On Monday 06 August 2007, Bill Thurston wrote:
I think the important part of solving a problem like this comes way before the algebra steps that Dan and Gareth describe. It's the part about going from a disoriented and muddled state to a notion of what to do, or even, what to sketch. This is hard for people like us to dig into, since so much about problems of this sort has become internalized.
Right. (Though I feel pretty presumptuous accepting the designation "people like us" when one of the other "us" is as eminent as Bill...)
In my case, I worked it out in my head. I had a mental picture of the two people running in opposite directions around a circle, and then I thought of using moving coordinates as in high school physics class so that one of the runners moved at the sum of speeds, which I said to myself was one lap per minute. The same picture said the difference of speeds was 1 lap per hour. Then I thought of a picture of an interval of length 60, with two points (the two speeds) averaging at the halfway mark but being 1/60th apart: from the picture you see the ratio as 61:59. I have an aversion to writing things like this down in symbolic form, because when I do they become denatured in my head and it's hard for me to keep focused on the whole picture. Actually, I tend to get distracted going back and forth between algebra and the actual situation, and I tend to make algebra errors.
Interesting. If I'd taken so geometric an approach then I'd have been worried lest I confuse 61:59 with 60.5:59.5 or something of the sort. So I have to do the algebra because it *stops* me making dumb errors.
But: I've gone through so many math problems that this doesn't reveal much about the important part of solving the problem --- you'd need to go back to algebra students.
Depends what you're after. I think it's interesting to compare the different ways people "naturally" leap to -- cf. the story about von Neumann and the problem about the fly. (Of course it's not really "natural" at all; it's the result of a long history of doing mathematics.) -- g