Teachers and authors of textbook should not forget that the intelligent student and the intelligent reader are not satisfied by verifying that the steps of reasoning are correct but also want the motive and the purpose of the various steps. The introduction of an auxiliary element is a conspicuous step. If a tricky auxiliary line appears abruptly in the figure, without motivation, and solves the problem surprisingly, intelligent students and readers are disapointed; they feel they are cheated. Mathematics is interesting in so far as it occupies our reasoning and inventive powers. But there is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous steps remain incomprehensible. To make such steps comprehensible by suitable remarks or by carefully chosen questions and suggestions takes a lot of time and effort; but it may be worth while.

– George Pólya (1945) – How To Solve It : a new aspect of mathematical method

Pour un livre qui a été écrit avant la naissance de la didactique des mathématiques, on y trouve des passages fort intéressants.

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