JB: How do you teach humanistic math?

HERSH: I'm going to sidestep that slightly, I'll tell you my conception of good math teaching. How this connects with the philosophy may be more tenuous.

The essential thing is interaction, communication. Only in math do you have this typical figure who was supposedly exemplified by Norbert Wiener. He walks into the classroom, doesn't look at the class, starts writing on the board, keeps writing until the hour is over and then departs, still without looking at the class.

A good math teacher starts with examples. He first asks the question and then gives the answer, instead of giving the answer without mentioning what the question was. He is alert to the body language and eye movements of the class. If they start rolling their eyes or leaning back, he will stop his proof or his calculation and force them somehow to respond, even to say "I don't get it." No math class is totally bad if the students are speaking up. And no math lecture is really good, no matter how beautiful, if it lets the audience become simply passive. Some of this applies to any kind of teaching, but math unfortunately is conducive to bad teaching.

It's so strange. Mathematical theorems may really be very useful. But nobody knows it. The teacher doesn't mention it, the students don't know it. All they know is it's part of the course. That's inhuman, isn't it?

Here's an anecdote. I teach a class, which I invented myself, called Problem Solving for High School and Junior High School Teachers and Future Teachers. The idea is to get them into problem solving, having fun at it, feeling confident at it, in the hope that when they become teachers they will impart some of that to their class The students had assignments; they were supposed to work on something and then come talk about it in class. One day I called for volunteers. No volunteers. I waited. Waited. Then, feeling very brave, I went to the back of the room and sat down and said nothing. For a while. And another while. Then a student went to the blackboard, and then another one.

It turned out to be a very good class. The key was that I was willing to shut up. The easy thing, which I had done hundreds of times, would have been to say, "Okay, I'll show it to you." That's perhaps the biggest difficulty for most, nearly all, teachers-not to talk so much. Be quiet. Don't think the world's coming to an end if there's silence for two or three minutes.

JB: Earlier you mentioned the word beauty. What's with beauty?

HERSH: Fortunately, I have an answer to that. My friend, Gian-Carlo Rota, dealt with that issue in his new book, "Indiscrete Thoughts." He said the desire to say "How beautiful!" is associated with an insight. When something unclear or confusing suddenly fits together, that's beautiful. Maybe there are other situations that you would say are beautiful besides that, but I felt when I read that that he really had something. Because we talk about beauty all the time without being clear what we mean by it; it's purely subjective. But Rota came very close to it. Order out of confusion; simplicity out of complexity; understanding out of misunderstanding; that's mathematical beauty.