I agree; "Don’t just sit and stare at it: think hard; until you’re exhausted; then come back the next day and try again." is bad advice. Don't just think about it, whatever that means to you; just thinking tends to get people's minds running in smaller and smaller circles-which is why going away and coming back the next day helps. But thinking, and working, differently will help you more and help you more quickly. Try diagramming the problem, try working a simplified or more specific example, try generalizing the problem to see if it fits something you learned in another context. There are many things you can do to work on a problem.
Polya's How to Solve It is mostly simpler types of math, it is intended for teacher training after all, but the general methods he demonstrates can often help with much harder and more complicated problems.
Wickelgren's How to Solve Mathematical Problems (originally titled, How to Solve Problems, the new title is more accurate) has both basic and more advanced tactics for problem solving.
The best way I have found to use these types of books, after reading them through quickly for an overview, is to stop and browse in one of them when you get badly stumped on a problem. Then go back to the problem; if you still can't make headway, stop and browse a bit more.
ADDED: Mathematics involves three distinct types of learning and work: learning the mathematical theory, which I generally find fairly easy. Learning and applying problem solving methods to apply theory to actual problems, which is much harder. And doing the calculations to solve the problems once you have worked out how to apply mathematics to the problem, which I find really, really hard, fortunately this is the easiest aspect to automate (calculators and Mathematica, for example).
I might not have been the best at math but I've sometimes been considered a "math whiz" - I took the undergraduate math seminar at UCLA when I was a High School senior.
I've always had the impression that what made people bad at math is the exactly the "grind" attitude - "focusing" on a problem only reduces your creativity. In fact, whenever I took this attitude, I became bad too.
Playing with a given problem every way you can is good. Enjoying a problem and finding it interesting is important. Grasping the concepts is good. Letting a given problem go whenever you can't solve it is good - I think there's a place below conscious awareness where problem solving can happen well.
But don't just grind on mathematics, that's poisonous.
Polya's How to Solve It is mostly simpler types of math, it is intended for teacher training after all, but the general methods he demonstrates can often help with much harder and more complicated problems.
Wickelgren's How to Solve Mathematical Problems (originally titled, How to Solve Problems, the new title is more accurate) has both basic and more advanced tactics for problem solving.
The best way I have found to use these types of books, after reading them through quickly for an overview, is to stop and browse in one of them when you get badly stumped on a problem. Then go back to the problem; if you still can't make headway, stop and browse a bit more.
ADDED: Mathematics involves three distinct types of learning and work: learning the mathematical theory, which I generally find fairly easy. Learning and applying problem solving methods to apply theory to actual problems, which is much harder. And doing the calculations to solve the problems once you have worked out how to apply mathematics to the problem, which I find really, really hard, fortunately this is the easiest aspect to automate (calculators and Mathematica, for example).