The problem is that the teachers themselves often do not have this "factored" perspective.
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Replying to @StephenPiment @St_Rev and
Yes… that became my suspicion in retrospect… MIT math professors didn’t really understand the subjects they were teaching.
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Replying to @Meaningness @StephenPiment and
I don't think that's quite it. The professors understand the material, but have trouble conveying those understandings because they're so internalized. It's easy to explain individual facts but hard to convey the implicit framework that ties them together.
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Replying to @ProfJayDaigle @Meaningness and
Turning implicit knowledge into something explicit and articulable is difficult under the best of circumstances, and many professors don't even think of it as something that needs to be done. Much easier to work through problems in the book and get students to pass the test.
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Replying to @ProfJayDaigle @StephenPiment and
That makes sense! Maybe another way of saying this is that the kind/degree of understanding required to teach material is greater than that required to use it fluently?
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Replying to @Meaningness @StephenPiment and
And it seems that the kind/degree of understanding required to write a textbook with a novel structuring of the material is greater still. (I’m attempting this now…)
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Replying to @Meaningness @StephenPiment and
The hardest part probably is the structure. When I think about linear algebra, I have a family of densely interconnected concepts to play with; but when I teach I have to explain one thing at a time. So I can't explain how I really think about it until the last week of class.
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Replying to @ProfJayDaigle @Meaningness and
stat mech was like that for me- felt like i was learning a bunch of random things until i studied for the final and realized everything was the same constrained optimization problem
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Replying to @The_Lagrangian @StephenPiment and
In a parallel life, I led a heroic reform movement that fixed STEM education and thereby saved the universe. In this life, I’m too old, so it’s your job
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Replying to @Meaningness @The_Lagrangian and
To rain on this parade, I think people are making a universalist mistake -- assuming there's a correct central insight that could be communicated!
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Think of a subject as a maze of twisty little passages, all alike. There's no shortcut that lets you skip months or years of wandering around blind and lost, before you get a sense of the total structure.
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.