My question is: how should one go about studying graduate level mathematics when they have a weaker background?
This is probably a weird question since most of the people go in to graduate mathematics will have a solid background in mathematics. I searched around S.E. for a while and haven't found a discussion that suits my particular case so I decided to post my own and look for advices.
A little introduction to my situation: I am a first year graduate student in mathematics and will be taking my qualification exams soon. Now that a full semester has almost passed and I have to admit that it was a complete fiasco. I struggle a lot doing the homework and barely managed to pass my exams (sometimes not even passing). I was not able to make any friends with my fellow colleagues because I couldn't find time to do anything else but my coursework (and this was only three classes, the bare minimum of any graduate studies). I constantly feel drained and tired because I am sacrificing my sleep to do the work, which obviously has turned into a vicious circle. The list could go on, but I think I have got my point across.
However, even with all of these stress and negativities, I still have a passion and love for mathematics. While I am having doubts about if I am suitable for studying mathematics (whatever this means), I never regretted my decision to go into the field of mathematics. This is what lead me to post this question, in the hope to seek advice on improving and getting better at doing mathematics.
Education Background and Motivation: I was a double major in economics and mathematics back in my undergraduate studies. While I don't think having a high GPA does not equate to being a good student, I did manage to maintain a 4.0 GPA till the end of my undergraduate studies. I am a diligent student and I was taking one to two math classes and two economics classes each semester. While I have to study hard for my math classes, economics classes were like a breeze since I could easily understand the math behind the economics models. I barely even need to study for economics except from doing the homework and some practice problems for the exam. Whereas on the other hand, I would grind for mathematics hours a day to understand it and participate in study groups to discuss it. While the grinding is real, I enjoyed it immensely, a lot more than doing economics. This has lead me to pursue my graduate study in mathematics.
Study Strategies and its Problems: The study strategies that I have used for my undergraduate mathematics classes seem to have lost its mojo in my graduate studies, judging from my performance in this very first semester. What I did for my undergraduate math classes is that I would carefully work out every details of the proofs on the book before the class and most of time try to reproduce it after/during each class. I do this even for the non-proof based classes. It worked well for me most of the times. I was able to retain a lot of information from the proofs (memorizing it by understanding it, so to speak). After I made sure myself have completely understand the proof of each theorem (to the degree that I could reproduce it down to every detail), I would start to work on problems.
As you could probably already tell, such a way of study cost a GREAT deal of time. I would be spending 2-3 hours per day of good quality studying on each of my math courses just to do all of the things I have described. It is simply not realistic to do so in my graduate classes and worst of all is that I seem to have lost the ability to retain the information that I have learned even after working through every detail. It just feels like that I am lost in the details and ended up overloading myself with details and missed the bigger picture. I could perhaps remember everything after a day or so, but after a week, I would literally forget every single detail, or even the structure of the proof for the theorem. It also take me a long time to work out a single homework problem. There are times that I could work them out fast, but many other times I would just stuck on one problem for 2 to 3 hours and gotten absolutely nowhere.
Sometimes I might be stuck on a trivial question for an hour and then realize it is trivial! As a result, I wouldn't be able to finish my to-do lists everyday on studying for the class, and then I would go into the next class without properly understanding the last class. The knowledge that I am forgetting and missing would accumulate and they accumulate fast. This inevitably caused my failure this semester.
I was given suggestions by my professors to do extra exercises. But as you could probably guessed, I simply do not have enough time for extra exercises. I barely even have time for the homework! It has became a vicious loop and I am unable to break it.
I apologize for the long post and any advice would be appreciated!