Reading this post in which Mark Palko quotes from the classic “How to Solve It” by the legendary mathematician and math educator George Polya, I was reminded of my decades-long aversion to Polya, an attitude that might seem odd given that (a) Polya has an excellent reputation, and (b) I’ve never read more than a paragraph of anything he’s written.
Why this attitude? Because I did the training program for the high school math olympiad one summer, and Polya was, like, the god of the people who ran that program. The olympiad program had all sorts of attitudes I couldn’t stand—the one I remember most was that using calculus or Cartesian coordinates to solve geometry problems was considered a form of cheating (see the last two paragraphs of this post, also this), and so I ended up with a negative attitude toward Polya and everyone else associated with school mathematics competitions. Completely unfair, I’m sure. I bet if I were to read Polya now, I’d think he’s great.
I’ve been lurking this blog for a while :)
I’m a software engineer and read How to Solve It recently. I do some interviewing for software jobs (the typical 90’s-Microsoft style software problems that are very similar in structure to math olympiad problems), and after reading the book, a lot of the problem solving skills that interviewees have or lack can be directly tied to the high level ideas in that book. It’s a very quick read – there are essentially two pages breaking down and naming all the steps people tend to take (and often skip) naturally while solving a problem, and then the rest of the book is just elaborating with examples and definitions. It’s well worth it just because it gives names to things that everyone who solves problems already implicitly knows.
It frames everything with geometry problems, but that is pretty arbitrary. I suspect geometry was chosen because it was safe to assume people had a solid grounding in it when the book was written (maybe not so much now).
Polya’s “How to solve it” was one of the best books I ever read. The high level structure it imposes on problem solving has been of immense help ever since.
A related book “How to solve it by computer” by Dromey was also a very good undergrad read.
I think that book is all about how to teach.
> The olympiad program had all sorts of attitudes I couldn’t stand—the one I remember most was that using calculus or Cartesian coordinates to solve geometry problems was considered a form of cheating…
?!? Wow. I would not have guessed that.
A very belated comment on your “Confusing reliability with validity” post linked to above – you wrote “My theory is that students at top universities have succeeded pretty well by being able to solve problems quickly; they haven’t really needed to develop the tools to solve problems systematically by brute force…” I agree. In industry – DoD work in my case – you absolutely need to be able to work systematically (“slog”) through problems which don’t lend themselves to fast, elegant, or even efficient solutions.
Over thirty years ago when I headed off to college I and all the other entering freshmen had to go through a “Writing and Thinking Workshop”. More than anything else it was about developing strategies for overcoming writer’s block. Principally, that amounted to getting over ones self-consciousness about writing crap and just writing. Once you get a bunch of thoughts out then you can review what you wrote, identify ideas which might have merit, and expand upon them. I take a similar approach to technical problems where the solution isn’t apparent to me. (I read a lot in order to avoid reinventing the wheel. Probably 95% of problems I get handed have already solved elsewhere – not literally, but the problems are similar enough that published methods are readily adapted.) When I encounter a problem where I can’t adapt a solution developed for a similar problem then I identify as many seemingly plausible approaches as I can and start trying them out. It can be a slog but eventually you find what works and you get the job done. My Ph.D. advisor used to say, “When you don’t know anything do what you know.” That wasn’t an invitation to do nothing. His suggestion was that if you didn’t know the right way to solve a problem then apply the methods you do know. Observe how those methods fail. Hopefully be able to identify the root cause of the failure which in turn will enable you to develop a proper solution to the problem.
In high school I fell in love with Polya’s Induction and Analogy in Mathematics (http://www.amazon.com/Mathematics-Plausible-Reasoning-Volume-Induction/dp/0691025096). I find How to Solve It too meta for my taste.
Well put. HTSI is nice but over-rated. The longer books are masterpieces. I was never sure what to make of Polya-Szego.
As to Andrew’s memories from MOP, I’ll hazard the guess that the enthusiasm for Polya, and especially the advocacy for classical Euclidean style of proofs in geometry problems, came from Greitzer, who was both a geometry evangelist and a math-education guy. The people who ran the program after him weren’t averse to those things, but did not have overt methodological preferences as far as I could tell.
There was however a practical reason to try a geometrical solution when possible on competitions: the grading of computational solutions is more all-of-nothing, in that they may not have an identifiable idea that can be given partial credit, and if the algebra goes awry somewhere in the chain, all is lost scorewise. Similarly, olympiad grading was unforgiving about using sledgehammers like calculus or Lagrange multipliers that basically trivialized a lot of hard problems, if all the required assumptions and boundary cases were not fully stated and checked. This is also intended to make the IMO (and USAMO) more fair as not everyone knows calculus.
I didn’t have any experience with the Math Olympiad — I don’t even recall hearing about it when I was in high school. (So I checked Wikipedia, and found that it started in 1959, when I was in high school; apparently it hadn’t made it to the wilds of Michigan by then.) But there was the Michigan Mathematics Prize Competition, which I participated in at least twice. (I looked that up, too — it started in 1956, which was before I was in high school, as a project of the Michigan chapter of the Mathematical Association of America) But there weren’t any prep courses, and there wasn’t a competitive atmosphere — just fun.
In college, there was the Putnam Exam (which started in 1938). I took it a couple of times, but don’t think there were any any prep course for it. However, there did seem to be an attitude that was too competitive for my tastes, so I stopped participating because of that.
As best I recall, my first experience with Polya was when I decided to teach a problem solving course as a freshman seminar for an honors liberal arts program — I would guess around 1980. I did find Polya a good source for “outlining” the problem solving process, and a good source of examples. I also found Alan Schoenfeld’s book Mathematical Problem Solving helpful, especially when I later taught a problem-solving course for future math teachers. It drew a lot from Polya’s books, but gave additional examples, and also emphasized that the problem solving process didn’t always go through Polya’s steps in lockstep manner; that cycling back was often necessary. There was also a myriad of other books that were good sources of problems.
My approach in teaching problem solving courses, for whatever audience, did not push competitiveness — I encouraged students to work together (but to give credit when credit was due and write up solutions individually — no copying!), and also emphasized that it’s not enough to just solve a problem; you also need to think about explaining your solution. I also deliberately tried to give problems that had more than one means of solution, to get students out of thinking that there is always “one right way” to solve a problem — especially the future teachers. I had students present their problems in class, and asked the class if they thought the solution was correct and if they understood each step. I also asked, “can anyone do it a different way?” Inevitably, there were occasions where two students presenting different (but correct) solutions would each say of the other’s solution, “I never would have thought of doing it that way!”
“using calculus or Cartesian coordinates to solve geometry problems was considered a form of cheating”
Differs from my memory, but we weren’t there at the same time. I _always_ used coordinates and didn’t see the point of mucking around trying to find the right microfact about the circumcenter to use when the algebraic manifestation of the problem was right there to be worked on
I think the graders hated to check long algebraic calculations, especially if a lot of students went the brute force route, and some of the instructors felt that solutions that don’t build up a series of intelligible geometric lemmas indicate a lack of geometric intuition or experience that MOP hoped to remedy. I shared your disinterest in “microfacts”, partly because it was hard to judge what was important and what was not when such facts were presented, but I found it almost always illuminating to approach the problems geometrically.
Vector algebra (dot and cross product — almost nobody in the 1980’s had heard of multilinear algebra) and complex numbers were a middle ground that allowed solution-by-calculation, but more intelligibly and retaining more of the geometry of the problem than Cartesian coordinates.
Yeah I don’t know. I’ve had the opportunity to view his papers from his lecture notes and such at Stanford Library and find him to be endearing. He’s great.
I think we should focus all our hate on Hilbert.
It should be mentioned that Polya is viewed differently in math and math-education circles. As a mathematician he was an accomplished guy and one of the last major exponents of analysis in the classical style that he lived long enough to see go out of fashion, for which Polya-Szego is a kind of textbook. Pretty well known but not a superhero or major historic figure, more a solid, well known and well liked mathematician who stayed active for a very long time.
Whereas in math education circles, Polya seems to be a sort of god, maybe the only one to have that status, a founder of the field and the one who set the fashion for decades even unto the present. Not only was MOP in your day run by a math-education guy (Greitzer), but Polya was one of the few to write about mathematical problem solving as a subject, often using competition-style exercises as cases to study. I think he was involved in bringing the Hungarian math competition tradition to the Stanford area by setting up a regional olympiad. So in various ways he was an important precursor and philosophical influence on MOP as well as an author whose books should be read simply because no alternatives existed. These days there is a much larger literature focused more directly on math contest problems and Polya’s stuff might be a bit quaint. But back then it was a kind of high culture behind the play.
Pingback: Jordan Ellenberg’s new book, “Shape” « Statistical Modeling, Causal Inference, and Social Science