The woes of the unit circle — a rant by my 15 year old daughterpic.twitter.com/y7UhTxiVp5
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Replying to @stevenstrogatz
She's perfectly correct, of course. But the key question is: why is the usual textbook presentation of K-12 mathematics so inept? We've had thousands of years to arrive at optimal approaches. And we have, and they're known. So why aren't they in common (or even universal) use?
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Replying to @MathPrinceps @stevenstrogatz
Publishers want textbooks to be acceptable to the majority of buyers, without being objectionable to anyone, even people who might be wrong. Things get watered down. Authors are encouraged to make the safest choices. Change is difficult. And yet, we fight...
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Replying to @mathgrrl @stevenstrogatz
But the earliest methods of presentation were often the best. The Babylonian approach to quadratic equations is superior to the modern one. Euler's textbook on elementary algebra is vastly superior to any now in use. We have regressed.
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And make no mistake about it: regression is change. We seem quite capable of changing for the worse; why can't we change for the better? Why can't we even return to superior pedagogical practices of past times? Ptolemy's approach to trigonometry remains hard to improve on.
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Replying to @MathPrinceps @stevenstrogatz
Sounds like you should write that book! I am sure many would appreciate it. The best authors have a fire in their belly
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Replying to @mathgrrl @stevenstrogatz
A very near approach to pedagogical optimality may be made simply by cribbing shamelessly from the great past masters (who, pretty much by definition, knew exactly what they were doing.) If it were up to me, I'd probably mostly just try to update Euler's notation and language.
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It is amazing how many ostensibly modern math textbook tropes can be found in Euler. Here's a random example from Euler's Elements of Algebra:
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Replying to @BanjoTragedy @MathPrinceps and
"Twenty persons, men and women, dine at a tavern; the share of the reckoning for one man is 8 shillings, for one woman 7 shillings, and the whole reckoning amounts to 7l. 5s. Required the number of men and women separately?"
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I am particularly impressed by Euler's INTRODUCTIO, which Andre Weil quite correctly maintained is (far) superior to any modern text on the same material; from it I learned (embarrassingly recently) things about partial fractions that I ought to have learned as a tenth-grader.
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Polya was quite correct about Euler: he preferred to enlighten, rather than dazzle, his readers. Not for nothing did Laplace urge: "Read Euler, read Euler, he is the master of us all." But Lagrange is perhaps even better: he's more systematic, more sympathetic to the beginner.
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