Wednesday, June 2, 2010

Who Was George Polya's Intended Audience? (Or More Mathematically Correct Lies)

George Polya, c. 1973

One of the more difficult aspects of wars, even ones where the main ammunition is words, is separating lies from facts. Every side in a war has a proclivity for propaganda. Inconvenient facts are brushed aside. Inaccuracies, petty or gross, become the coin of the realm. The Big Lie rules. 

Of course, sometimes, it is possible to sort through the fog of war to arrive at what appears to be incontrovertible truth. It may take years, even decades, to find the facts, even when they are readily available to anyone who bothers to look in the right place for them. Sometimes, they've been staring everyone in the face for a very long time. 

Thus, it is with no small embarrassment that I present a long-overdue and clearly definitive retort to one of the lies frequently promulgated a decade or so ago by Professor Wayne Bishop and some of his Mathematically Correct and HOLD anti-progressive allies, namely that George Polya's work on heuristic methods (from the Greek "Εὑρίσκω" for "find" or "discover": an adjective for experience-based techniques that help in problem solving, learning and discovery) was intended only for graduate students or perhaps undergraduate mathematics majors, not for the general student of mathematics, and certainly not for high school students or younger children. 

Of course, in the Math Wars, it is of the utmost importance to the counter-revolutionaries and anti-progressives that nothing that broadens access to mathematics be allowed to stand unchallenged or unsullied. Any curriculum, pedagogy, tool, etc., that is brought forward by reformers as "worth trying" must be smashed. That has been the tireless task of members of groups like Mathematically Correct and HOLD: to undermine any and all efforts to change what they view as immutable approaches to the teaching and learning of mathematics.

It's almost as if they were the American Medical Association, fearful that if too many people get into medical school - indeed, if there are no arbitrary, meaningless gates, such as requiring a full college calculus sequence, put up to block the pathway to the profession - some of their members who managed to get through the gauntlet but who in fact are not all that good at being actual doctors might suddenly be threatened by "others" who happen to have all the requisite skills, including some that these doctors lack. Such folks are inclined to argue that the established path is absolutely correct, the ONLY reasonable one that could possibly be allowed. Anything else would clearly be "fuzzy," "unscientific," "watered-down," etc. Even when, in fact, the main difference might be an emphasis on people skills, psychology, and perhaps looking critically but with open-mindedness at various sorts of holistic, non-Western, and other alternative medical approaches. If the goal is to help as many patients as effectively as possible, what would be the harm in looking scientifically at alternatives? It has been known to happen that methods once dismissed by mainstream science turned out to be highly effective (for one such example, look at the work on treating infantile paralysis by Sister Kenny). 

Instances of complete dismissal of a wide variety of innovations or, as in the case of lattice multiplication, the return to an older, mathematically valid algorithm (see "Looking Further At Multiplication" and "Who Invented Lattice Multiplication?" , are legion in the 'work' of these educational reactionaries and conservatives. But one particularly amazing instance surrounds the work on heuristics by Polya, author of several books on the subject, most famously HOW TO SOLVE IT: A NEW ASPECT OF MATHEMATICAL METHOD, first published in 1945, followed in 1954 by Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics, and Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Reasoning, and in 1965 by the two-volume Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving.

I am hardly alone in suggesting that the above work, while many of the examples geared to undergraduate and graduate students of mathematics, has many deep implications for earlier mathematics education. I cited both the books of Polya and the more famous of the videos of his teaching, LET US TEACH GUESSING in support of the notion that K-12 teachers and there students would gain much from considering and making use of Polya's approach to problem-solving, and that grounding K-12 curricula in this approach would be an improvement over business as usual. (See David Bressoud's 2007 MAA column "Polya's Art of Guessing" for more details on part of what Polya is up to in that video). 

My notions were fiercely rejected by Wayne Bishop and others. They denied that Polya was thinking in any way about seeing his methods used in K-12 education and that it would be disastrous to introduce such methods into the public school curriculum, particularly in lieu of teaching traditional algorithms (it's remarkable how everything in the Math Wars comes down to 'either/or' in the hands of the MC/HOLD crowd. The notion of "as well" seems unknown to them). 

Well, let me cut to the chase. Here is what Polya himself says at the end of the introduction to HOW TO SOLVE IT: 

We have mentioned repeatedly the "student" and the "teacher" and we shall refer to them again and again. It may be good to observe that the "student" may be a high school student, or a college student, or anyone else who is studying mathematics. Also the "teacher" may be a high school teacher, or a college instructor, or anyone interested in the technique of teaching mathematics. The author looks at the situation sometimes from the point of view of the student and sometimes from that of the teacher (the latter case in proponderant in the first part). Yet most of the time (especially in the third part) the point of view is that of a person who is neither teacher nor student but anxious to solve the problem before him. pp. xx - xxi [emphasis added]

Well, slap my face and call me "Susan," but it surely appears that Polya is at minimum open to seeing his method used in high school, though it's not outlandish to suggest that he is suggesting that it's appropriate in some form for any student and any teacher of mathematics, as well as those who are neither.

But perhaps I'm just overstating the case. Maybe heuristic methods are just too daring, too advanced, too non-traditional, surely TOO SOMETHING! to be risked with our K-8 students and faculty. Skies may fall, dams may break, heads may explode, should we try making problem-solving methods a major foundation of our mathematics curriculum, rather than calculation, as has far too long been the case in this country.

Well, here's an interesting bit of evidence from a wonderful article by Tibor Frank, "George Pólya and the Heuristic Tradition Fascination with Genius in Central Europe" in which he explores the general intellectual traditions of Hungary in the period from which Polya, Von Neumann, and many other brilliant mathematicians and physicists emerged:

[He]uristic thinking was also a common tradition that many other Hungarian mathematicians and scientists shared. John Von Neumann‘s brother remembered the mathematician‘s „heuristic insights” as a specific feature that evolved during his Hungarian childhood and appeared explicity in the work of the mature scientist.

Von Neumann‘s famous high school director, physics professor Sándor Mikola [note: another of Mikola's students was the Nobel physicist Eugene Paul (Jeno) Wigner], made a special effort to introduce heuristic thinking in the elementary school curriculum in Hungary already in the 1900s

So it may not in fact be stretching anything at all to say that the intellectual and educational tradition out of which came thinkers like Polya favored the heuristic approach to mathematics education for ALL students, at any grade level or stage of growth.

Generally, of course, the rebuttal that is leveled by educational conservatives to mention of such things is that the students that Polya and others had in mind were "math people." That is to say, they were students who came prepared to do serious mathematics, already had mastered the basics, and were already showing the necessary mathematical interest and 'talent' for doing higher-level thinking and mathematical problem solving.

The problems with such a claim are two-fold: first, there is no evidence that Polya or Mikola or anyone in Hungary or anywhere else who promoted these notions was looking at a narrow, highly-gifted group. But second, and perhaps more importantly, even if such were the case, that does not prevent American mathematics teachers, teach-educators, and researchers from considering how to implement heuristic approaches into their teaching. It does not mean that such approaches are "verbotten" other than in the conservative minds of anti-progressives. And given that such people seem utterly closed to ANY innovation, any change, any departure from what they assert produced a "Golden Age" where all American kids learned WAY more of EVERYTHING than do today's kids ("See?" the self-serving story goes, "Back in MY day, teachers really taught rigorous content and EVERYONE learned it; today's teachers are slackers and today's kids even more so.")

Can we afford to buy such arrant nonsense without close examination of the facts and of the motives of those who are purveying this bizarre version of history? I, for one, think we cannot. I believe that George Polya, John Von Neumann, and many of their contemporaries and instructors would agree. But do read the source material and decide for yourself. Then check out Dan Meyer and other teachers who aren't waiting for the approval of Mathematically Correct, HOLD, or any other dinosaur or nay-sayer.

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