Once you understand something, you can't ununderstand it. For example, you can forget a quadratic formula, but if you understand how it was derived you can derive it again yourself, or just solve a quadratic equation by completing the square.
This is the point. Once you understand something, you may not be able to recall it immediately, but you are confident that you'll be able to work it out from simpler principles. In an exam however, memory recall may be more effective sometimes because of its speed.
Hm, I think that's my problem, I tend to memorize how things work rather than learning HOW they work. My fundamentals are off I think.
For example, I know that the derivative of 3x4 is 12x3, but if you were to ask me "why".. idk if I could explain it very well, even though it's a really easy problem.
So what should I do, go back and try to understand everything I've learned so far? I'm behind in my uni class and I need to get a good mark but I also need to go back and review stuff from the beginning but I can't spend too much time on that since my final's in 1 month. What do I do?
So let's take this example. You memorized the rule that (kxn )'=knxn-1. It's simple and lets you solve some problems immediately. But if you don't understand where this rule came from, you might have problems later on understanding how other derivatives are taken. What if you don't know the rule for a new situation?
So the other approach is to ask "where did this come from?" You start with the definition of the derivative, f'(x) = (f(x+h)-f(x))/h where h goes to zero. This is one of the few thing you have to remember. Not the formula, but what it stands for, the slope of the tangent line.
And then you can say 3(((x+h)4 -x4 )/h), open the parentheses, let h approach zero, and realize that the only term remaining is 12x3 . You won't have to do it every time, it's the ability to do it that makes a difference.
Hm that makes a lot of sense, thank you! Do you also have any strategies for solving proofs?
I use the same approach. Instead of remembering the steps I'm trying to understand the main idea, then work it out myself from the axioms and theorems that I already proved.
Oftentimes there's more than one way to do it. For example, if someone asks me now to prove the Pythagorean theorem now I won't use the proof I was taught in school. It'll probably take me quite a long time to reconstruct it. Instead I would use another proof which I find easier to understand.