9 ways to get out of being stuck on a problem

Me and friends (from left to right) Sam, Riley, and Kayla, at the annual Northwest Undergraduate Symposium of Mathematics at Willamette University this November.

Pretty much my life lately has been being stuck on math problems.

The transition to graduate school hasn't been quite the wallop in the face I feared it would be. Luckily, my undergrad professors prepared me fairly well for graduate study by giving me problems that I got stuck on frequently. The main difference between graduate and undergrad seems to be that I am stuck for a much longer period of time now, and the progress I make on a homework set or problem in a day seems minimal (and thus extremely discouraging).

So what can you do when you're stuck on a problem? Give up? Well . . . no. At least you should try the problem for a while first. Here are a few things that have helped me when I'm stuck, or have helped me get started on a problem that looks intimidating:

  1. Writing down definitions and theorems. This allows you to examine the tools you've got for solving the problem. If the proof is a follow-your-nose type problem (and not many of those exist the farther you go in your study of mathematics), writing down definitions may be all you need to do for your “light-bulb” moment.
  2. Making up examples. Algebraic-type mathematics have always been the bane of my existence. In particular, linear algebra seems to come to me quite slowly, but I always had a much easier time writing proofs after looking at a lot of definitions and theorems and subsequently verifying that the examples in the book fulfilled the definitions. Then I'd make some examples of my own. Strangely, after doing this, proofs seemed to flow much more easily.
  3. Drawing pictures. When I independently studied select chapters of Walter Rudin's Principles of Mathematical Analysis last spring, I noticed that one of the easiest ways to gather intuition about a problem was to draw a lot of pictures (usually of neighborhoods and limit points). 
  4. Trying proof by contradiction. 
  5. Trying proof by contrapositive. There are some proofs where directly showing something to be true is extremely difficult (such as the second half of the proof that sequential characterization of continuity is equivalent to ε-δ characterization of continuity). I've found on occasion that switching to proof by contrapositive yields the proof quite easily.
  6. Being cynically optimistic. One of my downfalls as a mathematician is how much I question myself (some people call this a good trait, but I'm not sure I believe them). Sometimes, forging ahead regardless of what your skepticism says can be helpful, though. What would you like to have happen? Alternatively, finish this sentence: It would sure be nice if . . . .
  7. Taking a break (as in, not doing math for a while) or working on another problem. Let your subconscious sit for a while. As a senior in undergrad, I found that once I started taking regular breaks, my creativity got so much better.
  8. Appeal to your intuition. This is sort of along the lines of #6. What do you think is true?
  9. Free-write about math. I mean, writers who get writers' block are told to free write about the topic they're stuck on. Why not do it with your math problem too? Set a timer for eight minutes and free-write about the problem you're working on for that amount of time. Don't stop writing until the eight minutes are up!
However, none of these methods are substitutes for the all-important 
* When you need it, ask for help. If your professor is fine with you Googling questions, do it. Or ask your professor for help on the problem during office hours (of course, ask your TA, classmates, an older student, etc. -- but your professor will have the clearest idea of what you need to do to solve that problem). I always know exactly when I'm spinning my wheels on a problem -- I start losing focus and I get very frustrated. When you get to this point, stop and ask for help. 

Comments