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.)
2 thoughts on “The instinct to respect, the instinct to expand”
Go for it. (blogging about the article that is)
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With regard to the idea that when kids are following procedures they don’t understand, they aren’t doing math, I think at least sometimes, that complaint is made because kids end up doing things that really aren’t math. For example, one of the objections to teaching kids “tricks” like cross-products is that then they go and apply that in situations where there aren’t two equivalent ratios, e.g. when asked to multiply fractions. Something analogous to trying to make a basket by following the steps, but ending up with something that’s not a basket. But it’s worse, because if you’re a kid and you make a basket that’s not a basket, you can tell right away. But if your page of fraction multiplication problems is returned covered in red ink, because you used the wrong procedure…
If I had to frame in social language, I’d say both students and teachers desire…competence. Kids want to be competent (and acknowledged as such) in all kinds of activities, in school and out of school, and teachers find it rewarding when we see kids do things that they couldn’t do before.
As math teacher, my specific focus is on helping kids develop mathematically–in other words, to develop competencies they probably wouldn’t achieve if they didn’t take math at school. And as a teacher my first go-to move for developing competence is usually direct teaching, followed by some kind of assessment and feedback. But I’ve recently benefited from reading (and rereading) Jenna’s posts on *assigning* competence and *presuming* competence:
…maybe what she writes about gets at the same tension?
Last unsolicited thought–teaching arithmetic is way underrated…it should get more respect. 🙂