I’ve found that there are two general groups of people when it comes to subjects like mathematics and physics. There are those who memorize, and those who internalize the material. Both can bring understanding to the student, but they are much different.
One of my mathematics professors illustrated this when he said, “As a mathematician, I like to do the least memorization possible.”
At first, this struck me as a little ironic, because a staple of mathematics exams is just remembering the truckloads of formulas for various situations. I know that at least in my classes in university, we get no formulas, no regular expressions (such as the trigonometric identities), no unit circle, or anything else. Everything needs to stay in our head, which means we have to memorize some things.
However, the deeper point I think he was trying to get at was that mathematics isn’t about remembering formulas and knowing when to use them. Sure, that’s what happens when we work on these ideas for a long time and get used to doing them, but the point is that these steps and procedures we take shouldn’t necessarily feel foreign. At the very least, they need to be logical and consistent. Doing a double integration by parts with say components $x^2$ and $e^x$ by choosing the former as $u$ and the latter as $dv$ but then doing the opposite after the first integration by parts isn’t logical.
I try to keep this in mind when working on both improving my skills in a mathematics or physics class and while trying to tie everything together for the end of the semester. I don’t want to remember a thousand different formulas. Instead, I want to remember the intuitive and powerful principles that I learned throughout the semester and be able to apply them when I get to problems, without necessarily memorizing everything.
However, I do want to point out one final thing: a lot of the teachers are being a bit disingenuous when telling you, “It’s not the end of the world if you don’t remember a formula. You can easily re-derive it.” Sure, that’s true and you can do that, but most students do not have the time to re-derive a formula and then answer the question that was troubling them on a test. This is particularly true if the test has a short duration (as mine were) or if there are multiple questions in which you have to do this. Given enough time, I’m sure I could get the formulas I needed, but that kind of time isn’t typically available during tests. That’s why it bugs me when teachers say this, because it’s true but not practical.
In a broader sense though, there’s something nice about being able to remember a few principles and working from there. I’m not saying you have to reinvent calculus for your test, but it might not hurt to try and “compress” the number of things you need to remember into more broad categories that can adapt to your specific situation.