I'm an undergrad at a top US university and I'm currently taking an Introduction to Analysis class. I really really enjoy this class and it's actually getting me more interested in math in general. I feel like I'm able to understand the material really well, and often notice that I understand some of the nuances better than my classmates (let me also add that I'm not saying this to be arrogant, but simply to say that I think I'm doing pretty well conceptually). Yet, I find that I'm not able to do too well on exams and tests. It's not that I don't know the problems or don't know what to do in them, I'm just not able to produce my best work on test day.

Again, I want to add that I am actually putting time and work into this class and feel like I'm understanding the material, I just think I'm not approaching the tests correctly (in terms of preparation and how I actually do them). This is a general question, but has anybody else faced something similar? What would you recommend I do? Any other general pieces of advice on how to approach a rigorous math course?

  • $\begingroup$ Can you elaborate if you struggle with proof writing or coming up with a proof on the fly? If it is the former, then I would recommend rewriting every proof presented in class/textbook until you are confident that you can replicate the argument given a blank canvas. If it is the latter, then I would recommend doing exercises and reviewing examples so that you get an understanding of what is considered "interesting" within the topic, and so get an idea of what to expect on an exam. $\endgroup$ – Alberto Takase Feb 13 at 19:28
  • $\begingroup$ I would say it's a bit of both. Problems on the exam generally require arguments already discussed in class (or through homework) and I can definitely understand what needs to be done to provide a solution. I'm just not able to write the best solution on the exam, even though I broadly know what has to be done. Sometimes I just forget to write some finer details of an argument which I know in my head but somehow just miss out while writing. Thanks for your comment! $\endgroup$ – gtoques Feb 13 at 19:39

For many such math courses, it is important to distinguish understanding and identification. Just because you know all of the material doesn't mean you can spot it in an obscure problem, and exams are often intentionally obscure in this way. Often, the tricks are similar based on the type of problem. For example, if a problem involves Concept X, then usually Theorems A, B, C are the most useful for that type of problem. Trying to apply these first will save you a lot of time on exams, since unlike homework it is important how quickly you find the right order in which to do things. Often, these are just the theorems used to solve similar problems in the homework.

Hopefully this helps!

  • $\begingroup$ Thank you for your comment. I'm actually not at Princeton (I suppose Math 218 is one of your introductory analysis classes there?) but I'm at a school not too far away! Anyway, I agree with what you're saying. I just think I need to do more written practice. I've realized that it's problems that require annoying computations to get to the result that I'm not able to do really well, as opposed to those that have shorter and more "clean" solutions. Any tips on those? $\endgroup$ – gtoques Feb 13 at 20:19
  • $\begingroup$ Hm, unfortunately I have the same issues with “bashier” problems. One thing I can say is that especially with analysis, if you’re doing more bashing than just multiplying matrices, there might be a theorem you can use? Sorry I’m not of more help in this department :/ $\endgroup$ – Michael Gintz Feb 13 at 21:33
  • $\begingroup$ Yeah I get it. Plus I think getting comfortable with tedious computations is just a matter of practice. $\endgroup$ – gtoques Feb 14 at 6:29

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