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I seem to find it takes me a lot of time and work compared to my classmates to solve problems or "get" certain proofs. To add insult to injury, often I find a question not asked in the textbook or homeworks, but feel the need to answer it, and create an "unassigned" problem for myself. I tend to be very persistent at solving problems, even if it takes me while, but that also costs me my grade in the course,. I face a decision of whether to quit the problem or not, and ask myself how much time should I be spending on my problems?

My question in general is, what is the maximum amount of time you spend on a math problem before deciding to take a "break"? (which may be a < 1 day break, or weeks). What's the longest break you've ever taken from a math problem,? and when would you (or have you) ever decided to completely give up on a problem?

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up vote 4 down vote accepted

Alan Schoenfeld has written a fair amount about mathematical problem solving; you may be interested in some of his work. This particular question falls under the category of what he would call "control decisions." The question should not only be how much time you spend on a math problem, but also how much time you spend on a particular approach as you try to solve said problem.

Schoenfeld also writes about beliefs, and how many students think that math problems should be solved quickly or not at all. This is (as you might well guess) a belief that associates negatively with mathematical problem solving performance.

My advice to you is to try different strategies when you are stagnating. George Polya has a long list of strategies ("heuristics") for problem solving, including: try a special case; try a simpler problem; try a similar problem; generalize; draw a picture. For every question that you cannot solve, there is a slightly easier question that you can't solve either! Mess around with things until you gain some traction and have a better sense of how to broach the problem at hand.

With regard to time spent: don't let your grades suffer too much, since this will probably lead to undesirable consequences down the road. The amount of time I spend on a problem depends on why I'm trying to solve it. If it's just for fun, then I work on it when I have free time; often you need a period of incubation (a break from work) before you can have a major insight. If it's a problem set for a class or perhaps a take-home test, then you probably should spend more time on it (though breaks are still necessary!).

Persistence and curiosity are important characteristics for any math student or mathematician, but don't lose focus and let your course grade suffer too much as a result. (And as for your classmates who are seemingly solving things much more quickly: pay them no mind.)

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A deserverved upvote from me. Just waiting to see if this will get any other answers before choosing. –  Not a NaN notha Oct 26 '12 at 1:00
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If you start reading Polya/Schoenfeld and find some of their stuff particularly interesting, let me know what and I'll try to find some more relevant literature for you. –  Benjamin Dickman Oct 27 '12 at 4:30
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Just to be concrete -- if you're in a situation where it's difficult to finish homework assignments on time, I think it seems about right to work on a problem for an hour or two before getting help from classmates, office hours, or the internet. (When you don't have to worry about deadlines, you can spend more time on problems if it seems fun/worthwhile.)

I could be completely wrong though.

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The question of how long to spend on a problem before asking for help did cross my mind, but I decided not to make the question more general. –  Not a NaN notha Oct 25 '12 at 4:54
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