I am a first-year graduate student in a US university pursuing a PhD in mathematics.
I am a bit frustrated in trying to balance coursework with original research. I saw students who spend most of their time with classes, but only produce one or two mediocre papers by the end of their PhD careers (I am not saying that not publishing a lot is a bad thing, as that depends on the research group in which the student is involved).
In my opinion, one should learn by reading papers rather than taking courses at the PhD level. If one wants to study spectral theory, for example, one should read the classic papers by Weyl alongside modern treatments, and should talk with his advisor (hopefully he or she is at an institution with strong research group) with a view towards trying to contribute something new to the field. One should also try to keep up with the new contributions to the field (sometimes a graduate program is blessed the resources that enable it to bring people from the US and/or world to give talks on active topics).
Unfortunately, with teaching 4 times a week and taking 3-4 math courses with long homework assignments, the idealized scenario that I just described is very hard to follow. My life has turned into finishing homework assignments, and I am not really convinced that spending my time in this way is going to help me. Perhaps, there is a reason why some graduate math programs (such as Princeton, for example) do not require the students to take courses and get grades.
In doing research, if one encounters some new concepts he or she is not familiar with (perhaps, due to not taking enough courses), then he or she should develop the skills necessary to efficiently learn that field/ concept to move forward with his or her research; this can come through with having conversations with fellow graduate students well versed in the field or self-study. Unfortunately, a lot of the graduate programs do not stress this enough in the early stages (and one cannot always a read a book or paper(s) front-to-back); however, Harvard's math department does require the student to write a minor thesis ("The minor thesis is complementary to the qualifying exam. In the course of mathematical research, the student will inevitably encounter areas in which s/he is ignorant. The minor thesis is an exercise in confronting gaps of knowledge and learning what is necessary efficiently."), which is crucial.
Has anyone else felt this frustration? Obviously, complaining about this is not going to do anything, and I want to find some way to cope, work around or come to terms with this frustration.