If you just memorized the proofs, but cannot actually recreate them, it means that you did not actually learn it, and the perfect score on the exam doesn’t matter. The point of learning mathematics is to be able to transfer this skill into new domains, not to just regurgitate it.
Not necessarily. Remembering all of the multiple representations of the beta function, for example, is probably aided through the use of flash cards. You can still use such representations without necessarily having to go through and derive them from scratch, whilst still understanding what the beta function is. Ditto for the many trig identities.
Similarly, there are often underlying assumptions that can be tricky to remember in the moment, e.g. certain log laws only holding for the absolute value of the argument. There's a combination of both understanding a tool to begin with, and remembering various equivalences, representations, and underlying assumptions that makes math difficult.
Part of memorizing proofs is also just increasing your exposure to certain ideas, because maths is one of those subjects where there's no real substitute for time spent thinking about something (i.e. mathematical maturity).
>If you just memorized the proofs, but cannot actually recreate them, it means that you did not actually learn it, and the perfect score on the exam doesn’t matter.
I wasn't tested on the exact proofs and problems I memorized. I was tested on variations and novel combinations of them I haven't seen before. That sure sounds like learning to me.
Further you can't just memorize a proof straight through, there isn't enough space in your brain for that - rather the act of mentally walking through the theorem over and over via an Anki flashcard prompt will eventually just... Change your logic, invisibly, to be correct. Which, again, sounds a lot like learning to me.
>The point of learning mathematics is to be able to transfer this skill into new domains, not to just regurgitate it.
I am far more confident in both my intuition and conscious reasoning around e.g. Abelian groups or the enumerative combinatorics applications of group actions than whatever I learned in real analysis, where I studied in the "usual" way. Indeed going back to learn Haskell a few years after that AA course was much easier than earlier attempts because I had a considerably stronger background in what kinds of things to look for in that domain.
But more importantly homework problems are rigged [1] and transfer learning is close to non-existent in every domain we've seriously looked at [2], so this is awfully close to moving the goalposts on what "really learning" something is by setting an unreasonably high bar to start with. Math certainly can transfer to new domains, but I would never call that "the point" of math, and that's also a totally different endeavor to be performed in addition to learning the math itself.
> But more importantly homework problems are rigged [1] and transfer learning is close to non-existent in every domain we've seriously looked at [2],
Indeed it is, but that’s because most people just learn to pass exam by redoing the same exact problems with different inputs! Bringing up this fact does not support your argument in favor of memorization, instead it is closer to my view, which is that most of schooling is just cargo culting education, people just memorize the exam problems, pass, move on and forget. What’s the point of the whole thing in the first place if you can’t transfer?
What I'm actually getting at there is your standards are inhumane. Transfer is pretty poor across the board in pedagogical studies and we don't know how to reliably get more of it. Indeed it's a tough thing to even rigorously define, since it's basically creativity finetuned on crystallized intelligence. You might get more of it out of people by massively upping the difficulty and number of homework problems. That's a huge cost to put people through, especially when a significant proportion of them really do just want to study the thing for its own sake, and couldn't care less about something as nebulous as "transfer". I don't need my knowledge of the Kan extension to have to inform how I play tennis.
