I plan to discuss, summarize, or comment several sections of G. Polya’s How to Solve it: A New Aspect of Mathematical Method; I may do all of the above in one article because it seems to me that one tends to involve the others. I will start with pages 1 to 5 in this article and continue in subsequent articles. In the first section of the book, the author presents the role of a teacher as a very tricky business that requires much attention. It seems for the author a teacher is more than a dispenser of knowledge but is someone who needs to be careful about how much he or she gives to their students, lest that they not learn anything. In the second section, Polya makes the point that certain questions and suggestions can lead to mental processes that may help a student to solve a problem; for example, “what is the unknown?” seems to make the student look for the unknown in the problem, and such question, or any variant of it, can also be used by anyone who wants to solve a problem. In the next section, the author goes on by characterizing questions and suggestions that can lead someone to solve a problem. One of their properties is “generality,” in that they may be used for different mathematical and non-mathematical problems; some examples are “What is the unknown?,” “What are the data?,” and “What is the condition?.” He later explains that there are questions, such as the ones above, that are more appropriate to “problems to find” rather than “problems to prove,” where the latter kind tends to leads to questions about the hypothesis and conclusion. Another property is “common sense” in that those questions and suggestions are not different from what someone would ask or would do to solve a non-mathematical problem; he gives the suggestion “Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.”
In the next section, he elaborates on the role of the teacher and the importance of the questions and suggestions. He thinks that, when a teacher asks such questions and uses such suggestions, which lead the student to solve the problem, the student may realize they are effective and ultimately use them on his or her own to solve other problems. Finally, he mentions that by imitating other people who solve problems and by practicing, the student will be able to solve many more problems.
From those sections, Polya seems to make a point about problem solving ability: there is no magic about solving problems; it is a skill that can be learned with imitation and practice. In the other sections, we’ll see what more he has to say.