This is entirely embarrassing to admit, but I'm realizing, one year into my doctorate program, I don't know how to effectively study math. I feel like a failure and a fraud for even having to come here.
I've been one of the students who got by first by being smart and not having to work hard, and then by swallowing a lot of material wholesale before an exam as I juggled two majors in undergrad. I thought I had the chops to handle the pace and volume of grad level classes on my own, but the truth is I relied a lot on my fellow first year students to explain proofs and subtleties in our classes.
Granted, I've been juggling some light research interests as well as I lean toward applied mathematics (they like the applied people to get into research as soon as possible at my institution), but I feel like that's a poor excuse as I'm not the only one.
Whenever I attempt to work through problems, I'll come to points where I'm at a total loss: I can't even identify what I don't understand or what other approaches to try. And even if I can, I get sucked down rabbit holes trying to fill in the gaps in my understanding. I want to develop good habits to help me for these exams and for future endeavors when I need to become familiar with a new field for research.
So I guess I'm asking if there's any agreed upon protocol or good practice for learning advanced math? Like a reliable troubleshooting method? Mine seems to be grossly ineffective.
All perspectives are appreciated. I just really want to pass these exams.
I'm a third year graduate student who had to pass three qualifying exams. I used the same strategy to study for all three. This is what worked for me.
Step one (general comprehension): Spend two to three hours a day reading proofs and important results for the exam you're studying for. Try to work examples as well as understand theorems. Memorize the proofs of important theorems. Spend two more hours working problems on your own that are related to what you were just reading. If you're getting nowhere on a problem for 20 - 30 minutes, ask someone how to solve it, understand the solution and move on. Carry on for a month or however long it takes for you to feel like you're somewhat comfortable with the subject.
Step two (getting good at solving problems): Every day, take a practice qualifying exam under test conditions. Hopefully your university has a collection of old exams you can look at; otherwise, use a different university's. So, don't use your notes and continue trying to solve the problems even if you are stuck and have an hour left. You will be surprised what you can remember and make use of if you force yourself into a situation where you cannot get hints. After the time is up, find the solutions to all the problems you didn't get. Keep doing this for a month.
Step three (tying up loose ends): Maybe a week before the exam, stop taking practice tests and just rest up. Continue to work new problems, but don't spend too much energy trying to figure them out if you get totally stuck. Instead, ask for help shortly after you get stuck (for example, on stackexchange). Memorizing the solutions of a lot of problems shortly before the exam, combined with the huge amount of problem-solving practice you just did, you should be in pretty good shape.
If you're lucky, with the huge amount of problems you have encountered, one or two of them will appear on your exam and it'll be free points.