Friday, June 4, 2010

Mathematical Habits of Mind

Avery (Without Geometry, Life is Pointless) pointed out a very interesting article, Habits of Mind: An Organizing Principle for Mathematics Curriculum* (pdf here). The authors ask "how do we decide what mathematics to teach?" and then suggest that's the wrong question to ask.
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. ...

There is another way to think about it, and it involves turning the priorities around. Much more important than specific mathematical results are the habits of mind used by the people who create those results. ... The goal is to allow high school students to become comfortable with ill-posed and fuzzy problems, to see the benefits of systematizing and abstraction, and to look for and develop new ways of describing situations.
 They identify lots of mathematical habits of mind, and give good examples to explain them. Some seem pretty similar to me, so I combined them here.

Students can learn to be...
  • pattern sniffers
  • experimenters/tinkerers
  • describers
  • visualizers
  • inventors
  • conjecturers/guessers

Mathematicians ...
  • talk big and think small,   (Trying to understand a new idea? Start with a simple example.)
  • talk small and think big, (Start with a simple example, and build a big web of mathematical structure.)
  • use functions,
  • use multiple points of view,
  • mix deduction and experiment,
  • push the language,
  • use intellectual chants.
They go on to list habits of mind that geometers and algebraists use. I was comfortable with most of their examples in the first half of the article, but often got lost in the second half.

In Avery's post on the new Common Core Math Standards, he noticed two things missing from the standards:  pattern sniffing and problem posing. (A good book on that last is The Art of Problem Posing, by Stephen Brown.)

If the standards were about habits of mind, instead of particular content, maybe I could get over my anti-standards attitude. As I said at Avery's blog,  I'd love to turn the ideas from this article into suggestions for standards; I think that would help us focus our lessons in good directions.

*I believe the proper word there is curricula, the plural.

1 comment:

  1. Thanks Sue. I think it will take a long time for these ideas to hit mainstream, but in the meantime, I suspect the more adventurous charter schools with good teachers will benefit a lot and be the impetus to change the current system (I hope).

    I think the habits you mentioned are useful even/especially as general life philosophy. :)


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