In the sixth section, Polya expounds more on the value of the questions and suggestions mentioned in the previous sections. He opens the section by mentioning that kind of thinking involved in solving a problem is rather nonlinear, where an idea at the beginning of solving a problem may be revised or changed several times upon arriving at the solution. Then, he claims there are four steps that are vital in solving a problem. The first one is to “understand the problem”; by that, he means a student needs to know what she needs to solve, and a question such as “What is the unknown?” may help at this stage. I suppose Polya considers “problems to find” for this step, but it seems such a step could also be useful for “problems to prove.” The second step is to see the relationship between the “data” and the unknown so that a plan can be devised. For this step I wonder if knowing a clear relationship between all the data and the unknown always is possible; perhaps, even this step might need revisions as the student progresses to a solution. Maybe a question that could be asked to see a relationship is to ask why a detail is provided in the problem. The third step is to implement the plan and the fourth to revise and discuss the solution.
In the same section, he explains why each step should be taken. To me, the first step seems crucial and the third very important and practical. For the second, I still suspect that all the relationships may not be clearly established, not because of laziness from the student but from a rather superficial acquaintance with the problem, which may change over time. In fact, maybe this is why Polya urges that at least the “main connection” be established.
In the next section, he reiterates the importance of understanding the problem. Another point he makes is that the problem needs to interest the student not just the teacher. I wonder what Polya would think of teachers who readily conclude that students are lazy or retarded (or both) if they fail to quickly provide a solution to problems that may be of no interest to them. A way he suggests such understanding can be checked is for the student to be able to answer to “What is the unknown? What are the data? What is the condition?.”
In the eighth section, Polya presents an example to illustrate his method; it is a problem that asks to find the diagonal of a parallelepiped, given the length, the width, and the height of the solid. An answer to “What is the unknown?” is the length of the diagonal; for “What are the data?”, an answer is the dimensions of the solid, and the condition is that the diagonal and the dimensions are for the same solid. Then, he claims the condition is sufficient to find the unknown, which is true, but I suspect an answer about the sufficiency of a condition of a problem may not be automatic for some problems. Furthermore, Polya mentions two interesting points in this section: one is that some knowledge is necessary to solve some problems, and I suppose that knowledge could be acquired by reading mathematics (or maybe hearing); the other is that a dialogue needs to be taken place between the teacher and the students.
He presents more examples in the subsequent sections, and I may discuss some of them in the next article.