How to Solve it (Part III)

In the tenth section, Polya starts illustrating how his questions and suggestions can be used. He uses the problem of finding the diagonal of a parallelepiped. In the section, he fully demonstrates how a dialogue needs to be taken place in the classroom. He seems to present the teacher as someone who is very attentive to her audience, seeing when the students seem to not being able to make any progress and being able to find the appropriate ways to encourage them. It seems he presents the teacher as a skillful orator and performer. By pushing the students to remember a related problem, she successfully helps them to make the connection between the diagonal and the hypotenuse of a right triangle; she starts by asking “Do you know a related problem?”.

The next four sections are the continuation of the process of solving the problem, which involve devising a plan by the student and implementing it, checking each step while carrying the plan, revising the solution, and making sure all the given information was used. In all those stages, Polya seems to suggest that the teacher needs to help but not too much, lest that the students actually learn something from the experience.

In the sixteenth section, Polya explains how helping the students needs to proceed: the teacher needs to start with more general questions and suggestions and use more specific ones if the students do not make much progress with the more general ones. He further prescribes that the list of questions and suggestions be “short,” so that they can be easily remembered by the students, “simple and natural,” and “general” enough so that they can be applied to various problems.

In the seventeenth section, Polya notes that some suggestions and questions are far from helping the students solve a problem. An example of a bad question related to the problem of finding the diagonal is “Could you apply the theorem of Pythagoras?” For Polya, such question is bad because it tends to cause disaster at different levels of attempting to solve the problem: if the students are far from a solution, they may not see the relationship between the question and the problem; if they are close to a solution, all their effort has been disregarded by giving them the key to the problem; more importantly, giving them too much of a specific question tends to deprive them from any opportunity to learn how to solve other problems and may unfortunately make them regard the teacher as the magician who knows it all.

At this point, I suppose the gist of all this writing by Polya is that learning how to solve mathematical problems may be similar to learning other skills. The same way a physical activity may require that someone know certain moves, solving problems also requires some specific mental moves (or processes), and the questions and suggestions, according to him, can initiate such processes.

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