[analog31]

Regarding memorization and busywork, maybe a certain amount of it is valuable. I don't know if it's a great analogy or not, but memorization and busywork have a central role in learning to play a musical instrument. You can't learn to play music, solely from attending lectures and reading books. You have to program that knowledge and understanding into your ears and hands. This is done through memorization and busywork. To get past the most basic beginner stage, you either have to force yourself to do it, or derive some pleasure from it. If you do get past that stage, you can "hear" something in your head, and it comes out of your instrument automatically, so you have some bandwidth in your brain left over for thinking about higher level things, such as: How do I want to interpret this music? What are the other musicians doing? Are drinks on the house?

Is there something like that in math? I'm thinking of getting from A to B in a proof or solution by introducing things like definitions, theorems, common algebraic manipulations, and so forth. If you can't just see those things in your head, and write them down, then you won't have a reserve of mental bandwidth to think about the higher level structure of the problem that you're working on, and you'll reach a level of complexity and abstraction where the cumulative effect of small mistakes prevents you from ever getting to B.

Is it poorly taught? That's certainly a possible problem. One thing I noticed when I taught math, was that the kids were never given a higher level explanation of what they were doing. Memorization and busywork are necessary but not sufficient. My students did not understand what "show your work" means. What it means is that math is a mixture of theory and performance art, like music. Math has a weird social role too... by the time they are in high school, most kids know that they are learning math with no expectation that they will ever use it. Parents treat it as some sort of obedience training.

[klibertp]

> If you can't just see those things in your head, and write them down, then you won't have a reserve of mental bandwidth to think about the higher level structure of the problem that you're working on, and you'll reach a level of complexity and abstraction where the cumulative effect of small mistakes prevents you from ever getting to B.

The big difference between math and music is that the latter is a real-time applied form of the former, with "real-time" being the key. You can waste as much time as you want thinking over things and research solutions to the given math problem; the solution you arrive at after a week is as valid as someone else's solution written down in minutes.

[analog31]

I think that's true to a certain extent. However, being able to write down a solution in minutes might be beneficial if it's a small part of a larger effort, because it doesn't interrupt your "flow," to borrow a popular term. That kind of facility is also beneficial if you're working with other people. "Hold on, give me a week to come up with an explanation for my idea" has much less impact than "here is how it works."