What if at the end of class, they wanted it to be longer, and kept the teacher in the hallway answering questions?
What if they learned that coupling their imaginations to their powers of reasoning would give them a tool of awesome power for exploring the cosmos?
What if an 11 year old got so excited by his insights that he yelled out
OH WOW! I get this now!
What if all this happened in math class?
Suppose you wanted to learn to play the piano, and, at the first lesson, all you got to do was repeatedly tap middle C, and that, when you asked the teacher what was going on, she said that, for the first few months, you would learn one note a week, then you'd spend 10 years or so playing scales, and, after that, maybe a song?
Would you go back? And, if someone MADE you go back, would you learn to love music? If, five years into this drudgery, someone told you that playing the piano gave them intense joy and satisfaction, and was a tremendous outlet for their creativity and spontaneity, would you believe them? Or would you think they were crazy?
Suppose you went to a different teacher, and, at the first class, you didn't even get to touch a piano, but simply to watch video of the fingers of great pianists. And suppose the curriculum called for doing this for a decade or two before ever sitting down and playing. Would you learn to play?
It's a Tuesday evening in Boston. The five year olds are figuring out how to find the area of a circle (one of them is doing this sitting on her mother's lap and occasionally sucking her thumb). The 7 year olds are exploring different bases. The nine year olds are doing group theory. The big kids (11 year olds) are proving the Bolyai-Gerwien theorem (if two polygons have the same area, one can be cut up with a finite number of straight cuts and reassembled to form the other). No one is doing any drills, no one is getting bored, and no one is getting put down for wrong answers or bad guesses. This is math class, but Bob and Ellen are teaching. And it's not like anything I've seen before.
Who teaches area to kids who don't know how to multiply?
Who thinks you can teach bases to 2d graders?
Who thinks that kids who successfully prove the Bolyai-Gerwien theorem should go on to one of the Hilbert ProblemsWho thinks 11 year olds can be guided through great theorems in math?
The same sort of people who know you can sing before you learn a scale (or even what middle C is). The sort of people who know that math isn't multiplication and division, or even differentiation and integration, but one of the most beautiful and interesting creations of the human mind. The sort of people who know that once you turn a kid on, you had better get out of the way, because they move fast.
A lot of people hate math; and almost no one does math just for fun. After a hard day's work, relax by trying to prove a theorem in a new way; or play around with Goldbach's conjecture? No. Not likely. And, if you're one of the few people who do that sort of thing, you probably keep quiet about it, lest your friends think you mad. If you play the piano for fun, you can tell your friends.....they may envy you (or not), or admire you (or not), but they won't likely think you crazy. Not even if they don't like music. If you tell them you think Bach's music is beautiful, they make not agree, but they won't look at you oddly and shake their heads. And that's true, even if they know you will never give a recital, much less play Carnegie Hall.
If you play in an after-work basketball league, no one expects you to be Michael Jordan. And if you're proud of yourself for playing better than usual, no one says "why do that? That's for professionals!"
Why this difference? And what can we do about it? Bob and Ellen Kaplan have some of the answers. And those answers are about more than math. They're about reasoning, about learning, about joy, and, in a real sense, about being human. They call their group the Math Circle
They take any kid who comes, but not everyone who comes loves math at the start. Some come because they have a friend taking the class - and, so, Bob and Ellen get to light fires. (Talking with them, I think they'd almost prefer to take ONLY kids who think they hate math). While there is self-selection in the kids they teach in Boston, they've done similar classes in public schools and in other countries, and people are now using similar methods in prisons, and will soon be using them in Cameroon.
Bob and Ellen didn't originate the idea of a math circle. Similar things have been done for a long time in Russia, and, in the United States, similar things were done by Robert Lee Moore, about 100 years ago. But Moore was a combative, competitive type, and his aggressiveness turned a lot of people off. The Kaplans marry some of Moore's ideas to an utter lack of competitiveness and a liberal sensibility about kids. But the essential idea in the Moore method, and in the Kaplans', is that people learn by doing, whether it's piano or math, and that this sort of learning can bring joy to people, even if they will never play Carnegie Hall or win a Fields Medal.
They've been doing this now for 14 years; this year, they have more than 100 students, and, now, in the summers, they train others to follow their methods. They've detailed how they do it, and why they do it, in a book: Out of the labyrinth: Setting mathematics free, and the Oxford University Press has its own blog, which links to the book here
Some of what they do is, of course, dependent upon their personality. But some of it, quite a bit of it, is transferable. Bob and Ellen have had success training people to do what they do; they've trained people who are already teachers in math, so that they know enough to teach this way, and they've trained graduate students in math in the ways kids think, so that they can teach this way. Many people who love math want to share that love, and spread the word that math is not a boring, dry subject - some of those people can learn to do this.
Some of you may be saying that the idea of playing scales for ten years is a straw man. Of course no one insists that a musician practice scales for ten years! That's precisely the point. We DO insist on that, or its equivalent in math. What do musicians do? They play music. What do composers do? They write music. What do mathematicians do? They prove theorems. They play math. But many students of math don't get to do any of this until college. They may see a proof, especially in geometry. That's like watching Vladimir Horowitz play piano. It's not bad, but no one ever learned to play the piano by watching. You learn to play the piano by playing - and, at first, you play badly. When a 5 year old guesses (as one did while I was watching) that 9 x 9 is 25, because 25 is 'big', he is doing what a first year piano student does when he butchers a basic piece. Both will get better by practicing, especially if that practicing is guided by a teacher.
How can we change math education?
Well, when that 9 x 9 problem came up in class, one of the five year olds said
LET'S FIGURE IT OUT!
Published by Peter Flom
I am a statistician, working with a wide variety of clients, mostly researchers in psychology, education, medicine, social sciences and other fields. I also have given talks and written articles on learning... View profile
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