How to Solve It

indirectly, mental operations typically useful for the solution of problems. (Location 308)

What is the unknown? What are the data? What is the condition? These questions are generally applicable, we can ask them with good effect dealing with all sorts of problems. (Location 310)

Take the suggestion: Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. (Location 318)

First, to help the student to solve the problem at hand. Second, to develop the student’s ability so that he may solve future problems by himself. (Location 328)

Moreover, when the teacher solves a problem before the class, he should dramatize his ideas a little and he should put to himself the same questions which he uses when helping the students. Thanks to such guidance, the student will eventually discover the right use of these questions and suggestions, and doing so he will acquire something that is more important than the knowledge of any particular mathematical fact. (Location 346)

Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution. (Location 350)

First, we have to understand the problem; we have to see clearly what is required. (Location 353)

Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. (Location 354)

Third, we carry out our plan. (Location 355)

It is foolish to answer a question that you do not understand. It is sad to work for an end that you do not desire. (Location 362)

The student should understand the problem. But he should not only understand it, he should also desire its solution. If the student is lacking in understanding or in interest, it is not always his fault; (Location 364)

We have a plan when we know, or know at least in outline, which calculations, computations, or constructions we have to perform in order to obtain the unknown. (Location 395)

main achievement in the solution of a problem is to conceive the idea of a plan. (Location 397)

We know, of course, that it is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge. (Location 401)

Thus, it is often appropriate to start the work with the question: Do you know a related problem? (Location 406)

Here is a problem related to yours and solved before. Could you use it? (Location 411)

If you cannot solve the proposed problem try to solve first some related problem. (Location 416)

Did you use all the data? Did you use the whole condition? (Location 419)

To devise a plan, to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, good mental habits, concentration upon the purpose, and one more thing: good luck. To carry out the plan is much easier; what we need is mainly patience. (Location 452)