I asked my question in another branch of stackexchange (https://academia.stackexchange.com/questions/158563/what-can-i-do-with-my-failing-analysis-class-failing-graduate-study), and the people there referred me to this site. I extracted the part more relevant to math here, but I would really appreciate it if you can help me with the bigger question about my graduate school plan and options listed in the other link.
I am a first-year master's student in a math graduate program. Out of the four classes I am taking at the moment, I think I might be failing my analysis class.
Precisely speaking, the letter grade won't come out until Xmas, so it's not the final result yet. I still have another exam about one month later, which I THEORETICALLY can max out and get 100% on it - this might flip the situation putting me at somewhere around A- or even an A, but I'm not optimistic about this.
66.6% of the course has passed, and in the first 2 exams I got a below-average score and another lowest-in-the-class score. I am not sure what's precisely a fail in this class because my professor said he hasn't drawn the lines, nor looked into any grade stats of the class by far. However, I don't see a good chance to get an A from this class - I might get a B or somewhere around/below this level.
I have never been a big fan of analysis ever. I very much enjoyed algebra and want to apply for a PhD and do some research in it. Yet this probably upcoming fail will put me at a significant disadvantage in the application: I will apply to some PhD programs next year. This means I only have a graduate-level transcript of 2 semesters (including this one that I am having trouble with) to present to the admission committee. Having anything around or below a B might as well just block me from going anywhere, even though I am not sure about the validity of this idea either.
I read the textbook, read the notes, watched all lectures, and finished my homework very carefully. I asked my TA for office hour appointment every single week to ask him questions and go over my write-ups, to an extent I think I might be taking too much time from him. I also checked with my professor to confirm if there is any more things that I can do. He thought there is no problem with my strategy, but he confirmed my awkward situation where I have correct intuition for every problem, but I can't seal up the proof nicely. He went over the rubric for me, and very often I didn't realize some of the items require a proof. He doesn't seem to know how to deal with my situation either except assigning me a bad letter score in the end. On the other hand, this is a graduate-level class where no solutions to HW are posted at all - so even if I have a chance to confirm with my TA on 1 or 2 HWQs, there is no way for me to know whether I am proving every single thing as nicely as expected BEFORE the exams make punishment on me. (Or is there a hidden way to know how well my proofs are that everybody knows except me? If there is, please comment below. Please...)
I would really appreciate any suggestions on how to study analysis because I always get panic in an analysis exam. I immediately get panic whenever I find the exercises behind the book to be very different from what appears on the exam, and they always are. Sometimes even understanding a theorem is not easy to me. After I finally understood them, I am still unable to quickly make connections between the theorem that I just learned and the ones I learned 2 months ago, UNLESS I have seen some exercises around their connections before the exam, which never happened actually.
I usually would just suck it up all by myself when I was in undergrad, but now since I'm in a master's program which is significantly shorter than any undergrad program, I am very concerned about what this indicates or where this will put me at. Analysis always makes me feel like I am inferior, and the study strategies I have always been using are not working at all. No matter how hard I try and how much time I am spending on it, I'm just not getting ahead of the class. I always feel gaps between my knowledge system that hold me from excelling in an analysis class, but I don't exactly know how to fill these gaps. I would LOVE to find a way to master analysis, but I just don't know how after I have tried going over the proofs in the class, understanding the theorems, etc. From next week we are going into a new chapter that is not cumulative upon the previous material which I am uncomfortable with, so this literally is my last chance if I want to flip my situation in this class. Any suggestions for studying analysis, please...?