G. Polya 《Mathematics and Plausible Reasoning》1954 年初版。Volume II Patterns of Plausible Inference, 最后第 XVI 章，Plausible Reasoning in Invention and Instruction > 9. A few words to the teacher.

Mathematics (下简称 math) has many aspects. To many students, I am afraid, math appears as a set of rigid rules … To some instructors, math appears as a system of rigorous proofs …
To a mathematician, who is active in research, mathematics may appear sometimes as a guessing game: you have to guess a math theorem before you prove it, you have to guess the idea of the proof before you carry through the details.

It may appear a little more surprising to the layman that the mathematician is also guessing. The result of the math’s creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by guessing.

…there should be a place for guessing in the teaching of math. Instruction should prepare for, or at least give a little taste of, invention. At all events, the instruction should not suppress the germs of invention in the student.

The teacher should show that guesses in the math domain may be reasonalbe, respectable, responsible.

I address myself to teachers of math of all grades and say: Let us teach guessing!

On the contrary, we should teach both proving and guessing, both kinds of reasoning, demonstrative and plausible. More valuable than any particular math fact or trick, theorem, or technique, is for the student to learn two things:

  • First, to distinguish a valid demonstration from an invalid attempt, a proof from a guess.
  • Second, to distinguish a more reasonable guess from a less reasonable guess.

… I would say that talented students of engineering are usually more accessible to, and more grateful for, well-presented plausible grounds than for strict proofs.

In the meantime, I reiterate my hope that this book, as it is, may be useful to some teachers, at least to those teachers who have some genuine experience in problem-solving. The trouble is that there are so few teachers of math who have such experience. And even the best School of Education has not yet succeeded in producing the marvellous teacher who has such an excellent training in teaching methods that he can make his students understand even those things that he does not understand himself.

后感

娃天生会猜，大人需要引导这种天性，让娃认识到，哪些方面更值得猜、如何更有效地猜。

不仅数学。