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 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...)

Currently I have 3 ways to choose 1 from:

  1. Withdraw from this class and see if I can do it better next year. This will put a Withdrawal on my transcript, but it will allow me to postpone taking this core class next year before which PhD admission committees have already received my transcripts.

  2. Try my best to finish the rest of this class, and then postpone taking the second-in-a-series class in analysis 2 years later until right before I graduate from this master's program. This doesn't help me avoid a B/C on my transcript this time, but it will make sure that the transcript for my next semester is no longer unappealing. Two years later when I need to take this second class in analysis, PhD admission committees have already received my transcripts before any bad grade like this one happens again.

  3. Try my best to finish the rest of this class and the whole sequence this year. This is what my graduate chair has advised me to do, because he think the grades from these core courses will provide more information for PhD admission committees to evaluate. I am not too sure about this piece of advise from him because he personally is a very smart professor - I doubt he had experienced this failing moment ever in his life. Is more information to evaluate always a good thing? What's the tradeoff between (more mediocre information) and (less but very appealing information)?

Out of these 3 options, which one should I go for at this moment?

More mathy question: How should I study for analysis?

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, and 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.

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...?

  • Your university should have advising staff that can help you. – Anonymous Physicist Nov 7 at 9:39
  • Any suggestions for studying analysis, please Maybe ask in Mathematics Stack Exchange? Pick the most significant topic you've had trouble with (or the troublesome topic which, if you can learn it before the end of the course, would be of most help to you) and say exactly what it is ("analysis" could be proving convergence of real sequences and series, signed measures and Radon-Nikodym theorem stuff, Banach space theory, etc.) along with the specific textbook you're using, and ask for alternative reference suggestions for understanding it. – Dave L Renfro Nov 7 at 15:58
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    If "correct intuition ... but I can't seal up the proof nicely" is an accurate summary of your problem, it might help you to imagine, when writing up proofs, that you need to convince Dirichlet (with his nowhere continuous function), Weierstrass (with his everywhere continuous but nowhere differentiable function), and Vitali (with his non-measurable set). Would your proof withstand attacks from folks who can create really weird examples? If not, clarify your proofs until they do. (And it might also help if you think about creating your own weird examples.) – Andreas Blass Nov 7 at 20:02