How to Solve Mathematical Problems

What Hacker News says

1. Jun8Jul 2015
   The idea of a world class mathematician as as weird genius is common and may be said is not totally without base (Perelman comes to mind). Tao's interactions on his blog is always unassuming, trying to help, "super normal" as this article puts it.

   But to me the even more amazing thing is how good an educator he is. Most mathematicians I have encountered were of the "proof is left as an exercise" type, where if you didn't see how things done would't or couldn't help you in learning how to proceed. Tao is a big outlier in this regard. His How to Solve Mathematical Problems (http://www.amazon.com/Solving-Mathematical-Problems-Personal...) is more focused than Polya's famous book but is a much better help in learning how mathematicians think. You can get the first chapter for free (http://www.math.ucla.edu/~tao/preprints/problem.ps), an excellent way to finish off a Friday.

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2. billswiftApr 2011
   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).

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