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I am not sure if this is the appropriate place to ask this question so forgive me if it is not.

I am a first year graduate student in mathematics. Often times when I face a relatively difficult problem or exercise, the first thing that I think of is "I can't do this!" which leads to "why am I so dumb?". This kind of unhealthy monologue is preventing me to actually attack the problem, and I have to spend a lot more time on problem sets than the others because of this.

I am wondering if anybody else has this kind of problem as well, if so how did you overcome it? Any advice would be appreciated.

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closed as primarily opinion-based by D_S, Strants, Lord Shark the Unknown, Dave, John B Apr 4 at 2:54

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I just leave it as a comment. The main thing is that you don't have the required background to do the class yet and you still want to take it. Or you don't like the teacher ? or both ? or you don't even like the subject that the class teaches. $\endgroup$ – DeepSea Apr 3 at 2:31
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I am not sure if this is the place, but let me try. I am a graduate student in physics who had second major in mathematics during undergraduate so I somewhat can understand your feeling (hopefully).

First thing first, if in graduate school you don't face difficult exercises or problems, generally you will not grow --- or rather, not quickly enough. During undergraduate, because exams are often the end of everything, our learning is very exam-oriented, which often does not help as much in graduate school. That said, graduate studies benefitted from undergraduate studies in terms of literacy. A lot of difficult concepts in graduate studies become somewhat easier to penetrate or even read when you have faced or seen the basics at least once (even if you do not fully understand them).

I suppose you will face many exercises involving proofs. I like the following sentences by John Howie in the preface of his Complex Analysis textbook:

It is, however, possible to appreciate the essence of complex analysis without delving too deeply into the fine detail of the proofs, and in the earlier part of the book I have starred some of the more technical proofs that may safely be omitted. Proofs are, however, given, since the development of more advanced analytical skills comes from imitating the techniques used in proving the major results (emphasis mine).

While of course it is true that you cannot live off mathematics with just copying proofs, I think in mathematics critical thinking and "healthy imitation" are essential. A lot of technical strategies developed over the decades and centuries are extremely clever and ingenious, and mimicking them might give you a sense of how these older problems were dealt with, and oftentimes new problems can be bent similarly. Of course, some require completely new ideas, but usually very strong novel ideas come from experience, knowing what worked and what did not. That's why doing a lot of exercises, simple and hard, are essential even if you cannot deal with them at first. At least personally, I understand better the problem if I first know why I could not deal with it in certain ways, and then found out how to deal with it later (by myself or with some help).

In that sense math.stackexchange platform like this also helps; it gives new perspectives that some online users have, and surprisingly they can be really cool. For that same reason, do not do mathematics in isolation --- unless you have very strong reason not to (e.g. you are extremely introverted or find yourself really unable to talk to people).

Last but not least, at the beginning you should not overthink about "how fast" you should get the stuff. In fact, a lot of times you can learn or understand new things only way after you "finished". I first learn differential geometry in a course I took during study exchange, I learnt a lot, but I am pleasantly surprised that some concepts were actually misunderstood until I tried to use it elsewhere more than a year later (more specifically, when I was dealing with something in general relativity). And it occurred again 2.5 years later, when another part I thought I knew better was right but somewhat sloppy. I think not getting everything at first shot may be something one needs to be comfortable with. You only have finite time to learn anything.

A lot of things I said here are probably either common-sense or well-known, but I personally need to remind myself that graduate studies is hard. Of course some find it easier than others, but I guess we can work it out if we don't give up.

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