# Methodology for solving graduate mathematics problems

Recently, I have been introduced to graduate mathematics, and I am doing graduate courses, in these courses we have homework and the problems tend to be challenging.

In comparison to undergraduate mathematics I find that for a reasonable amount of problems I usually think a little bit on them, say $$10$$ minutes, and if I don't reach any conclusion I go see references to see maybe propositions or theorems or other similar exercises that may help me solve the homework. (I don't search for the actual resolutions)

My question is : Is going for references a bad strategy to solve problems, pedagogically, or should I happily continue with this practice?

• Depends how confident you feel with the course material required to solve the exercises. At the start of graduate studies that's usually limited. 10 minutes of thinking about homework is nothing. Learn patience. Go for a walk. Personally I prefer to find all solutions myself rather than seeing them from others. Feb 5 at 20:43
• My opinion is that if these are exercises from a textbook, then looking back in earlier sections of the textbook, even right away, is fine (since part of the process is becoming familiar with the topics), but looking in OTHER references/books should wait for much longer than 10 minutes, maybe not until your 2nd or 3rd sit-down attempt at the exercise (going on to other exercises when you can't solve one, and coming back to the one you couldn't solve at a later time, such as the next day). Feb 5 at 20:44

The answer will of course depend on the kind of problems being assigned, but this is not the worst strategy - especially if you solve many problems in this way. However, here are some suggestions for adapting it.

First, you should not need to consult references for routine exercises.

This is deliberately vague - it is not always clear what a "routine" exercise is. So interpret it yourself.

But the point of routine exercises is to get to the point where you've internalized the method so that you can build on it to solve more complicated problems. If you haven't internalized the method - if you need to follow along with an example every time you encounter a problem of this type - then you will have difficulty later when you're solving a more complicated problem where this one is just a single step.

Of course if you cannot solve a problem, you should review until you can, rather than stare blankly at it. But the goal here is to be able to solve the problem without review.

For trickier exercises, read references first (in addition to what you're doing).

I am imagining here a serious graduate-level book which presents some new techniques, some examples of using them, and then some hard problems: such as Alon and Spencer's Probabilistic Method, for instance. Even with the book right in front of you, it will not be obvious how to solve a problem - but finding a similar example or a relevant-seeming theorem is often a big hint.

What I suggest is going through the preceding material carefully before you begin working on the problems. The goal is not to memorize every theorem and every example so you never need to look at them again. However, the goal is to become familiar with the theorems and examples.

When you get to a tricky exercise that you cannot do right away, your thoughts will - ideally - not be "let me look through the references to see if I find something similar", but rather "this looks like the example with the smeerps somewhere in the applications of the Hyperbolic Fibonacci Theorem, let me look at that again".

• Why is this what you want? Because that's how you'll use this knowledge in the future. Years later, the advantage of having studied this graduate course will be that when you encounter a similar example, you'll think "Oh, I bet we can apply the Hyperbolic Fibonacci Theorem here" and have a good idea of what to try - or where to look if you're stuck.
• Why read first? Because otherwise, you'll get a disjointed view of the topic skewed towards the exercises you happened to do. Usually, one cannot write (or solve) enough exercises to cover everything uniformly. Think of the exercises as a test to see if you read carefully enough: if one of them looks completely unfamiliar when you get to it, perhaps you didn't absorb the relevant material after all.

Avoid consulting outside references until much later.

This doesn't mean "never do it". But looking at other sources may confuse you in a number of ways:

• Maybe, despite your best efforts, you find something that trivializes the problem;
• Maybe the other source presents material differently and you'll end up with a mixed and incorrect understanding;
• Maybe the exercise meant to guide you towards coming up with an idea yourself, so that you internalize it better.

Before you get to this point, you should work for considerably more than 10 minutes, employing advanced techniques such as "put the problem down and do something else for a bit first" and "skip ahead and come back to this one later".