Meet Ana, a child who weaves baskets. From a young age, Ana had watched her family weave, and slowly she learned to make baskets herself. Eventually, she was taught how to make baskets in several different styles. There are distinct patterns to learn, and she has learned them to the point where she can quickly and efficiently make a variety of designs.
If you ask Ana to explain how to make baskets, she could explain to you how to do it, step by step. First you raise this strand, then put it under, and so on and so on.
If you ask her to explain why this works? Well, that’s a very abstract question. Her knowledge is essentially practical, and it’s worth as much respect as some other mathematician’s theoretical knowledge. There are, after all, multiple ways of knowing, each worthy of respect.
But later, Ana goes to school. Her teacher announces on the first day of class that it’s not enough to solve a problem — you have to be able to explain why you’re doing something. In fact, what he really cares about is the why — that’s what mathematics is about, anyway. I mean, anybody can compute stuff. A computer can compute stuff! If a machine can do it, heck, a machine should do it.
But can’t a machine weave a basket? Does it count as math if you can’t explain why? Does practical math count?
There are two ideas out there in math education-land, and I wonder how they work together:
Idea #1: A lot of kids just get taught how to do stuff without understanding it. That’s not really mathematics.
Idea #2: A lot of activities — like basket weaving, hair braiding, making change — are mathematics, even if they don’t look like school math.
So the only thing valuable about school mathematics is understanding stuff, but outside of school the standards seem different.
Maybe those two ideas are really attempts at expressing two educational instincts. There’s the instinct to respect what a person already knows, and there’s the instinct to expand their horizons. We’re trying to find ways of putting those essentially social instincts into language, but it’s awkward
I feel the tension between those two instincts whenever I’m listening and talking to a student. (Not just with students, though.) People want to be understood — teachers need to understand. But teachers want to help expand what people know, and people want to learn. Those instincts are also desires.
So as instincts or emotions, I understand what’s going on. But as ideas, I’m not exactly sure how to put the pieces together. Why shouldn’t you treat pure computation as beautiful, practical mathematics?
(This post was partly inspired by Two Cultures of Greek Mathematics, a really cool article that I might blog about, especially if somebody wants me to.)