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Math Break |
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Copyright © 2001 by Dave Badtke |
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In The Mystery of the Aleph, Amir Aczel reminds us that Pythagoras probably coined the term mathematics, which in Greek means “that which is learned.” The Pythagoreans attributed mystical significance to numbers. Their holiest number was ten, tetractys, not because they counted digits as the French still do—80 is quatre-vingt, four-twenty—but because 10=1+2+3+4 was the generator of geometric dimensions: 1 is a single point with dimension zero; 2 points determine a line, dimension one; 3 points determine a plane, dimension two; and 4 points determine a tetrahedron in three dimensional space. (An infinite number of planes intersect a single line, but as soon as you pick one point off the line, you’ve defined a single plane. And then, using the three points that define the plane and one additional point off the plane, you can construct a four-sided pyramid, a tetrahedron.) That the Greeks made the leap from the obvious to the meta-obvious at a time when most everyone else was counting fingers and toes was truly amazing. Even today, more than 2000 years later, most of us fall short of an understanding of mathematics some Greeks had, a realization that became painfully obvious to me 30 years ago when, as a new teacher at Kenwood High School near the University of Chicago, I was assigned five Algebra I classes—five identical classes! Though the school was supposed to be one of Chicago’s best, teaching a course that most children dislike made me wonder if we had progressed at all since the Greeks. By the end of the day, I was going absolutely bonkers, answering the same questions about the same problems over and over again. Reading Aczel’s book, published last year, reminded me of this time, for if it hadn’t been for Pythagoras and infinity, I’m sure I would have lost my mind, explaining again and again why associative, distributive and commutative properties of numbers are important. Having just completed a set theory course at the University, I decided that we’d start each class with a short discussion of an interesting (to me) mathematical topic. Some days we discussed Platonic solids, the gems of Pythagoras’ group, and I’d show them stellations I had constructed from tag board. Or I might propose that we think about Bertrand Russell’s Barber of Seville Paradox: If a barber shaves all those who don’t shave themselves, does the barber shave himself? The kids would roll their eyes and groan. I’d explain that this paradox went to the heart of a fundamental problem in mathematics. More groans and blank stares. I’d explain that if the barber does shave himself, then he can’t; but that if he doesn’t, he must. Telling them variants of the paradox, waving my hands, pacing in front of the class, going to the board to diagram sets and relationships, I would have a wonderful time as the paradoxes tied my mind in knots. I’d tell them that Russell sent the paradox to the German logician Frege, who had just completed his book on set theory: Frege would be so perplexed by the paradox that he would struggle another two years to complete his book. Invariably, on each day in each class, someone would finally raise his hand and grumble: Why do we have to learn this stupid algebra anyway? I would stare at the student, wishing I could strangle him, sigh, and return again, for the thousandth time, to an explanation of how “+” and “-” were operators and regions on the number line. But there was still one topic that was sure to capture their attention, that would make them truly want to understand mathematics. I didn’t really understand algebra until I needed it in calculus, and calculus didn’t become clear until I studied physics. Exposure to mysterious mathematics, I reasoned, would forever bind my students to numbers. I knew that history teachers used murder and mayhem to keep their students awake, that biology students would sit straighter at the merest mention of sex, and that mathematics did indeed have a salacious underbelly—infinity. I began talking about countable and uncountable numbers, rational, irrational and transfinite numbers, the alephs in Aczel’s title. I introduced them to Georg Cantor, the inventor of set theory, who would eventually go insane trying to determine the validity of the continuum hypothesis, which relates countable to uncountable infinities. I told them of Kurt Gödel’s proof that systems are inherently incomplete, that there are some things that cannot be proved to be either true or false. Finally, up a hand would shoot. I would anticipate an insightful, penetrating question now that my students had been exposed to the wonders of mathematics. Do we have homework for tomorrow? the student would ask. If you were that student, perhaps you’ll find Aczel’s book a better teacher than I was. |
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- Dave Badtke can be contacted at: www.CarquinezReview.com; Dave@Badtke.com; PO Box 763, Benicia, CA 94510; or by calling 707-745-5540.
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