Polya, G. (1973) How To Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey: Princeton University Press.

George Polya, or Pólya György, was a Hungarian mathematician who spent much of his career at the Federal Institute of Technology, Zurich (ETH Zurich) and at Stanford University. His is a family which suffered, or courted, difficulty. His father, perhaps best known for his Hungarian translation of Adam Smith’s The Wealth of Nations (Polya, 2006), was a lawyer whose academic ambitions were thwarted by institutional anti-Semitism, until he converted to Roman Catholicism. His brother Eugen (Jenö) was a surgeon, famous for a type of gastrointestinal bypass surgery for the treatment of stomach ulcers, who died at the hands of the Nazis; another gifted brother, Laszlo, was killed in the First World War. His great-nephew is the controversial artist and biochemist, Gideon Polya. George seems to have limited his misfortunes to an early punch-up with a student with royal and high level political connections in Göttingen: fortunately for mathematics, this incident partly led to his appointment at ETH Zurich.
Polya performed poorly in mathematics at school, which he later attributed to bad teaching. It was not until he was at university when, following qualifications in literature, he began to be interested in philosophy, that he began his mathematical studies. Despite this late start, he went on to make contributions in a range of topics so diverse as to call to mind the polymaths of ancient and mediaeval times, in a career that extended well beyond his retirement in 1953: remarkably, he was still teaching at Stanford University in 1978, at the age of 91. From the first, his real interest in mathematics seems to have lain in proof and mathematical discovery (Albers & Alexanderson (1985), in O’Connor & Robertson (2002)). Indeed, it is arguably that this underlies his contributions to mathematical education, for which he is most remembered. This legacy comprises a series of books on problem-solving in mathematics, the first of which, How To Solve It, is the subject of this review.
How To Solve It is an unusual book. There are in fact only 36 pages in all which outline Polya’s method: these form Parts I and II of the book, In The Classroom and How To Solve It – A Dialogue, respectively. These chapters are preceded by a two-page outline of the method, presumably intended as an aide memoire, and an Introduction, which outlines and explains the book’s rather unique structure. Part III, the largest section of the book, is entitled a Short Dictionary of Heuristic, essentially an appendix expounding in detail on the problem-solving techniques and ideas mentioned in Parts I and II, together with potted biographies of a few mathematicians and definitions of a few pertinent terms. The final section, Part IV, is the aptly-titled Problems, Hints, Solutions. Throughout the book, subsections within the chapters are numbered, rather more like a textbook or business document. Initially a little disturbing to the reader expecting something a little more literary, this does however reinforce the fact that this book is very much a guide: a textbook in mathematical guidance.
Of the book, Part I – In The Classroom – is the most important. It is this chapter that sets out the purpose of the book: to help improve and develop problem-solving in students. Five main issues are raised and explored briefly:
  • Helping the student
  • Questions, recommendation, mental operations
  • Generality
  • Common sense
  • Teacher and student. Imitation and practice.
The emphasis is very much on finding problems suited to, yet challenging at, the student’s level of knowledge and ability, and then on unobtrusively guiding the student to a solution. Polya then explains that his model, for reasons of convenience, is split into four phases: understanding the problem; making a plan; carrying out the plan; and looking back. Each phase is then examined in detail, with a running example. Polya recommends a method of a quasi-Socratic, positive questioning – posing a variety of leading questions, in a variety of ways, to steer students gently towards finding ‘their own’ solution. On completion, the solved problem can then be examined – mined, as it were – in the same manner to explore connections to other problems and concepts under the guise of checking, incidentally reinforcing the learning that has just occurred. 
A significant difficulty with the book is the language in which it is written. Originally published in 1945, it has something of the flavour of F. Scott Fitzgerald and the more serious works of P.G. Woodhouse and Dorothy Parker – a convoluted phrasing, an emotionally-detached remoteness, that makes for difficult reading. Polya’s literary and philosophical background shows strongly, although the fact that the text is readable at all by the layman indicates that he must have tried to play both down considerably. Some of the mathematical terminology is outdated: in the first running example, he uses a problem concerning a parallelepiped. Fortunately, I vaguely remembered seeing the –epiped suffix somewhere as meaning a 3D figure, so I did not have to interrupt my reading to look up a dictionary. There is a diagram, but this appears later in the exposition of the problem-solving phases – perhaps too late for another reader. However, I had a difficulty with the printed text: the suggested guidance questions were identified by italicised font, but unfortunately, so are all Polya’s emphases. For the teacher skimming through for ideas, this is unhelpful. Since the edition I read was published in 1973, it is odd that some alternative was not found.
In all, this is an immensely useful book which fully repays the endeavour of reading, despite the difficulties of language. Polya makes a heroic attempt to explain, in albeit condensed and simplified terms, a difficult task. As anyone who has ever attempted to solve a difficult mathematical problem can attest, the thought processes that lead to an answer are proof (sic) against analysis: occasionally lightening fast, at other times tortuously slow, and then again, suddenly productive after perhaps weeks of drought. Polya takes this amorphous thing, and supplies a structure, a logic, and a process from which something might reliably emerge. It is not a book to be read through, short as it is. It is a book that should be kept on a convenient bookshelf, to be referred to, dipped into, and mulled over. Frequently.

References
J J O’Connor, J.J., & Robertson, E.F. (2002). MacTutor: George Pólya. [online]. University of St Andrews: MacTutor. [Cited 26/03/2010].

Polya, G. (1973) How To Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey: Princeton University Press.
Polya, G. (no date). Personal Profile. [online]. Media With Consciousness News . [Cited 26/03/2010].
Polya, G. (2006). Global Avoidable Mortality. [online]. Blogger. [Cited 26/03/2010].
Who Named It? contributors (no date). Eugen (Jenö) Alexander Pólya. [online]. Oslo: Who Named It? [Cited 26/03/2010].

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