How to succeed in upper-level math

[KingTut]

This question will essentially be more of a how-to plea or general help request.

I'm currently studying math and I'm at the point where I've transitioned into upper-division classes, most if not all of which are proof based.

To be blunt, I currently feel discouraged at the prospect of being able to succeed in "upper-level" classes. Last quarter I was able to get decent grades in my classes, but nothing like the success I enjoyed in lower-level math. My discouragement stems from the fact that in the past if I studied I did well. Now I feel as if I'm studying and not doing well at all.

Obviously this is a reflection of either my study methods or frequency with which I review the required material, but either way, I'm starting to become disillusioned with the good ol' saying that if you "practice, practice, practice" then you'll get better.

My question to whoever cares to comment, and believe me I greatly and truly appreciate all advice, is how did you transition into proof based classes and succeed in them? Obviously the material is different for each course, but in general how did you approach absorbing the material? While everyone is different, what are review methods that you found helped you to best understand the material and also retain your understanding?

I don't know if others have this same issue but I find that for proof based classes, since problems tend to all be different, I don't quite retain a sense of how to tackle problems even after completing an assignment whereas for classes such as calculus in which e.g. I had to learn integration, after so many integrals I had a general sense of how to solve them. Any advice on how to remedy this deficiency?

I apologize if this question is inappropriate in any way, but I'd love some perspective from those that have gone through a similar process.

 

[micromass]

Congratulations, it seems you have thoroughly analysed the issues you have with your class. It is indeed true that eg solving integrals in lower-division calc classes tend to be "drilling" problems, in the sense that if you solve a lot of them, after a while you know how to solve them. It is also true that upper-division classes do not contain such drilling problems anymore. The problems are of an entirely different nature. So while "practice, practice, practice" might have worked before, it doesn't now. In upper division, you still need to practice, but you need to be "smart" about it. Just mindless computing won't get you anywhere.

So what kind of skills are useful in succeeding in upper-level math?
1) Knowledge.
Knowledge in lower-division classes was mostly techniques. How to solve this or that. This is much different in upper division. Now you will need a ton of knowledge about the behavior of different concepts. It is no longer true that you can ace exams by "having an intuitive feel" of the material. Now, you must know the material in excruciating detail if you want to score high. You need to know the theorems and definitions by heart and you must be able to apply them. You must also know plenty of counterexamples and examples. If you lack a piece of knowledge, you cannot score well.

2) Visualization
The biggest secret that math books don't tell you is that all mathematicians visualize stuff to some degree. Whether it be groups, categories, topologies, everybody visualizes those. A lot of my time goes to finding exactly how to visualize certain results, and constantly reworking how to see things. Having pictures in your mind about concepts is extremely helpful.

3) Hard work
Hard work is still very important but does not consist anymore out of doing mindless drillings. Now it consists out of being very critical towars the material and really "fighting" the book. In my insight, I have given a detailed way of studying mathematics. This way might not work for you and you might have to tweak it. But it makes it clear how much work is necessary to digest the material. https://www.physicsforums.com/insights/how-to-study-mathematics/

I have known people who take a real analysis book and read it completely in one month, and then claiming they got it. It doesn't take many questions to find out they know nothing at all. You cannot read math texts quickly. You need to go very very slow and digest everything thoroughly.

 

[IGU]

... how did you transition into proof based classes and succeed in them?

Your first proof based class was probably geometry in high school. There's really no difference in later such classes. The way to tell if you "get it" is if you can re-derive a proof from first principles. To get to that point, it helps to try to prove things yourself before looking at the proof. And by try I mean spend serious time at it, not just a couple of minutes. It helps to look at variants of proofs (what if I eliminate this condition? can I still prove it? if not, why not?).

But before coming here, I hope you've read the classic work in the field, Polya's How to Solve It. You'll learn more from that little book than from anything I can post. Meanwhile, don't expect the transition from training to learning to be easy. Actually understanding stuff is quite difficult, and that's what is required of you at this point.