Summary of a position paper from the Connected Geometry project, to be published by the Journal of Mathematical Behavior.
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.
(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?)
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 (All Typos are Mine) Manes
Education Development Center, Inc.
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