What is the proper way to study (more advanced) math?

Here's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I think that I'm being lazy, I'm not thinking things through, or I'm just not thinking clearly. This approach is not practical because I only end up falling behind on other work. I don't like to look up solutions because I feel like, given enough time, I would be able to come up with the answer (or some good reasoning). But maybe I should start looking for answers after a shorter period of time. I don't know what the right thing to do is.

Do you guys have similar problems? Should I feel bad because I have to look at solutions? Or is this just part of learning?

This is a great question and I think I may have responded to a similar one months ago on the companion site Math Overflow. I don't recall exactly what I said there,but to answer your question:

Firstly, although I firmly believe mathematics has to be learned actively and looking up answers should be something you try to avoid, there has to be a practical limit. Most of us have gotten a stubborn streak with a particular homework problem as a student where we literally waste days trying to solve it-we're not gonna let it "beat" us. Part of it is just stubbornness, but beneath it is a deeper fear that our inability to solve a problem with no help is the dreaded "wall" that shows we ain't as good as we think we are and it's the first step towards ending up mopping floors outside our more brilliant classmate's office at Princeton. This is a lie, of course-with the exception of the truly gifted, all mathematics students struggle with proofs and computations.

More importantly,since we live in a Real World where there are deadlines on assignments and time limits on exams, such thinking will be very self destructive if it's not controlled. Finding oneself when time runs out on an exam having spent all the time on a single problem and getting a grade of 7 out of 100 for such stupidity is not a good day.

Personally,I think all textbooks regardless of level should come with complete solutions manuals. I know,I get a lot of flack for that,but I think having access to the solutions is a very good thing for students to have because it allows them to set a limit as to how long they'll work on a problem by themselves without solving it. "Oh,but then they'll just look up the answers and get an A." That's a facile argument to me because even if their professor is irresponsible enough to grade them solely on work they can look up, sooner or later, they will be required to find answers to even more difficult problems without access to solutions.

A middle ground solution to the corundum is to have textbooks with good, detailed hints. In my experience, a good, well-worded hint is usually good enough for a hard working mathematics student to point him or her in the right direction,they usually don't need more then that to get unstuck. But I'm getting off topic here.

My point is that although certainly you should make every possible effort to try and work things out yourself, there comes a point where it becomes self defeating and you have to either look up the solution or ask for guidance. How long you're willing to work before considering asking for help is a decision you have to make for yourself, but after being a student for some time,it's not hard to work out a reasonable boundary to set for yourself on this.

I'd like to close by telling you how I study. I generally make up a large, detailed batch of study cards-2 kinds for mathematics; one containing theorems and sample exercises and the other definitions and/or examples.The definition and example cards are the critical ones for studying. This is what furnishes the basis for understanding mathematics. For example, to understand the Cayley group isomorphism theorem, you have to understand what it means to have a permutation on a group. You can then try and re-express the result in terms of other concepts. For example, you can think of the Cayley theorem as stating there is a fundamental group action of every group on itself. However you do it-absorbing the definitions and what they mean is absolutely critical. Test yourself numerous times to see if you've absorbed and understand them.

For the theorem cards, this is where things get creative. Study them with a pen and paper in hand.I generally write the statement of the theorem on one side and the proof on the other. Do not look at the proof-try and determine the proof yourself directly from the definition. Wrestle with it as long as you possibly can before turning it over. Then if you can't reproduce it-put it on the bottom of the deck and move on to the next one. Do this until you have 2 separate piles: the ones you can prove and the ones you can't. Then do it again until you can prove it. This works,trust me.

I've often wondered what the matter is with the following approach:

1) Write up what you CAN do for the problem/proof, pointing out as clearly as possible where you are stuck. 2) Coming back to fill in the gap when you figure it out at some time down the road.

You can write all of this in a blog, for example, and it will be searchable. Perhaps this is a remedy for the infuriating fact that much of mathematical discovery happens in the subconscious, and it is healthy to keep moving during the incubation period following apparently unsuccessful hard work.

Although I imagine it is fine to look at an answer once in a while, it is certainly important to learn to properly cope with being stuck. The above approach effectively emulates what often happens in mathematical research, where it is often healthier and more productive to focus on and develop what you CAN do rather than obsessing too much about what you cannot. The above also emphasizes trying to create conditions where an answer can be more readily "seen": Think of carefully writing what is known as analogous to putting together all the pieces of a puzzle around one missing piece...it has the effect of making what the missing piece is more visible when it is stumbled across. Looking at a solution, on the other hand, has the danger of stopping you from thinking about the problem, perhaps out of a sense of defeat.

A nice way to go was suggested to me by a mentor: you try to prove the theorems yourself, and peek at proofs when you are stuck. I like this because it handles the fact that there are some simply brilliant ideas out there that took quite a long time to come up with. It is crazy to expect that you could re-create these things without excellent planning on the part of the writer of a textbook or a mentor's hints. Unfortunately, we often look at these things too early and inadvertently "cast pearls before swine"...ourselves being the swine. If you haven't struggled sufficiently long with a problem, you will not appreciate what the brilliant idea has done for you when you find it. This is why I find myself wondering again about the above approach. If you write up a bunch of problems that are stuck on the same sort of thing, you may even guess the form of the miraculous result you need to solve the outstanding problems. This is almost as good as coming up with the construction yourself!

The bottom line is: keep trying to do as much as you can by yourself. See the answer to my related question here.

Edit: It occurred to me that what is more important than learning a bit of mathematics is, as Thurston put it once, "coloring" it. Thinking hard about some mathematics and not proving it successfully can be more valuable than "knowing" the proof. The mental model and approach you develop is more likely to differ from the standard solution, which may allow it to work in a setting where everyone has already tried the standard solution.

This said, if you believe the above then it doesn't matter if you read a proof if your goal is to find a sharp, vivid mental model for a mathematical construction. If your goal is to have a better understanding of something, you will not stop thinking about it after reading a proof of it. Instead you might try to prove the theorem many different ways (only think of Gauss and Quadratic Reciprocity if you doubt that good mathematicians `waste time' doing this). Even the best proofs of a statement provide a one-sided explanation of a fact. I remember my advisor giving a very long proof of something for which a book had a much simpler proof, however my advisor's proof showed a way to use different intuitive basic assumptions to get to the proof.

I think the competetive nature of mathematics publication can be dangerous to learning mathematics (if not to mathematics research itself) because of its emphasis on quickly resolving the truth of a given statement. For an analogy, if a theorem is a peak in a mountain range then learning a textbook proof of the theorem is like following the shortest path to the summit. There often are many, perhaps longer, ways to the summit, but we stop looking for these once we know the short way. What is good about this is that quickly learning a proof allows us to move along after attaining familiarity with at least one new effective tool. I.e. it is essential to passing on a subject to future generations that ideas be easy to assimilate and verify. Another way to look at this is: textbook proofs are meant to be read. What is bad about the fast method is that other new `trails' are not considered, which may provide new techniques for solving other unsolved problems.

I think that, in supplement to my earlier answer, that reading should be used to "locate" a result that is interesting to research deeply. Once such a topic or problem is found, the objective of the game radically changes. The focus should change from getting to the summit" toknowing the mountain".

It is tempting to propose that subjects admitting glib treatment are already pretty mature and should be viewed as waypoints to more green areas of research. In an active research area it is important to find as many proofs of each fact as possible, in order to uncover which facts are essential. Only after these facts are uncovered is it possible for expositors to go back and arrange things for the purpose of assimilation. I wonder to what extent mathematical exposition is historical work.

The view in the above paragraph is supported by the fact that "Moore method" students are quite good at working on new research problems but not as good at knowing the landscape. For a difficult open problem, it is better to build a complete understanding of the landscape than to try to rush to an answer.

So my suggestion is: read mathematics (proofs included) in order to locate a place to begin a comprehensive study of an area, then sloooow down and work everything out yourself many different ways...building your own proofs and learning those of others, perhaps in the way suggested in my original answer.