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Monday, October 20, 2008

Buried Treasure

In Out of the Labyrinth: Setting Mathematics Free
, a book I have turned to a dozen times, I found this passage about "teaching by indirection" that informs all of my lesson planning. I don't remember ever seeing it before, though I suspect I must have:
Means to an end seem to be more readily absorbed than the end itself...So if you want your students to learn some topic A, find another topic B, which A entails - and have them work on B. You'll see that A is effortlessly taken in stride (although B itself may escape).

Let's say for some reason or other you thought it important for your young students to learn their times tables. Set them instead to puzzling over which of the first 100 or so integers are prime. This is always a heady pursuit for our pattern-hungry minds, and in the course of it such "facts" as that 51 is (disappointingly) 17 x 3 are hit on -- and stick. Once they dope out how the times tables are made ("once you know 3 x 4 your get 4 x 3 for free"), a very few significant encounters tamp down the actual values. You could also have had them figure out how to add fractions: finding a common denominator is tense and rewarding, and puts times tables in their right (ancillary) place. Or if you want them to master deductive reasoning, put in front of them some intriguing topic in Euclid. If the subject is calculus, embody it in a particular physical problem. It takes some thought to find a good B for each A - fortunately no system of facile entailments is ready at hand.

(Emphasis mine.) It's that "fortunately" that undoes me. That the Kaplans expect me to know my subject with such intimacy that I can set my kids on a path where they will necessarily overcome the thing I want them to learn anyway, without even being aware of it. And I can't look it up which B goes with my A. And I shouldn't even try to compile a list. And the idea we can hand the uninitiated a script, or a binder, and expect real mathematics to be learned, would be laughable if it weren't so sad and terrifying.