Monday, January 10, 2005

Math Pedagogy

If you've gone through American public education like I have, you will probably agree that math pedagogy in the United States is a total mess and in serious need of repair. Here's a study that confirms that sentiment (via Isabel). The study, instead of using teacher self-evaluations, uses a random sample of videotaped lessons in the U.S., Germany, and Japan. The authors seem to think the most interesting differences are between the U.S. and Japan.

In Japan, for example, kids are doing proofs in 8th grade and this is a natural extension of the way they learn mathematics all through the previous grades. I think that the primary flaw in American math pedagogy is the lack of emphasis on intuitive understanding. Students are not taught *why* some particular method works or how to derive some law, just that these things are true and they must be memorized. The study reports that in Japan the class is conducted by posing a "rich problem" to the class, taking suggestions on how to solve it, and guiding the class to a solution, making explicit along the way the thought process involved. Because of this, students learn to look intuitively at the structures underlying methods and laws, so when they start to do proofs it is perfectly natural.

But it seems like this, like so many other things, is culturally influenced. The culture in Japan is that you care about your own education. At home parents act, not as drill-masters, but as sympathizers, trying to get kids to watch T.V. or take a break. The students are the ones who motivate their education. In the U.S. classrooms, 31% of lessons were interrupted by public address, in Japan none were. In fact Japanese professors who watched the tapes were shocked that the classes were interrupted at all. In U.S. classrooms, use of blackboard and overhead projector were about equal. In Japan, only the blackboard was used. Notes were left on the blackboard for the whole class period, not just for a moment and then erased. In the U.S. these "little" things are only seen in college classrooms, not secondary or middle school.

The study also lends evidence against a couple panaceas of education which I have always thought were fundamentally silly. First, classes are much larger in Japan, on average. The whole push for "smaller classes" in the U.S. is misguided. Perhaps it would be better if teachers spent more individual time with students, but to do this in a significant manner would require huge amounts of spending and I don't think it would be very helpful anyway. After all, the teachers still don't know what they are doing. Also, the Japanese classrooms were not homogeneous. They did not split up advanced and "regular" students. Everyone was taught the same material at the same level (high). This is something I think has been destructive in the United States. Falling back into classes that are easy and unchallenging make you lose any interest at all in the subject. Calculus should be a requirement for graduation.