Re: [math-fun] How we "do mathematics"
Dan, I'M AFRAID I didn't express myself clearly.I wasn't asking for a solution but rather about the mechanics of finding it. Did either of you use pen or pencil or paper at any point? I had to draw two pictures before I could see what the equations had to be. Maybe you were able to do all these things in your head in which case my speculations are off base. [SPOILER]
David writes:
<< It's often said that among the sciences, supporting mathematics is a great bargain because unlike other sciences it does not depend on using advanced and expensive equipment. This is true. of course, but I claim mathematics is nevertheless technology dependent on a a quite marvelous invention called a pencil (or perhaps a pen, or whatever Euclid and Pythagoras used). Try to imagine how one could do mathematics if there were no such devices. Their usefulness is as a means of storage. If we couldn't write things down we'd have to keep everything in our heads at the same time which is hard. Example. An extra credit problem in my high school algebra book went something like this.
Two racers run around a circular track at constant rates. If they run in opposite directions (one clockwise, the other counter clockwise) they meet every minute. If they run in the same direction they meet every hour. Find the ratio of their speeds.
If you don't already know the problem maybe some of you can do it in your heads but if you can't, save the paper you used and see how many symbols you wrote down? How many pictures you drew?
Let's see:
1 / f-s = K 1 / f+s = 1 f-s = 1/K f+s = 1 f/s = 2f / 2s = 1+1/K / 1-1/K = K+1 / K-1 = 61/59
which makes 61 symbols. (This seems a lot more than ought to be necessary.)
--Dan
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On 8/6/07, gale@math.berkeley.edu <gale@math.berkeley.edu> wrote:
Dan, I'M AFRAID I didn't express myself clearly.I wasn't asking for a solution but rather about the mechanics of finding it. Did either of you use pen or pencil or paper at any point? I had to draw two pictures before I could see what the equations had to be. Maybe you were able to do all these things in your head in which case my speculations are off base.
I think those of us like me, with lots of math contest experience and high school teaching experience, do this kind of stuff in our heads all the time. But I think that's a result of having gone through the steps on paper so many times for problems similar to this one! --Joshua Zucker
On Aug 6, 2007, at 7:57 AM, gale@Math.Berkeley.EDU wrote:
I'M AFRAID I didn't express myself clearly.I wasn't asking for a solution but rather about the mechanics of finding it. Did either of you use pen or pencil or paper at any point? I had to draw two pictures before I could see what the equations had to be. Maybe you were able to do all these things in your head in which case my speculations are off base.
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. In this case: it's the kind of thing that throws people because it sounds like there's not enough information. What is it that supports the idea that when they're going opposite directions, the they meet at a frequency determined by the sum of their speeds, and in going in the same direction, the difference of their speeds? There's also the more commonplace issue of getting straight the reciprocal relation between speed and frequency. 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. 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. Bill
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
David Gale wrote:
I'M AFRAID I didn't express myself clearly.I wasn't asking for a solution but rather about the mechanics of finding it. Did either of you use pen or pencil or paper at any point? I had to draw two pictures before I could see what the equations had to be. Maybe you were able to do all these things in your head in which case my speculations are off base.
I thought Dan *was* describing the steps he went through to solve the problem on paper. I did it in my head, but I'm quite certain that if I'd happened to have a pencil in my hand then I'd have written down precisely the formulae I gave. -- g
participants (4)
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Bill Thurston -
gale＠Math.Berkeley.EDU -
Gareth McCaughan -
Joshua Zucker