Attemping Qualifying Exam Problems -- and failing

Question

My question is concerning learning strategy. I can solve the majority of the exercises in a typical graduate mathematics textbook like, say, Dummit/Foote's Abstract Algebra. To supplement my education (and, hopefully to increase my problem solving ability) I have been attempting many qualifying exams questions. Many are available online — as well as complete solutions. I find myself not being able to solve these — although I have the general idea in some cases. I am not particularly discouraged because I know these problems are not designed to routine; they are designed to ensure the student has an extremely deep understanding of the material and is ready for research mathematics.

My objective is to get to be better. I want to be able to attack these problems and solve them — or at least get closer than I am now.

The Question: To get better at problems like these, I would like to take a problem, give myself a time limit, say $3$ hours, to see where I get. Then, see (and study) the solution. Is this a reasonable strategy? I am looking for responses/advice from current PhD students studying for their exams and, of course, those who have passed such exams. My idea is if I implement the strategy above every day, there must be a point where I start getting better at these problems, right? (I should note that working on these has certainly made end-of-chapter problems look easy!)

Answer 1

I know this is way late to the party, but I can lend you my experience.

For a given subject, say a month prior (or longer if that time will be interrupted), commit to a full review of all the relevant material before attacking problems. It's important to keep all the relevant theorems and ideas front-loaded and relatively fresh. You'd be surprised how much rust builds up and how easily you can lose sight of otherwise handy results.

As you're reviewing, visualize. Draw pictures. For a given result, build the intuition through visualization that would lead someone to think it was true. Then note the correlation between the visualization and the machinery used to prove the result.

Different schools do things differently, but it's possible that if you can explain intuitively why something is true but can't muster the machinery, having the insight alone may grant partial credit. It's a hell of a lot better than leaving a blank space, anyway, and sometimes through the process of writing your intuition down (or reciting if oral), ideas for the machinery may come. Mathematics is at least as much about the insight as it is about the machinery.

Then, train in the format that the exam will be given. If oral, recite orally. If written in silence, practice writing in silence. Absolutely, work problems, but perhaps save the qual-level ones for when your review of the subject is complete. Once you start in on the qual-level problems, if written, obviously time management will be a factor, so identify quickly which problems you can handle and then handle them, leaving the ones which will require more effort for later.

Once you do groups of these, you'll start to see areas where your understanding isn't as strong, and that will prompt you to re-review those areas. Wash, rinse, repeat.

Good luck!

Answer 2

I am currently a mathematics Ph.D student.

My approach is a bit different than Ken's.

A lot of time may be wasted just reading a book.

Getting your hands dirty is the best option.

In my opinion, you don't know what you don't understand or have forgotten until you attempt a problem.

With that being said, I just do a lot of problems from old hw's and old prelim exams, make packets and revisit things that I got stuck on and thought about for at least an hour.

If you can, meet with other students or your advisor (or professor).

Generally it will serve as a distraction to just arbitrarily look at exams from other schools since they may have used different books which lends itself to different notation, definitions and bunch of other things.

If you can, look at exams from other schools which are very similar to your own university.

There is not guarantee (if ever) that you will be able to do every problem, but that's not the point. Getting the idea or having some feeling of how a problem is suppose to go is more than half the battle.

Also, there is no formula for success in mathematics. I've tried to imitate a lot of people's study habits, but you must see what works for you and then you'll see improvements.

Again, attack problems, build intuition and sketch rather than being rigorous all the time, consult with other students/professors and make notes of general trends (tricks) that you see occurring in problems.

''The first time you see it, it is a trick, but the next it is a technique'' haha. Hope this helps!