## 03 January 2005

### Math lessons, here and there

I was blown away by this document: How Culture Shapes Math Instruction in Japan, Germany, and the United States. (1997.)

Researchers videotaped a few hundred real 8th-grade math lessons in three countries. The results are fascinating:

• “We measured how many times the lesson was interrupted by someone coming into the classroom or an announcement coming over the public address system. This happened during 31% of the American lessons, 13% of the German lessons and none of the Japanese lessons.”
• American classes spent 62% of their time on homework (either going over the previous day's homework or starting on the next assignment). This number was 40% for Germany and 10% for Japan.
• In America, 8th-graders learn pre-algebra arithmetic. In Germany and Japan they learn algebra and geometry.
• “Fifty-four percent of the Japanese lessons included proofs. None of the American lessons included proofs.”

And on and on. But I'll stop here, because this last topic is of particular interest. A proof is simply a step-by-step, defensible logical explanation of why a particular mathematical statement is true. All of higher mathematics is entirely about proofs. Isn't it a little weird that we never teach this to schoolkids?

As I mentioned in a previous post, the first proof I can remember encountering showed that the area of a circle is πr2. I found it in my third-grade textbook. (Here is a rough version of this proof online.) I was fascinated. The teacher never tried to use it in class.

Why do you think American and Japanese classes end up being so different? The document answers that question, too. There are two apparent reasons. First: Japanese teachers care about and labor incessantly to increase lesson quality. They intentionally teach large classes (40 students or more) so that they'll have time to critique and improve their lessons, which they do constantly. In the U.S., individual teachers make their own lesson plans or get them out of textbooks. Classes are small. Teachers spend most of their day teaching classes and, typically, none of it looking back. Second: the goal of a Japanese lesson is to teach the kids math concepts and how to think about math. In the U.S., the goal is to teach the kids how to solve problems. I think this translates to “how to solve the homework problems”, all of which are similar for any given lesson and can be solved without much thought. (If you read the paper, you'll get a much clearer idea of how profound all these differences are.)

The paper argues that the underlying cause of these differences is cultural, which might be true. Or it might be economic or political.

(Found on Isabel's math blog.)