Here is an interesting perspective given to me by Keith Devlin via the MAA via SMF quaker lady. I agree with most, but not all of it. In particular, he does a great job of capturing my disdain for the two-columns proofs from high school geometry. Here are a couple of excerpts.
"What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species."
And more:
"Instead of a witty and enjoyable argument written by an actual human being, and conducted in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic form-letter of a proof. And what a mountain being made of a molehill! Do we really want to suggest that a straightforward observation like this requires such an extensive preamble? Be honest: did you actually even read it? Of course not. Who would want to? \\The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious, by insisting that it be “rigorously proved” (as if the above even constitutes a legitimate formal proof) is to say to a student, “Your feelings and ideas are suspect. You need to think and speak our way.”
Finally:
"Could anything be more unattractive and inelegant? Could any argument be more obfuscatory and unreadable? This isn’t mathematics! A proof should be an epiphany from the Gods, not a coded message from the Pentagon. This is what comes from a misplaced sense of logical rigor: ugliness. The spirit of the argument has been buried under a heap of confusing
formalism.
No mathematician works this way. No mathematician has ever worked this way. This is a complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping simple things simple."
As a mathematician, I wholeheartedly back up the statement that no mathematician would ever write like this. The theme of the entire paper is "there is no actual mathematics being taught in K-12; rather we are teaching a pseudo-mathematics that is only tangentially related to the subject of mathematics." The two-column proof is a good example of how we muck up the actual mathematics.
UNC won. Sibley won. It was a good day for family alma maters.
posted by Dirk Awesome @ 10:49 PM
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