[InclinedPlane]

In general the best way to learn anything is to start with a skeleton then add details (flesh) later. This is very true when it comes to math, especially at higher levels. You need to understand the concepts before memorizing anything will be worthwhile.

I'd add a few more things to your list. A lot of students hit a wall when they run into calculus. Partly this is the same old problem of mathematics being taught poorly in general, and there's also an annoying element of calculus classes being difficult on purpose (being as it is a topic which can involve an arbitrary amount of memorization and busywork) for stupid reasons. But additionally calculus relies heavily in having a firm grasp of advanced algebra, trig, and pre-calc, all of which equally suffer from the general problems of our educational system. Reviewing those subjects, probing for weaknesses in understanding, and shoring up any weaknesses can go a long way to making your life easier as you tackle more advanced subjects.

Also, as a general rule and a corollary to your third point: don't treat college as a race to get a valuable piece of paper, one that requires merely toil and busywork to acquire. Treat it as an opportunity to acquire new skills, understanding, and knowledge. The piece of paper might improve your earnings incrementally but the knowledge will not only change the way you look at the world it will unlock a great deal of potential in terms of what you can do (in your career and elsewhere).

P.S. since we're stating bona fides, here's mine: I got my math degree mostly by accident. It was something I was good at and a reasonable default choice, and I hardly had time to think about it before I graduated at 20. I don't "use" my math degree in my career typically but it's definitely made me a better developer and critical thinker.

[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."