How do high-achieving students in math study for their exams?

Question

I'm taking a real analysis course that is a bit more advanced than usual in that it introduces you to the Lebesgue theory and has some deeper concepts in metric spaces. This is an honors and advanced undergraduate course. I have always been curious how the best students would study for their exams, like the ones that go on to do a phd in math. Can anyone share what is the most efficient way to study for hard exams? My exam will be very tricky and often has problems where if you see the method, you can do it but if you dont, you are sort of out of luck.

Math isn't really like other fields where once you do enough problems, everything will start to make sense. Rather, right now I am trying to reprove all my previous theorems in the book but dont know where to go from there as I dont know if doing many problems will even be helpful and if my time could be spent better elsewhere.

Advice would be greatly appreciated! Thank you!

Answer 1

You say

Math isn't really like other fields where once you do enough problems, everything will start to make sense.

That's incorrect. Practice is key to success in math too. Once you have done enough problems, things will make sense to you. The question is whether you are doing these problems mindlessly, or perhaps analyzing and improving your approaches.

Suppose you are trying to get good at addition, but the only thing you learnt so far was transfering ones, i.e. $2+4 = 3+3 = 4+2 = 5+1 = 6 + 0 = 6$. You can add this way arbitrarily large numbers, and if you don't pay attention, then even doing zillion exercises won't you get anywhere. However, if you do pay attention, then you will notice that you can start from the larger number, $2+4 = 4+2 = 5+1 = 6 + 0 = 6$ halving the worst-case number of steps. Then you will notice patterns of higher numbers, then that adding numbers like $(4+6)+(3+7)$ is easier than $(4+3)+(6+7)$, and so on.

In other words, it is crucial for you to pay attention to what you are doing, not only to the problem you are currently solving. It's like solving a meta-probolem: how can I make my approach reusable?

To give a concrete example, there is a big class of inequality problems which can be solved using things like means inequality, Jensen's inequality, Muirhead's inequality and other similar, however, how would we know if some particular problem belongs to this class? There's a nice thumb-rule that can be used to spot these problems: check whether there is an equality if all the inputs are equal (or whatever that would make them equal from the formula perspective). It does not work all the time, but I was surprised how good it is (at least for the problems I was practicing on at the time).

To give another example, when solving a geometry problems, it's beneficial to know which points are fixed and which may be moved around, or to put it differently, how "rigid" the structure is and how points depend on each other. One approach that I learned by practice is to try to construct (e.g. with an edge and a compass) an instance of the problem (i.e. construct a diagram). Sometimes it is a hard challenge in itself, but then it often happens that your problem doesn't actually depend on the vertices from the formulation, but on some other unspecified points. These "hidden" points are usually significant and can be used to simplify or transform original task.

Of course, it is more than just some generalized approaches. The point is to notice patterns and use them when you spot them next time. It might be intuition, it might be some useful lemma, it might be whatever that helps you solve problems. After you have done enough problems in this way, things will start to make sense.

If I were to give you some advice:

• Get enough sleep.
• Ensure you understand all the fundamentals. That is, you have an intuitive feel why something works or is correct (and your intuition aligns with all the facts and theorems you know and problems you have solved). In particular, I would skip learning proofs if they don't help you with understanding (unless they are required).
• Make a summary (if possible, at least a full day before the exam) of things you have learned: important definitions (all the details), theorems (include full formulations), useful simplifications, general approaches, everything that is related. It has to be a summary, that is, short and concise, preferably in format supporting easy reference (e.g. bullet points).
• Get access to exams from previous years and solve as many as you wish (perhaps none).

I hope it helps $\ddot\smile$