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Tips for studying high-level math

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While it is laudible to figure out the solution to every assigned math proof on your own, it isn't always feasible. If you can still learn the methods of proof for an exercise, don't be ashamed to google the answer, or at least the steps that have you confused.


Alternatively, do your hw with peers in the same class, similar idea, but feels less like you are cheating.


You may not end up needing to do either, but it is frequent that math professors/grad students aren't the best at providing you with the tools that will make you successful on your exercises.


Too often, it is assumed that everyone in math is just some sort of savant, and that biases a lot of our outlook and instruction. Don't get caught up in that bullshit, do whatever you can to properly learn the material and not fall behind.
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From my experience teaching undergrads, i think one thing i can say is that it may be helpful to go against what your habits are.

For example, when i was an undergrad I loved reading books and literally memorized proofs until i could understand each step and reproduce it on my own. But i didn't really enjoy doing problems. At some point, i wasn't happy with my performance and i switched everything. I stopped taking detailed notes in class and tried to just listen. I stopped taking notes while reading and completely focussed on exercises, only going back to the text when i was stuck (which was often). It was a new way of doing things. These days i don't feel like i understand something until i can work out several examples and invent exercises for myself.

On the other hand, I've seen students who never read or indeed even own the textbook and just do practice problems. I usually advise them to read the "theory" part of the book more carefully.

Another thing is that i always thought of myself as someone who was "bad at analysis". Eventually my research required me to do a lot of analysis that involved Sobolev spaces and other "scary" topics. I forced myself to take a topics course in pdes and learned all the things i was scared of.

So maybe one thing you could do is just try to identify the aspects of studying math that youve neglected and really give them some attention. Idk if this is good advice but i really think i grew a lot by pushing myself out of my comfort zone.
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Just keeping studying. And this feeling of getting stuck is pretty common, just keep study one day you will realize how much you learned.
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Every now and then try to explain what you know about something you have studied. You will realize how much you have learned so far.

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