Finding a good question

January 9, 2009

I just met with my advisor, Sandor Kovacs. Meeting with him is great, and I walk out excited to get to work. No exception today, but my work feels particularly slippery at the moment.

The reason is that I don’t yet have a good question. I’m narrowing in on it, but it keeps eluding me. There’s a truism in math that finding a good question is half the work. I estimate that locating mine is going to take me pretty much this entire academic year.

I’ll be teaching a course this summer called Turtles All the Way Down, in which I hope to teach, among other things, the art of asking questions. Finding your question, in or out of mathematics, can inspire your creative work for years. Yet we devote so little time to learning how to do it well.

The fun thing about math is that the classic questions which begin math usually feel like they should be easy. Doubling the square is a good example. If I give you a length and tell you to double it, it’s really easy to do it (just take two of those lengths and put them together). Likewise, if I give you a square (imagine a one inch by one inch square if you need a specific example), and tell you to double that—by which I mean double the area it take up—it seems like it should be just as easy: just double the sides of the square. But then you have a two inch by two inch square, and that’s four times as big as a one inch by one inch square (draw it!). So how do you actually get the square that’s twice as big?

A seemingly innocuous question. The answer isn’t terribly hard, but it’s far from obvious, and requires some insight. Once you see it, though, there’s that aha! feeling. (Plato brings it up in the Meno). It also points in a very interesting direction. In fact, if we follow this question as fully and honestly as we can, we end up travelling the path that leads to modern mathematics. My old colleague Paul Lockhart once remarked, after a very bright second grader he knew cried in frustration when he was trying to work out an implication of this problem, that these were the tears where mathematics begin.

Off to work!

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