Developing the Question

I’m reading Polya’s Mathematical Discovering, Volume 2.  In Chapter 14, Polya puts forward rules such as Suggest it, do not force it down their throats, explaining…

[The teacher should] leave the students as much freedom and initiative as possible under existing teaching conditions.  Pressed for time, the mathematics teacher is often tempted to sin against the spirit of these rules, the principle of active learning.  He may hurry to the solution of a problem without leaving enough time for the students to put the problem to themselves in earnest.  He may name a concept or formulate a rule too soon, without sufficient preparation by appropriate material, before the students can feel the need for such a concept or rule.  He may commit the celebrated mistake of deus ex machina: he may introduce some device (for instance, a tricky auxiliary line in a geometric proof) which leads to the result all right, but the students cannot see their life how it was humanly possible to discover such a trick which appeared right out of the blue.

There are too many temptations to violate the principle.  Let us, therefore, emphasize a few more of its facets.

Let your students ask the questions; or ask such questions as they may ask by themselves.

Let your students give the answers; or give such answers as they may give by themselves.

At any rate avoid answering questions that nobody has asked, not even yourself.

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