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I am looking for strategies for catching mistakes in graduate exams eg. qualifying.

The more people suggest, the better because what is obvious to you, might not be to others.

Most of the advice in this site and others is about computational mistakes. But my most usual mistake is writing down a conceptual proof I thought is correct, only to later find it is not. Had I found it, I would have worked out the correct one.

Here are some strategies of mine

1)Attitude

"I have found that the best way to cope is to change my attitude. First of all, when I get an answer to a problem, I do not assume the answer is correct. I assume, rather, that I did, in fact, make a mistake and am now in the editing phase of the problem solution. How one edits depends on the problem to be solved. "

2) Example

Test your proof with an example. Draw pictures.

3)Hypotheses

Did you prove something stronger than you were supposed to? Draw any conclusions from your method of proof. Do you get any contradictions?

4) Condense

Condensing a proof, forces you to understand it better and thus to find mistakes.

5)Share

As you are preparing for the exam share your practice-work with others, so they can catch your mistakes.

6) Write in detail

When coming up with proofs, write them in detail so that the mistakes become more obvious (compared to when holding a proof in your head while checking for mistakes).

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    $\begingroup$ Sharing usually won't be welcome in exams ;) $\endgroup$ – AlexR Dec 8 '14 at 19:08
  • $\begingroup$ lol, I meant before. $\endgroup$ – TKM Dec 8 '14 at 19:08
  • $\begingroup$ I figured so, but this wouldn't really qualify for a "strateg[y] for catching mistakes in graduate exams" :P $\endgroup$ – AlexR Dec 8 '14 at 19:12
  • $\begingroup$ yeah you are right. I just felt it was important to mention. $\endgroup$ – TKM Dec 8 '14 at 19:13
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Some additional hints:

  • It's easy to oversee degenerate cases (edge cases). Are there any for your problem?
  • When forming a strategy: What could make the claim false (i.e. why should certain requirements be necessary)
    If your proof doesn't use these necessary conditions, it must contain an error
  • While working on the proof: Is the amount of work I put into this realistic given a specific time frame for the solution? If you take too long, there might be an easier way. If you were too fast you must either be a genious or you overlooked something
  • Double-check any requirements for theorem X if it seems too convenient. For example the directional derivative is only equal to the inner product with the gradient iff it exists and is continuous in a neighborhood of the point.
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