How to Solve It - A New Aspect of Mathematical Method

How much do I want to read more? 7/10

I'm curious to read on. This book sold millions of copies. The author was a successful teacher of Philosophy before to tackle mathematics.
I didn't read about maths so far, only pedagogy.

Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences may leave their imprint on mind and character for a lifetime.

The author remembers the time when he was a student himself,
"Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution?"

Foreword - byJohn H. Conway

it has taken me a long time to appreciate just how wonderful this book is.

It is one of the most successful mathematics books ever written, having sold over a million copies and been translated into seventeen languages since it first appeared in 1945. Polya later wrote two more books about the art of doing mathematics, Mathematics andPlausibleReasoning(1954) and Mathematical Discovery (two volumes, 1962 and 1965).

It is often said that to teach any subject well, one has to understand it "at least as well as one's students do." It is a paradoxical truth that to teach mathematics well, one must also know how to misunderstand it at least to the extent one's students do!

Experienced mathematicians know that often the hardest part of researching a problem is understanding precisely what that problem says. They often follow Polya's wise advice: "If you can't solve a problem, then there is an easier problem you can't solve: find it."

his work in geometry deemed merely "satisfactory" compared with his "outstanding" performance in literature, geography, and other subjects. His favorite subject, outside of literature, was biology.

Part 1 - In the classroom


1. Helping the student

This task is not quite easy; it demands time, practice, devotion.
The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.

The best is, however, to help the student naturally. The teacher should put himself in the student's place, he should see the student's case, he should try to understand what is going on in the student's mind, and ask a ques- tion or indicate a step that could have occurred to the student himself.

2. Questions, recommendations, mental operations

Trying to help the student effectively but unobtrusively and naturally.
we have to ask the question: What is the unknown?
What is required? What do you want to find? What are you supposed to seek?

3. Generality

What is the unknown? What are the data? What is the condition?

4. Common sense

5. Teacher and student. Imitation and practice

develop the student's ability so that he may solve future problems by himself.
If the same question is repeatedly helpful, the student will scarcely fail to notice it and he will be induced to ask the question by himself in a similar situation.

Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by prac- ticing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.

The teacher who wishes to develop his students' ability to do problems must instill some interest for problems into their minds and give them plenty of opportunity for imitation and practice.