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 @StephenPiment and
That’s really interesting and makes sense… I want to say “well, can’t you go top-down and explain the big picture first?”, but I can offer convincing objections to that approach…
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Replying to @Meaningness @StephenPiment and
Certainly you can try. For some subjects that works really well. In others it's more awkward. Linear algebra in particular is so interdependent that it's still hard to explain what you're doing at any one point.
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Replying to @ProfJayDaigle @Meaningness and
In fact I might argue that the entire first linear algebra course is an attempt to give the big picture, and then you get to start doing actual work in the last week or so. Everything before eigenvectors is just establishing language so you can start doing math.
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That is an interesting meta-systematic point in itself! “We are mostly just doing definitions here, the theorems come later” might be useful for students to hear…
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