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I guess most of us didn't meet Polya in person (this is the answer to the title)! Perhaps, it is much easier to guess that most of us have met one of his writings (or alike) on problem solving, and in particular, heuristic strategies. The question is: When did you first encounter the idea of problem-solving heuristics (cf. Polya) and has it affected you in your own problem-solving? (This version of the question has been suggested by Benjamin Dickman here. For the old version, see the revisions of the post.)

On the surface, this question is an attempt to get an indirect answer for my previous question: Is problem solving a subject to be taught? At a deeper level, this is an indirect attempt to shed light on an old debate (i.e. Is problem solving a subject to be taught?) without doing a debate.

PS. The underlying issue I try to understand is whether teaching heuristic strategies hinders or promotes the kind of thinking as described by Hadamard when he thinks of the infinitude of primes:

I see a confused mass...I imagine a point rather remote from the confused mass...I see a second point a little beyond the first...I see a place somewhere between the confused mass and the first point.

PPS. Another issue that is quite related to the first is whether using heuristic strategies is a conscious process while solving a problem. Perhaps a good way to approach this issue is to see if there is any serious-for-you problem that you solved while being consciously aware of heuristic strategies. For example, did you ever asked yourself "Could I imagine a more accessible related problem?". Of course, this is exaggerating the matter, but it may shed light of the role of wordy heuristic strategies.

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closed as off-topic by Mark Sapir, Steven Landsburg, Eric Wofsey, Chris Godsil, Andrey Rekalo Jun 29 '13 at 8:15

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I met Polya in 1972, at Stanford. –  Gerry Myerson Jun 28 '13 at 23:17
    
@GerryMyerson Wow! I guess this is the most concrete answer I would get for this question :) –  Amir Asghari Jun 28 '13 at 23:32
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Shouldn't this be CW? –  Todd Trimble Jun 29 '13 at 1:10
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Please read meta.mathoverflow.net/questions/285/… to find my reaction for being on hold! –  Amir Asghari Jun 29 '13 at 13:12
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4 Answers 4

I met Pólya once in person, at his home in Palo Alto in 1975, where he showed me some of his photo albums and discussed some mathematical problems. Richard Wilson was also there. Pólya was a very charming host. Even before then I was a great admirer of Aufgaben und Lehrsätze aus der Analysis. It was one of my inspirations for including lots of solved exercises in my books Enumerative Combinatorics, vols. 1 and 2.

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As a high school student, I read his "Mathematical Discovery". Then, as a freshman, I met my future adviser (A. A. Goldberg) and he recommended "Problems and Theorems in Analysis", saying that this book is "the basis of all his scholarship". For many years, he had a seminar for undergraduate students based on this book (in Lvov University, in 1970-s). He was a great practitioner of Polya's teaching methods.

Later I bought 2 volumes of Polya's collected works, and still looking for the third volume. Polya substantially influenced my own mathematics, and I am especially proud of proving one of his conjectures. I never met him personally.

But only concrete problems attracted me in Polya's books. His general considerations on "how to solve a problem" I always found boring, and never really read the second part of his "Mathematics and Plausible reasoning".

That's why I am very skeptical about a "course on problem solving" with any theory of "problem solving". I think one can learn solving problems only by solving concrete problems.

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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).

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@BenjamiDickman Dear Benjamin. Thank you for the answer. It is now for years that I am teaching a course called "theories of problem solving", an important part of which is reading Schoenfeld' book and his articles and also other people's article. Having said so, I really appreciate you have mentioned that literature since many in MO do not tend to read it. It was why I tried to use a jargon-free language to express my question. In particular I referred to Hadamard' work (The Psychology of Invention in the Mathematical Field) since he uses a language more familiar for MO viewers and –  Amir Asghari Jun 29 '13 at 17:09
    
more importantly, suggests a complete different viewpoint (from Schoenfeld and alike) that I think somehow shows the framework you have mentioned is not working for all. I am very interested in continuing this discussion. But MO is not a good fit for that. Perhaps we may keep in touch by e-mail. –  Amir Asghari Jun 29 '13 at 17:16
    
@AmirAsghari My email address can be found on my About page. I read all messages as soon as I see them, though I make no guarantees about response time! As for the MO-fit of Math-Ed questions: I hope that there will be a place here for well-worded queries. –  Benjamin Dickman Jul 3 '13 at 22:55
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I never met Pólya in person, but I met his and Szegő's book, "Aufgaben und Lehrsätze aus der Analysis," as an undergraduate at the University of Detroit, probably in 1964. The person responsible for the meeting was Prof. Časlav Stanojević, who taught me a reading course on whatever came to mind, and this book was an important part of that course. I don't remember how much of the book I got through, but it certainly influenced my general mathematical development.

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I also tried the book you mentioned as an undergraduate. But, it is not about teaching problem solving by teaching heuristic strategies. Am I right? –  Amir Asghari Jun 28 '13 at 23:39
    
I don't think heuristics are explicitly mentioned, but if one reads the answers in the book, one will pickup some heuristic ideas. –  Andreas Blass Jun 29 '13 at 0:16
    
(There you have the diacritics) –  Mariano Suárez-Alvarez Jun 29 '13 at 0:57
    
@MarianoSuárez-Alvarez Thanks. –  Andreas Blass Jun 29 '13 at 0:59
    
This tool shapecatcher.com will eventually be useful to find unicode characters. It still needs to be trained though... –  François G. Dorais Jun 29 '13 at 15:38
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