How to Solve It: A New Aspect of the Mathematical Method
by George Pólya
I, Michael Parker, own this book and took these notes to further my own learning. If you enjoy these notes, please purchase the book!
Part 1: In The Classroom
Purpose
- pg 1: Ideally, the teacher should ask a question or indicate a step that could have occurred to the student himself. In general, help discreetly, unobtrusively.
- pg 2: Focus the student on the unknown quantity. Curate such provoking questions that apply to “problems to find.” They do not apply to “problems to prove.”
- pg 3: Start with questions and suggestions that are natural, simple, obvious, and just plain common sense. But do so in general terms.
- pg 4: If such a question is repeatedly asked, it may be repeatedly helpful. The student may then ask himself this question, which may succeed in eliciting the right idea, at which point it is assimilated.
- pg 5: When a teacher solves a problem before the class, he should dramatize his ideas a little and he should put to himself the same questions.
- pg 5: The four phases of work are understanding the problem, making a plan, carrying out the plan, and finally looking back on the completed solution in order to review and discuss it.
- pg 6: The teacher should ensure that a problem is well chosen, not too hard and not to easy, and interesting, so that the student understands the problem and desires its solution.
- pg 7: The teacher can seldom afford to miss the questions: What is the unknown? What are the data? What is the condition?
- pg 9: Ideas for a plan are based on past experience and formerly acquired knowledge. So it’s often appropriate to start with the questions: Do you know a related problem? Could you use it?
- pg 10: If you can’t create a plan, try to restate the problem, or to solve some related problem. To ensure that you don’t stray too far, ask: Did you use the whole data, or the whole condition?
- pg 12: Prepare a gamut of more and more explicit hints for introducing a decisive auxiliary element. Do not simply reveal it.
- pg 12: Devising a plan takes formerly acquired knowledge, good mental habits, concentration upon the purpose, and good luck. Carrying out the plan takes only patiences.
- pg 13: When carrying out a plan, the student should be convinced of the correctness of each step. It should be seen. If not beyond the student’s grasp, it could be proved.
- pg 15: We have a natural opportunity to investigate the connections of a problem when looking back at its solution. Ask whether you could use the result, or the method, for some other problem.
- pg 17: Questions providing “experimental evidence” for a solution gives it new significance. It has a better chance of being remembered, and such questions can be easily transferred to similar problems.