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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?

TheProofIsTrivium
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    The short answer: practice. And remember that speed is not always the best strategy, but rather detail and clarity of argument. – Sean Roberson Jul 24 '17 at 20:50
  • A lot of test level problems and some assignments are precisely formulated, and are designed to check if you understand some topic. My general advice is to check if you used every hipothesis in the statement of the problem, and to develop some intuition for knowing what are they asking you to do. – omega-stable Jul 24 '17 at 20:53
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    chug a red bull and eat a theorem sheet before the test – Saketh Malyala Jul 24 '17 at 20:54
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    Mmmm ... chocolate covered theorem sheets! – GEdgar Jul 24 '17 at 20:58
  • @SeanRoberson I guess many MSE-answers are very short because people want to have an 'early answer' which sacrifices details: This is a speed example. – Felix Marin Jul 25 '17 at 02:13

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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.

JohnColtraneisJC
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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...)

paul garrett
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  • +1. "That is, to "speed up", the real trick would be to have already thought about things very similar to the question at hand." Great secrete that students seldom know! –  Jul 30 '17 at 19:55