The goals of mathematics education are frequently stated in terms of mathematics content (being able to factor polynomials, being able to solve quadratic equations, being able to use and apply the Pythagorean Theorem). However, as Cuoco, Goldenberg and Mark argue, it is much more productive to focus on “habits of mind” instead (“Habits of Mind: An Organizing Principle for Mathematics Curricula”, Journal of Mathematical Behavior 15, 375–402, 1996.) Some mathematical habits of mind include the predilection to sniff out patterns, counterexamples and generalizations, to conjecture, to guess, to estimate. I appreciate that the forthcoming Common Core Standards for Mathematics proposed by the National Governors Association begins by describing eight “Standards for Mathematical Practice” that differentiate novice and expert mathematical thinkers.
If I could start this year over again, I would start the school year with some more creative activities instead of diving straight into mathematical content. I would try to encourage students to get in the habit of looking for patterns, in particular. It would be so wonderful, for example, if when students see a bunch of problems like “(x-5)(x+5)=x2-25″ and “(4a-1)(4a+1)=16a2-1″, they would begin to wonder if there is something special that happens when expanding (a-b)(a+b). It’s not too late to work on these habits of mind now, but I should have focused on them more from the beginning. Oh well.
What kinds of creative activities? For example, the other day I heard some music by Tom Johnson (“V” from “Rational Melodies”, in particular) that would be great for getting students to develop a natural curiosity for pattern. And at the National Conference of Teachers of Mathematics Annual Meeting a few weeks ago I saw some great geometry problems for developing students’ reasoning abilities. Here is a GeoGebra applet for one of those geometry problems.