Paul Lockhart in a 2002 piece writes that mathematics is an art, like music and painting, but we don’t teach it that way. Mostly, he says, this is because math teachers are not mathematicians, are not themselves doing, or have ever done, mathematics.
So what, we might ask, is going on in math class? It would seem that at best the kids are learning some math vocabulary and some math syntax, but for the most part they will never go on to actually acquire the language of mathematics, and, in just a few years following school leaving, will have forgotten even the rudiments of the language they were forced fed while in school.
According to Lockhart what should be going on in math class? Well, mathematics, but what does he mean by that? Probably most of us wouldn’t be able to answer the question except by using the very terms we “learned” in school, —algebra, geometry, and for the few of us who made it that far the calculus.
Lockhart gives us a couple of examples of what he means.
With my students, “I might imagine,” he says, “a triangle inside a rectangular box, and I might wonder how much of the box the triangle takes up. Two-thirds maybe?
The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing, amusing yourself with your imagination.”
Or, another example, he takes the case of a triangle inside a semicircle. “The beautiful truth about this pattern is that no matter where on the circle you place the tip of the triangle, it always forms a nice right angle.
“Here is a case where our intuition is somewhat in doubt. It’s not at all clear that this should be true; it even seems unlikely— shouldn’t the angle change if I move the tip? What we have here is a fantastic math problem! Is it true? If so, why is it true? What a great project! What a terrific opportunity to exercise one’s ingenuity and imagination!”
One agrees that these problem-questions are beautiful, require imagination, if not magic for their solution. And we agree that students, everyone, should be exposed to these types of problems while in school, and out, and helped and encouraged to come up with their own solutions, never told that there is just one solution that should be memorized for testing purposes.
But are such as these, delightful as they are, even possible in the school environment? Perhaps that’s why there is so little music and art in the schools. For weren’t all artistic imagination and invention left behind, not allowed through the school doors? Perhaps classrooms are no more the proper environment for mathematics than they are for the fine arts.
Yet if real mathematical experiences, along with art and music, are not appropriate for the schools what learning experiences are? What does lend itself to interesting, and productive, and joyful learning in the classroom? Alas, probably not much.
And it’s probably true that schools could much better serve their students, and society at large, if they were much less preoccupied with what students should be learning, in order, say, to go on to college and eventually get a good job, and instead were much more taken up by creating lively and happy and exciting, and yes learning, but even more living and alive communities of, by, and for children.
Lockhart is right to say that what goes on in the math class is not mathematics, but he is wrong to think that mathematics any more than any other real and important and substantial and joyful human activity can prosper in the solid, mostly brick and concrete structures we call schools.