So… yes, communicating tacit knowledge is hard, and best done one-on-one through apprenticeship. (As it happens I’ve written a lot about that in other contexts here and there on the web!)
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Replying to @Meaningness @ProfJayDaigle
But the fact that so many people find the explanations of Rota, Thurston, and Tao massively helpful suggests that it’s possible and worth attempting. Those guys are all exceptional but they can’t be the only ones capable of communicating SOME of it.
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Replying to @Meaningness @ProfJayDaigle
I did a graduate seminar with Rota and mostly he just taught stuff straight, but the rare times he’d take five minutes to go meta and explain principles and how to think about things were SO valuable. All obvious in retrospect and not esoteric. Would be easy to put in writing.
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Replying to @Meaningness @ProfJayDaigle
“My cognitive tools work well in algebra, and much less well in analysis”— Just saying THIS might be hugely helpful at about the sophomore level. I figured out then, on my own, that my roommate could integrate by seeing/feeling the surface and the shape under it, and bam.
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Replying to @Meaningness @ProfJayDaigle
I couldn’t do that; I had to integrate syntactically. So I’m also much better with algebra. However, I did eventually develop visualization abilities (for abstract algebra and such). In retrospect, I’m pretty sure I could have learned to visualize surfaces much more.
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Replying to @Meaningness @ProfJayDaigle
If someone had said “These are fundamentally different ways of doing math, and some people seem to be naturally better at one or the other, but if you work at it you can manage both, and here’s how to develop your visualization ability” that might have been revelatory.
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Replying to @Meaningness @ProfJayDaigle
There’s stuff that applies only to broader or narrower subdomains of math, but the more general stuff doesn’t seem to get taught either.
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Replying to @Meaningness @ProfJayDaigle
Didn’t know until recently that many Math teachers & faculty are aware of big debates on pedagogical methods in mathematics. Called “The Math Wars” by some, the central contested point is how much Math ed should be simple Darwinianian struggle or something otherwise.
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Replying to @krowney @ProfJayDaigle
Does this controversy extend to the university level, or is it only up to that point?
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Replying to @Meaningness @ProfJayDaigle
High school, college, and grad school. It’s not everywhere at all levels, but it’s definitely active. Most intense, AFAICT in college math instruction. Many faculty apparently think it’s ok since that’s the way they made it.
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Wow, interesting, thanks!
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