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. ...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.
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.
Students can learn to be...
- pattern sniffers
- 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.
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.