How to Solve It · Lablypedia

Four phases

Pólya begins with a two-page checklist that identifies four phases to solving a mathematical problem:

Understand the problem.
Make a plan.
Carry out the plan.
Look back.

He emphasizes that the divisions between phases are not rigid, and it is important to be flexible in one's approach:

Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution.

First phase: Understand the problem

Pólya writes that students are often stymied in their efforts to solve a problem simply because they do not understand it fully. To address this situation, he recommends that the teacher prompt students with a series of questions which will lead them to a deeper grasp of the problem. He says the key initial questions to ask are: "What is the unknown?", "What are the data?", and "What is the condition?" (meaning, "By what condition is the unknown linked to the data?").

Second phase: Devise a plan

Pólya considers this the most important and challenging phase, acknowledging that the process of devising a workable plan "may be long and tortuous":

[T]he main achievement in the solution of a problem is to conceive the idea of a plan. This idea may emerge gradually. Or, after apparently unsuccessful trials and a period of hesitation, it may occur suddenly, in a flash, as a "bright idea". The best that the teacher can do for the student is to procure for him, by unobtrusive help, a bright idea.

The bulk of the book is devoted to offering suggestions to stimulate such an idea. Among the listed suggestions (and these are covered in more detail in the book's heuristics dictionary) are:

Guess and test
Solve a related but simpler problem
Use symmetry, i.e., look for interchangeable parts in the problem
Use a model
Look for a pattern
Examine special cases
Work backwards
Be creative

Pólya adds that applying these suggestions requires skill and judgment, which is best learned by solving many problems. He emphasizes the role of the teacher who should pose questions that help students devise plans. Pólya differentiates between asking a thought-provoking question such as "Do you know a related problem?" versus asking a leading question such as "Could you apply the theorem of Pythagoras?" In his view, the latter question, though well-intentioned, is not instructive because the student does not know how the teacher arrived at it.

Third phase: Carry out the plan

Pólya regards this phase as markedly easier than devising a plan. What is mainly needed is patience:

The plan gives a general outline; we have to convince ourselves that the details fit into the outline, and so we have to examine the details one after the other, patiently, till everything is perfectly clear, and no obscure corner remains in which an error could be hidden.

Fourth phase: Look back

After students solve a problem, Pólya urges them to pause and examine the result and the process which led to it. First, check the answer to ensure it makes sense in the context of the problem. Does the solution use all the data that was given? Students should explore whether an alternative strategy might have been used. They should verify the solution by double-checking their calculations and reasoning.

Pólya also advocates taking time to reflect on what was done, what worked and what did not, what could have been done better, and other problems where this solution might be useful. Doing so will enable the student to choose an appropriate strategy for future problems that resemble the current one.

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Heuristics dictionary

The book's largest section consists of a heuristics dictionary. It contains 67 entries with suggestions "for making progress on difficult problems". Many suggestions are aimed at finding a more accessible, related problem when a student is struggling to solve the original problem. For example:

Reception

Pólya's book was praised for its usefulness in mathematics education and in education more broadly. The Mathematical Gazette said the book was "primarily for students and teachers of mathematics, but it might interest any educated teacher of any subject which does not consist in the mere accumulation of knowledge. Its appeal would perhaps be greatest at the Training College level, and although it contains very little that a good teacher would not discover for himself, it might very well accelerate the process of discovery." E. T. Bell labeled the book "an instructive exposition of the heuristic method applied to the solution of problems in elementary mathematics.... If heuristic is no longer taught, How to Solve It may supply the deficiency. Every prospective teacher should read it." In a thumbnail review in the Chicago Tribune, How to Solve It was called "a disarmingly elementary book on the solution of problems. Largely illustrated by mathematics, but method is applicable in principle to problems in science, engineering, or social work. Of great value to every one."

In American Journal of Psychology, A. C. Schaeffer wrote:

Instead of allowing the student to search blindly and without a plan or method, Pólya shows how he may be led quite naturally to a solution through the kind of heuristic procedure which he found in a hint given by the Greek mathematician Pappus (ca. 300 A.D.), a hint which has served to solve, as the author says, many difficult and baffling problems. The work may be read with profit by students and researchers in mathematics and in some of the physical sciences. In fact, any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art.

Influence

Mathematics Professor Alan H. Schoenfeld wrote in 1987 that the increased interest at the time in the area of problem solving, and in finding ways to teach it effectively, owed a big debt to How to Solve It. He pointed out that in the prior five years alone, the book had been cited in journals as diverse as American Political Science Review, Annual Review of Psychology, Artificial Intelligence, Computers and Chemistry, Computers and Education, Discourse Processes, Educational Leadership, Higher Education, and Human Learning.

But not all were convinced of the practical benefits of Pólya's heuristic strategies. Schoenfeld did not consider the strategies sufficient for his undergraduate math students working on non-routine problems. He argued that domain-specific knowledge and "mathematical sophistication" were also essential. From his observations of students attempting to utilize How to Solve It, Schoenfeld noticed they were often confused as to which among the dozens of Pólya's heuristic strategies should be chosen for a particular problem.

AI pioneers Allen Newell and Marvin Minsky praised Pólya for reviving the field of heuristics. In his 1960 paper, "Steps Toward Artificial Intelligence", Minsky wrote that "everyone should know the work of George Pólya on how to solve problems."

In the 1960s and '70s, Soviet engineer Genrich Altshuller developed a methodology, known as the Theory of Inventive Problem Solving (or TRIZ from the Russian abbreviation), which in many aspects paralleled Pólya's work but was more oriented toward solving engineering problems.

How to Solve It became a valued debugging guide for computer programmers. In 1982, R. G. Dromey wrote How to Solve it by Computer, a computer science textbook dedicated to and inspired by Pólya. In his Preface, Dromey stated, "there was a definite need for a book written in the spirit of Pólya's work, but translated into the computing science context." In a 1980s interview at Microsoft, Charles Simonyi said that "Programmers get a couple of books on their first day here. One of them, called How to Solve It, is by George Pólya, the mathematician. [Simonyi takes the book from a bookcase next to his desk and opens it to a certain page.] These two pages are important [note: he's referring to the two-page checklist at the start of the book]. The rest of the book just elaborates on these two pages. This is like a checklist for problem solving."