Thanks for the distillation. Polya's been on my reading list for a long time, but is still not nearing the top, and I appreciate the useful preview.
Very cool. Some of those questions seem a little redundant, such as:
Have you seen it in another form?
An analogous problem?
Perhaps not the same, but reading the list made me wonder if it could be "simmered" a bit to distill the key points. In particular, I really liked the Looking Back section. Absolutely wonderful. It reminds me of my own post as well as many other LW posts: not attacking the strong points of a theory, but the weakest, being careful to avoid leaky generalizations, really knowing the purpose of your actions, internalizing vs. parroting, and not being so quick to assume you've thought of all the options.
I think the last section is a great set of questions to ask after coming to any decision and is certainly not isolated to mathematics! It, combined with the rest, seems like a nice recipe for both internalizing one's methods and data as well as trying to avoid duplicating efforts on related/similar issues. Thanks for sharing.
Very cool. Some of those questions seem a little redundant, such as:
Have you seen it in another form?
An analogous problem?
These aren't redundant in the context that Polya is talking about. In math, these are different. The first means the same problem but with different notation or some equivalent problem. The second means a problem that is similar in some way (say for example something over a finite field having an analog over the real numbers or rationals.)
I appreciate the explanation, especially when considering a math context (which is the intended context anyway, but I was thinking generally with my comment).
Related to: Tips and Tricks for Answering Hard Questions
In How To Solve It Polya describes methods and heuristics intended to facilitate the solution of math problems. These are mostly conveyed in the form of self-questions that are aimed at inducing useful mental procedures, and subsequently developing awesome problem solving dispositions. Ultimately we should work from these dispositions directly. Polya advises us to use the questions only when progress is blocked; at other times our thoughts should flow naturally from our dispositions. I expect that his methods are useful outside of mathematics, and thought they might be of interest to people here.
Below is the summary given at the start of How To Solve It (with the exception of a few added notes). He breaks the problem solving process into four steps, with each step having a set of self-questions and heuristics. I've bolded parts that I thought were particularly useful. This is not meant to be an alternative to reading the book; I expect that reading his illustrative examples is somewhat important. But more important is working with these questions on problems in order to develop your own dispositions.
Understanding the problem
You have to understand the problem.
Devising a plan
Find the connection between the data and the unknown. You may need to consider auxiliary problems. You should eventually obtain a plan of the solution.
Carrying out the plan
Carry out your plan.
Looking back
Examine the solution obtained.