I presented the problem [ from the TIMSS videotape of a math lesson in Japan (pdf file) ] to the class, saying I would like their feedback on whether such problem is appropriate for eighth graders. After my initial presentation of the problem I told them I would give them three minutes to work on it, but not to feel they had to solve it—I just wanted to reconvene at that time and then discuss it as a class. (This is in fact what they did in the Japanese classroom). All fell silent and worked at their desks. (Note to adherents of people-working-in-small-groups: In our class, when we are given a problem to solve, most of us like to solve it in isolation. When instructed to work in groups, one person in the group generally dominates. My mind becomes paralyzed and I crave being left to my own devices.)

After about a minute, I saw that people were perplexed, not getting anywhere, and I suddenly realized that in my excitement: I forgot to present the theorem they would need to solve the problem. I apologized and called for their attention and explained the key theorem they would need.

Now, I fully expected that no one would solve the problem in the three minutes and I would have to be “guide on the side” and coach them to see how to apply the theorem, thus proving to all who believe in constructivism that students can still “discover” when given information directly. I forgot that my classmates all have a math or science background and are not eighth graders. Three of my classmates solved it within a minute and others were on their way. Nevertheless, my oversight in not presenting the theorem did reveal something important: As smart and experienced as my classmates are, no one was having any great insights into a solution until I presented the theorem.

I led a discussion about the appropriateness of the problem for eighth graders. The people who solved the problem immediately thought that perhaps I should not give the theorem and let them “discover” it. Others who had a tougher time with the problem said, well, if you did that, maybe you should coach them to come up with the theorem rather than expecting them to do it on their own. Or maybe giving them the theorem wasn’t such a bad thing.

I suspect that the ones who had the easiest time were under the illusion that the theorem was superfluous and easily discovered. They forgot that a few minutes prior they were struggling until I told them what they needed to know. Just like people who in their memory believe they discovered all that was important in math. In short, anyone who was a constructivist at the beginning of the evening, was still a constructivist at the end of it.