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I was at Borders the other day looking through a workbook, and I remembered my brief flirtation with math back in the day -- I was excited by it, and thought it had real meaning, that was universally important. I still think this is reasonable -- the universe works according to rules that are expressed in this language. However, I thought it was meaningful/important in and of itself, like a philosophical system that said something about the universe beyond the laws of physics it described. I think I might have been a bit of a pythagorean phanatic. I wanted to major in it. Maybe I should have stuck with it, but the prospect of a college major that required some work was daunting. But then I sort of woke up from my trance, and decided that I was just young and overly-excited by a reality that actually functioned in a predictable way. Anyway, I was reading this workbook, and I was reminded of all of these algebraic tricks and shortcuts, and also reminded that I never really understood how any of them worked. Every time I saw a certain structure, like converting x^-1 to x^1/2 (i think -- won't I be embarassed if that's not even right). I didn't really understand why this worked, but I knew that it did, so every time I saw this structure and it was useful to change it to its counterpart, I did. And of course the examples get more complicated; that's one that's probably visible with arithemtic examples, and isn't itself a very good example of "a trick". But if you "see" it once (via arithmetic example, for instance), are you allowed to just go through the motions every other time? The concept of the limit was visible, and had some philosophical importance to me (just like some of the triangle stuff in trig did); it was a way of "looking at infinity": 1 + 1/2 + 1/3 + 1/4 + 1/5 + etc. Is the limit of this 2? I think so. I forget, and I don't know how to work it out algebraically anymore; I enjoyed my math-play more than 10 years ago. I think this became more interesting and I had a bit of a "deeper" understanding of limits only due to the providence of my graphing calculator -- I could see that line curving into a straight one, and how it never reached the axis of a coordinate plane, but only got infinitely close without ever getting there. Neat. But I doubt it would have ignited that spark of interest without a graphing calculator (or a sheet of graph paper, I guess, if I were in the 1970s). Anyway, I suppose my question is: what constitutes a "deep" understanding of mathematics, such that you're really tapping in to something larger, vs. A series of symbolic tricks and puzzles that happen to correspond with events in physics, engineering, etc, such that it constitutes nothing more than a tool we can pull out and use, without really understanding how it works? Is math no more significant than a crossword or sudoku puzzle? do people confuse a perhaps philosophical "significance" to mathematics with the satisfaction that comes with a meaningless symbolic puzzle working out the way it's supposed to? Me mum used to tell me a story: She was always bad at math. But then, it came time for some kind of state exam that had some determination in a student's future. A teacher of hers gave her a book, and said "just learn how to do the problems in here". She did -- she just memorized what to do to a symbolic construct if it appeared. And with that memorization came an extremely high score on the state exam. All of her teachers were flabbergasted. "how could someone as bad at math as you score so high?" I think it was a matter of my mother seeking the kind of "higher" understanding of math that she got with english, poetry, art, philosophy, etc. Something more romantic than "if you see this, do this. If you see that, do that." Problem is, in the case of math, a romantic, philosophical, "deep" interpretation/understanding is indistinguishable, from a practical standpoint, from the "if you see this, do that" approach. Both people score high on the test. The person with the "shallow", dull, human-trick understanding might even score higher. Or maybe she just wanted a good grade in math, and got flustered by the teacher's explanations. But once the patterns were there on the paper and could be imitated, there was no problem. Getting back to shallow vs. Deep understanding: it's possible that all of these algebraic rudiments (and even the ones in calculus -- you see d/dx and you do something specific to that symbol) are intended to be memorized in a shallow way; just absorbed without meaning, like the alphabet. Then, when you get up to, say, real analysis, you start applying these bits of language you learned to creative expression (proofs, I guess, the structure of which isn't really strict, I don't think). But ultimately, math ends up as symbols on a paper; even in real or complex analysis or topology or abstract algebra or whatever else there is out there, you end up expressing it on paper with symbols, and the question is still "Am I really understanding this, or am I just manipulating symbols?". -- Matthew Teigen |