Your list is missing a very important thing: experience. After all that learning, you need to go out and actually do stuff. And in the process of doing you discover many things which your previous education skipped or didn’t pay enough attention to.
but I think it rarely needs to be explicitly pursued
Depends on the need, doesn’t it?
It’s easy to underestimate that need, too. Recognizing you don’t understand a particular theorem is not hard, but realizing you need “fuzzy” background context is not trivial at all.
But it seems to me that a good way to get “fuzzy” background context is to bang your head against a variety of concrete problems (rather than explicitly being fuzzy). Example: you don’t know how to approach a math problem, so you solve a lot simpler/related/example problems and it gives you a better “feel” and intuition about the objects involved. Then you come back to the original problem. I’ve seen this pattern a lot.
Can you give some examples of the opposite being true?
But it seems to me that a good way to get “fuzzy” background context is to bang your head against a variety of concrete problems (rather than explicitly being fuzzy).
I agree. There is no need to be fuzzy “explicitly”. My point is that a lot of important learning will be excluded by the requirement for specific one-sentence summaries. For example, banging your head against a bunch of concrete problems :-) One-sentence summary: “My head hurts” :-D
I am not sure what do you mean by “opposite”—going from a more complex problem to simpler ones?
It’s OK, since you agree on the “not being fuzzy explicitly” point, I don’t have anything more to say about it.
Don’t treat this “one sentence summary” thing too strictly—it’s kind of a reflex/shorthand that is useful in many situations, but not all. I like it because it’s simple enough that it’s installable as a subconscious reaction.
Your list is missing a very important thing: experience. After all that learning, you need to go out and actually do stuff. And in the process of doing you discover many things which your previous education skipped or didn’t pay enough attention to.
Depends on the need, doesn’t it?
It’s easy to underestimate that need, too. Recognizing you don’t understand a particular theorem is not hard, but realizing you need “fuzzy” background context is not trivial at all.
But it seems to me that a good way to get “fuzzy” background context is to bang your head against a variety of concrete problems (rather than explicitly being fuzzy). Example: you don’t know how to approach a math problem, so you solve a lot simpler/related/example problems and it gives you a better “feel” and intuition about the objects involved. Then you come back to the original problem. I’ve seen this pattern a lot.
Can you give some examples of the opposite being true?
I agree. There is no need to be fuzzy “explicitly”. My point is that a lot of important learning will be excluded by the requirement for specific one-sentence summaries. For example, banging your head against a bunch of concrete problems :-) One-sentence summary: “My head hurts” :-D
I am not sure what do you mean by “opposite”—going from a more complex problem to simpler ones?
It’s OK, since you agree on the “not being fuzzy explicitly” point, I don’t have anything more to say about it.
Don’t treat this “one sentence summary” thing too strictly—it’s kind of a reflex/shorthand that is useful in many situations, but not all. I like it because it’s simple enough that it’s installable as a subconscious reaction.