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I wonder if anyone in graduate school in mathematics had managed to improve considerably his/her speed of problem solving. I had failed to get my PhD pass and obtained only Master Pass on my qualifiers. I have been trying to increase speed of solving problems but alas, achieved only slight improvement. I know that most problems on these exams are manageable and may be few are hard. Problems of similar level of difficulty would take me days or even weeks of solving. Any suggestions and especially real life examples of improving problem solving speed would be highly appreciated.

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Success in well-designed qualifying exams in mathematics will depend very little on "problem-solving" speed or talent, but, rather, will depend on whether you've already done problems nearly identical to the problems which appear.

That is, as you observe, it would be prohibitively slow to "solve" many of those problems "in real time". Thus, that is not the expectation. Rather, the "test" is whether examinees have studied sufficiently broadly so as to have seen examples resembling the instances occurring in the given exam.

And, no, it's not about "memorization", either, which tends to be insufficiently flexible to allow easy adaptation to slightly changed situations.

So, really, it's not about "speeding up in problem-solving", but to be able to merely "remember" instead of "solving".

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    Then I imply that about 3 to 5 thousand problems solved from major graduate textbooks on few main subject areas would do the trick of speeding up on exams:) wouldn't you agree? :) – TooOldToLearn Dec 14 '13 at 19:15
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    @TooOldToLearn, ah, then the point is that it is also infeasible to look at such a body of material in an unstructured way, the point being that one is compelled to take a "higher" viewpoint, rather than looking at seemingly disjoint individual problems. And, yes, it takes time to see the commonalities that are not obvious. Thus, yes, what is being tested is somewhat subtler and different from what might be perceived. – paul garrett Dec 14 '13 at 19:18
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    Yes I agree, the joy of reaching "higher" viewpoint is the real motivation behind solving math problems, it is just the relative scarcity of that joy that concerns me. But then again, there is a plain universal answer "do more problems" that in some case does not seem to work. – TooOldToLearn Dec 14 '13 at 19:35
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    @TooOldToLearn, indeed, "doing more problems" is itself not productive beyond a certain point, but people like that prescription because it is a simple answer (even if partly incorrect), and seemingly translates "failure" into the moral failing of mere "laziness". – paul garrett Dec 14 '13 at 21:42
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I don't think speed of problem solving is a criteria for graduate school in mathematics. You're doing research and inventing new mathematics, and that usually requires slowing down, not speeding up. If you're in a graduate program where not solving problems quickly leads to failure, I'd say you're not in the right program.

This is not to say that slower is better. If you're unable to solve problems relating to your subject area, that might point to deeper problems. But speed of problem solving should never be a goal (unless you're competing in a math competition like the Putnam)

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    Before creating new mathematics, graduate schools expect their students to show abilities to do so. This is what I refer to. – TooOldToLearn Dec 14 '13 at 19:13
  • indeed. But I'm still not sure where speed comes into the picture. – Suresh Dec 14 '13 at 19:27

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