Symbols are just names. You have a name, as do many other people. Some people even have the same name as you ! So really, saying "let x be a real number" is the same as saying "let us consider an employee from company X" (a theoretical company with infinite employees). If you want x to be in a certain subset of the real numbers then it's adding "let us consider specifically the janitors who work at company X".

So when you see a letter representing an element, think of it as an individual, relate it to something from the "real" world.

In order to truly understand your theorems and properties, what you must do is redo the proofs. Not copying them, but actually thinking and trying to come up with the steps after one read at most. After a while you will learn how to play with the words (symbols) and concepts. For example, the claim "a function that is differentiable at a point is continuous at that point" is not something you should simply admit but actually should prove.

Even if your professor gives you a demonstration for their claim, having it or reading it is NOT enough.

Also, try drawing stuff. For example, if you're working with sets and all, try making coming up with something that helps you visual it. It doesn't have to be "accurate" (everything can be drawn in 2D if you're brave enough, even if you're working on a space of large dimension, possibly infinite).

And finally, do exercises. Don't go directly to hard ones, start by the most basic things (example : prove that in a metric space, an open ball is open, if you've studied metric spaces; prove that a close ball is closed). The basic exercises and questions exist for a reason : they're the first natural thing you might think of and they are a direct application of definitions and theorems.

## dimview

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.

## ZedZeroth

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.

## Maltohbr

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 3x

^{4}is 12x^{3,}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?

## dimview

So let's take this example. You memorized the rule that (kx

^{n})'=knx^{n-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 youdon'tknow 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

haveto remember. Not the formula, but what it stands for, the slope of the tangent line.And then you can say 3(((x+h)

^{4}-x^{4})/h), open the parentheses, let h approach zero, and realize that the only term remaining is 12x^{3}. You won't have to do it every time, it's theabilityto do it that makes a difference.## Maltohbr

Hm that makes a lot of sense, thank you! Do you also have any strategies for solving proofs?

## dimview

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.