I read Kenny's essay about teaching mathematics so that it becomes obvious: not just how to arrive at the correct answer, but how to genuinely recognize the answer as obviously true. He wants his students to know why communitivity of multiplication is true. I like what Kenny's doing. But it also got me thinking about a different why question . . .
I had a fantastic math teacher at the NC School of Science and Mathematics, Dr. Steve Davis. People warned me before my first class -- "Oh, he has all these stories he tells that have nothing whatsoever to do with math. Especially golf." But then I got to his class, and listened to his stories, and realized all those people were wrong. All his stories were about life, and in his world, math was all about life.
The class he taught was called "Introduction to College Mathematics" or "ICM". (I take credit for coining the popular pronunciation of the acronym: "Ick-um".) The structure of the class was something experimental and new; in fact, the school was being given grant money by Digital Equipment to develop the class, to find a genuinely useful way to incorporate computers into teaching math. The class was replacing pre-calculus, so it had to cover the same sorts of pre-calculus material but also cover a lot of other ground, too.
The first day of class, the very first thing Dr. Davis said was: "Why have you been studying math all these years?" His question was met with stunned silence. Of all the questions we ever had encountered in a math class, this was not one of them. Nobody had the least idea of what to say. After thirty seconds of silence he got impatient and slammed his fist on his lectern. "C'mon, people, WHY? Why have you been studying math? You've been doing it for the past ten years, at least. Why? And don't tell me, 'Because it's good for you.' You are not five-year-olds being told to eat their vegetables."
This, also, was new. We weren't used to teachers yelling at us to answer questions -- at least, not on the first day, and questions we had never been taught the answer to. We were smart kids -- we were not used to not knowing the right answer, and perhaps too embarrassed to admit that we didn't.
"You mean, you actually let someone force you to do something for ten years, and you never asked why you were doing it? You actually let someone waste that much of your time?" His tone was mocking.
Another thirty seconds of silence. "I'm sorry," he said, tossing his chalk into blackboard tray, "But I can't go on. I can't teach this class until you answer that question for me."
"Well, it's useful," one girl timidly offered.
"Yes. WHY is it useful? Useful for what?"
"Well, there's money."
"You mean, making money? I can tell you from personal experience, mathematicians don't make much money." (smile)
"No, doing calculations on money."
"Good. Money's important. But we could have stopped at third grade if all you needed to do was add up your loose change. Why did you keep studying math?"
"Well . . . You have to know math to do engineering and physics."
"Are you going to be an engineer?"
"Ummm . . . no."
"I suspect that most of you will not become engineers or physicists. Most of you will never use calculus in your professional lives. Why do you study mathematics?"
And so it went, in a Socratic dialog, for the whole 45 minute class. I don't remember everything that was said, but I remember where we wound up: mathematics was a tool for modeling the world. It was "the queen and servant of the sciences" (quoting E.T. Bell), the most generalized way we understand the universe, and also the most universally applicable way of making predictions about how something will behave. That set us up for the material we would cover for the rest of the year: geometric probability, the graphing of functions, an understanding of limits, interpolation and fitting of functions to data . . . all of which naturally led us into calculus. More importantly, though, was the way that lecture made me feel about mathematics. What previously had seemed abstract, cold and useless now felt immediate, powerful, and alive. I felt like someone was showing me the secrets of the universe.
I also must confess: he was correct. I became a molecular biologist, but I never used calculus again. What I did do was a lot of algebra and a lot of fitting curves to data . . . the very skills we developed in that class.
I think of all this, after reading Kenny's essay, because Dr. D was trying to give us the other "why". We had spent years learning how to manipulate symbols, and never fully appreciated the value of the tool that we had. Every "word problem" that had been given to us was bizarrely arbitrary: we never knew, exactly, why it was important to know how many apples Johnny had. It was just more symbols, pushed around for no reason. Dr. Davis gave us the reason. No, better yet: he made us find the reason.
At the bottom of my "Math and the Obvious" essay, I have a link to something called "Lockhart's Lament." I would strongly recommend you check it out. Now, it's actually 25 pages long, but you don't have to read all that (unless, like me, you end up totally fascinated)...just read the first couple of pages, the musician's nightmare, and you'll get the idea. Lockhart takes Dr. Davis's idea absolutely seriously. He argues that we are doing more harm than good, and if we can't fix our math education, we should just scrap it.
In a wonderful follow-up to the Lament, Lockhart wrote: "Suppose the devil were to offer you this deal: your child will get a perfect score on the English section of the SAT, but will never again read a book for pleasure. I would like to believe that no parent would make that deal. But how many would gladly shake the devil’s other hand? Math is not something we want our children to enjoy, it is something we want them to get through."