, but relate these to familiar, specific situations).

• Mathematicians talk small and think big.
  (Seeing simple problems as special cases of deeper mathematical theories. Or developing deeper mathematical theories to work on seemingly simple problems, e.g. Fermat's Last Theorem.)

• Mathematicians use functions.
  (Algorithms, dependences, and mappings.)

• Mathematicians use multiple points of view.
  (Multiple approaches to the complex number system through algebra, analysis, arithmetic, and geometry.)

• Mathematicians mix deduction and experiment.
  (Mathematicians experiment with problems, but never stop there. They are always working towards proof. The experiment, if you're lucky, may point in a good direction for a proof, and the proof might bring up more questions that can be the bases of future experiments.)

• Mathematicians push the language.
  (Example: Definitions for zero and negative exponents come from wanting the rules for positive integral exponents to hold in other cases.)

• Mathematicians use intellectual chants.
  (When engrossed in a problem, a mathematician may alternately stare into space and scribble on paper. The staring has to do with a mental search for logical connections.)

Geometric Approaches to Things:

• Geometers use proportional reasoning.
  (Self explanatory?)

• Geometers use several languages at once.
  (Complex numbers, analytic geometry, turtle geometry, ...)

• Geometers use one language for everything.
  (The language of points, lines, planes, angles, surfaces, areas, and volumes, gets applied to seemingly non-geometric phenomena.)

• Geometers love systems.
  (Finite geometries, systematizing so that special cases are combined into one large result, etc.)

• Geometers worry about things that change.
  (Reasoning by continuity, as shapes are deformed, what happens?)

• Geometers worry about things that don't change.
  (Looking for geometric and numeric invariants in all kinds of situations.)

• Geometers love shapes.
  (Classification, analysis, representation, ... Check out Majorie Senechal's chapter in On the Shoulders of Giants.)

Why Habits of Mind?

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There is also a section on algebraic approaches to things, but I'm already running horribly late... I'd be happy to answer questions and engage in discussion based on this paper. Thanks for asking about it.

michelle
--
Michelle (All Typos are Mine) Manes
Education Development Center, Inc.
michelle@edc.org