Despite the admitted vagueness of your question, here is a modest (slightly wordy) attempt at answering it, and perhaps shedding some light on your linked query as well. My response comes from a background in Mathematics Education; so I look forward to reading answers from those with different backgrounds from my own (e.g., in Pure Mathematics).

**First**, allow me to say a few words about the "underlying issue" with regard to teaching problem solving heuristics. Since Polya, much of the work on the effectiveness of teaching problem solving (e.g., heuristics instruction) has found its way into the research of those in the field of Mathematics Education. A good early piece on this precise topic is Schoenfeld's (1979) "Explicit Heuristic Training as a Variable in Problem-Solving Performance." Shortly after, Schoenfeld (1980) published an article in the AMM entitled "Teaching Problem-Solving Skills". Finally, Schoenfeld (1985) wrote an entire book on this area, entitled "Mathematical problem solving."

I will not delve deeply into these works - instead, I include all references at the end of my response - but overall, there are at least two lessons to be drawn. (In fact, there are many more, but I don't wish this response to become overly prolix.) The first lesson is that teaching students problem-solving heuristics gives mixed results. The second is that part of the reason for the mixed results lies in what goes into problem-solving: Beyond heuristics (which is what Polya's "How to solve it" concerned itself with, principally) other areas of importance include what Schoenfeld refers to as *resources*, *beliefs and belief systems*, and *control and metacognition*. You can find a detailed treatment and definitions of these areas in the 1985 book mentioned above. At the least, it is clear that simply showing students different problem-solving heuristics will not suffice in making them better problem-solvers; to achieve such a goal, one's focus must extend beyond *what* strategies are helpful to, at a minimum, *when* one uses such strategies (often invoking more than one for a single problem).

**Second**, I "met" Polya as a graduate student at Columbia University Teachers College, where I worked as a Teaching Assistant for Phil Smith (cited as J.P. Smith in the Schoenfeld articles and book) who had, during his own graduate studies, worked as a Teaching Assistant for Polya. (If there were some sort of Mathematics Teaching Genealogy, then I suppose I would be one of Polya's grand-TAs.) Smith's Problem-Solving course, a perennial favorite among prospective teachers at CU TC, is based off of the work of Polya and Schoenfeld, but was clearly influenced by others in Mathematics Education as well. Two such researchers are Jeremy Kilpatrick (Smith's advisor; citations of his work can be found in Schoenfeld's articles and book) and Edward A. Silver (Smith's first advisee; his work includes both research on problem-solving and problem-posing). I include these names here since I am aware many in the Mathematics community do not read the Mathematics Education literature; if one should feel so inclined, then good authors to begin with are: Schoenfeld (PhD in Mathematics under Karel DeLeeuw), Kilpatrick (PhD under Begle and Polya), and Silver.

I think the exposure to the work of Polya (in particular) and other problem-solving theorists (in general) has affected both my ability to problem-solve and the way in which I go about teaching Mathematics. Probably this is due less to an increased arsenal of heuristics, and due more to the metacognitive aspect of problem-solving; that is, asking myself *what* I am doing and *why* I am doing it in the process of tackling a problem for which the method of solution is unknown at the outset. (See also Schoenfeld's "What's All the Fuss about Metacognition?") My experience is that the *why* question is sometimes lost in the problem-solving process, for both students and teachers of Mathematics, and that it is a question worth re-visiting as one matures mathematically. Unfortunately, there is an admitted tendency among some in the Mathematics Education community to overemphasize the *why* at the expense of the *what*: for example, to expect students to develop their own algorithms for integer multiplication, and dissuade instructors from teaching these methods explicitly. For me, teaching problem-solving is supposed to equip students with the ability to figure out *how* to broach a mathematical problem (one that is within their grasp or slightly beyond it - known as the Zone of Proximal Development in pedagogical parlance) by allowing them to strike a proper balance between the *whats, whys, whens* and other reasonable questions students might ask themselves should they become stuck.

Alan H. Schoenfeld. Explicit Heuristic Training as a Variable in Problem-Solving Performance.
Journal for Research in Mathematics Education, Vol. 10, No. 3 (May, 1979), pp. 173-187. http://www.jstor.org/stable/748805.

Alan H. Schoenfeld. Teaching Problem-Solving Skills.
The American Mathematical Monthly, Vol. 87, No. 10 (Dec., 1980), pp. 794-805. http://www.jstor.org/stable/2320787.

Alan H. Schoenfeld. Mathematical problem solving. (1985).

Alan H. Schoenfeld. What's All the Fuss About Metacognitlon?. Cognitive science and mathematics education, 189. (1987).