How to Solve It: A New Aspect of Mathematical Method
Since its original edition in 1945, this classic has not been out of print. With this reprint, Princeton University Press keeps the marvelous ideas of Polya alive.
How come people do like solving crossword puzzles, brain teasers, riddles, or nowadays why gaming in so popular. All these challenge the intellect ant require creative solutions. And yet, as Polya writes in his preface to the second edition: ``...mathematics has the dubious honor of being the least popular subject in the curriculum ... Future teachers pass through the elementary schools learning to detest mathematics ... They return to the elementary school to teach a new generation to detest it.''
So his focus is on the interaction between student and teacher. It is not only how to learn to solve the problem, it is even more so about how to teach someone how to solve a problem. The first section makes this clear: ``The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help or with insufficient help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.'' Note that this is a general rule that applies not only to mathematics, although most of the examples in the book are mathematical, and often geometrical. Another aspect that shows in this quote is that it is not a problem of the student or of the teacher, but that the focus should be placed on the interaction between both. This explains why most of the book is written in some kind of dialog, whereby the teacher is most often just asking an appropriate question, followed by a possible reaction of the student. This reflects on a meta-level as the author teaching the reader-student how to approach the solution of a problem and the reader-teacher how to help a student.
In the first part, Polya explains that the solving process goes in four phases, and that the proper questions have to be asked in each phase.
- We have to understand the problem. What is the unknown? What is given? What are the conditions? (introduce a figure if possible, introduce proper notation, check if a solution is possible)
- We have to make a plan. Is there a related, familiar, or simpler problem that we can solve? Are all the data used? Do we satisfy all the conditions?
- We have to carry out the plan. Can we check each step and prove that it is correct?
- We have to reflect on the solution. Can we check the result? Is there another derivation possible? Is there a shorter derivation? Have we used all the data? can we use the method in another problem?
These issues are illustrated with several examples.
The second part is very short (only 4 pages) and reflects a dialogue between a teacher and an ideal student in very general terms.
The bulk of the book is part 3, that is written in the form of an alphabetically ordered dictionary. It has entries that are names of mathematicians (Bolzano, Descartes, Leibnitz, Pappus), but most entries somehow give further reflections on the the questions asked above and their answers with more concrete examples.
Another short fourth part formulates some problems, then some hints and finally solutions. The first one has become famous among riddle lovers: A bear, starting from the point P, walked one mile due south. Then he changed direction and walked one mile due east. Then he turned again to the left and walked one mile due north, and arrived exactly at the point P he started from. What was the color of the bear? One might be tempted to think that the color is white because P is the North Pole, but there are other solutions!
The focus is clearly on mathematics and almost all examples are indeed mathematical, but the same principles can be applied in any other problem solving situation. It takes some time to go through all the examples but the time is not waisted. It pays to think about them and see the spark of the generality of the idea. Some might find it frustrating that either the examples are too specific, not touching on their own problem or they find that the advise given is too general, not helping to solve the particular problem they have at hand. In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let's hope this classic will remain a source of inspiration for several generations to come.