The modern Mathematical Heuristics movement began with the publication of How to Solve It by George Polya in 1945 [Po45]. The book is popular in nature, and the content is accessible to high school students. Its popularity is demonstrated by the fact that it’s still in print at the time this book goes to press, over half a century later.
Although his book pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of discovery. The book contains a list of questions that encapsulate his recommended methodology, followed by recommendations for use in the classroom and a short dictionary of heuristic concepts. He developed these ideas in a series of volumes over the following two decades. Mathematics and Plausible Reasoning was published in 1954 and Mathematical Discovery was published in two volumes in 1962 and 1965, respectively.
Polya’s “How to Solve It” list consists of four phases:
Understand the problem.
Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection can’t be found. You should obtain eventually a plan of the solution.
Carry out your plan.
Examine the solution obtained.
Polya sought to provoke understanding of a problem by urging his readers to consider the following questions and suggestions:
What is the unknown?
What is the data?
What is the condition?
Is it possible to satisfy the condition?
Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
Draw a figure. Introduce suitable notation.
Separate the various parts of the condition. Can you write them down?
Polya sought to enable development of a plan by suggesting his readers should consider the following questions:
Have you seen it before? Or have you seen the same problem in a slightly different form?
Do you know of a related problem? Do you know of a theorem that could be useful?
Look at the unknown! Can you think of a familiar problem having the same or a similar unknown?
Here is a problem related to yours and solved before. Can you use it?
Can you use its result? Can you use its method? Should you introduce some auxiliary element to make its use possible?
Can you restate the problem? Can you restate it still differently?
Go back to definitions.
If you cannot solve the proposed problem try to solve first some related problem. Can you imagine a more accessible, related problem? A more general problem? A more special problem? An analogous problem? Can you solve a part of the problem?
Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Can you derive something useful from the data?
Can you think of other data appropriate to determine the unknown?
Can you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?
Have you used all the data? Have you used the whole condition? Have you taken into account all essential notions involved in the problem?
Polya encouraged the execution of a plan by advising his readers to consider the following questions and suggestions.
Carrying out your plan of the solution, check each step.
Can you see clearly that the step is correct?
Can you prove that it is correct?
Polya motivated the review of results by recommending that his readers ask the following questions:
Can you check the result?
Can you check the argument?
Can you derive the result differently?
Can you see it at a glance?
Can you use the result or the method for some other problem?
Alan Schoenfeld [Sc85] makes the following criticisms of the Mathematical Heuristic movement:
“Faith in mathematical heuristics as useful problem-solving strategies has not been justified either by results from the empirical literature or by programming success in AI [artificial intelligence].”
“Despite the fact that heuristics have received extensive attention in the mathematics education literature, heuristic strategies have not been characterized in nearly adequate detail.”
“The number of useful, adequately delineated techniques is not numbered in tens, but in hundreds…. The question of selecting which ones to use (and when) becomes a critical issue.”
“The literature of mathematics education is chock-full of heuristic studies. Most of these, while encouraging, have provided little concrete evidence that heuristics have the power that the experimenters hoped they would have.”
It is fair to ask whether the use of heuristic methods as such have resulted in a meaningful, measurable improvement in student’s mathematical competency. The second half of Schoenfeld’s book contains empirical studies that support these criticisms, among others.