So if you want to communicate how to do, say, homological algebra, the audience of people who are interested is pretty small, maybe a couple hundred. And you'll meet nearly all of them at conferences. So it's not as useful to find ways to communicate that knowledge at scale.
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Replying to @ProfJayDaigle
Thank you very much for your reply. Your points are all excellent and appreciated and I basically accept all of them. I can't help wanting to push back partially against some.
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Replying to @Meaningness @ProfJayDaigle
By way of context, I think I may have had an unusually unhelpful math education (MIT SB math & PhD theoretical CS with a lot of graduate-level math courses).
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Replying to @Meaningness @ProfJayDaigle
Partly my fault, for various reasons, but I do think in retrospect the department’s professors were great mathematicians but in many/most cases terrible educators. From memory, but also I’ve recently watched a bunch of their lectures on the MIT MOOC, and they are objectively bad.
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Replying to @Meaningness @ProfJayDaigle
I may be overgeneralizing from my personal experience. I wish I had had you as a mentor instead! I’m confident that your students’ experience is very different. That said, whenever I gripe about this, many people say “yes that’s how it is” and nearly never “not my experience!”
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Replying to @Meaningness @ProfJayDaigle
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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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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