Problem-solving speed in undergraduate math courses I am an undergraduate student studying mathematics. So far, I've found that I enjoy assignments with interesting and difficult questions I can think about for a long time, but in a time-sensitive testing environment, I'm not able to perform at the same level.
It is my understanding that mathematics research (and by no means am I suggesting that I am capable of such work) in nature is longer-term and involves much more difficult problems as opposed to anything shorter or time-sensitive.
My question is two-pronged; how can I become better at writing proofs quickly? How can I demonstrate promise in mathematics research for graduate programs despite my inability to do fast work?
 A: I was in the same boat you were in, and I'm frankly still in that boat in terms of calculation speed and proof-writing speed. My recommendation is always ask yourself 'What am I trying to prove?' Always remember definitions and corollaries and work backwards, in the sense that if you are doing an epsilon-delta proof, work from the $\mid f(x)-L\mid<\epsilon$ step...then find your delta. 
A: The fastest resolution/solution/answer to questions is "recollection", not real-time solution. That is, to "speed up", the real trick would be to have already thought about things very similar to the question at hand. Pre-thinking surpasses real-time thinking enormously... and "even" many very "quick" people are all-the-more-quick because they've had the good fortune to have thought relevant things through far in advance.
As to your other question: most professional mathematicians understand that speed in solving little problems means very little. On the other hand, yes, one must provide some evidence for potential to make a worthwhile contribution. All the more if one is not a great test-taker (for whatever reason), one can/must/ought engage in other (presumably more genuine) mathematical activities, and generate corresponding letters of recommendation (not to mention the edification that would come...)
