The Art of Asking Questions

It started with an old (1848) journal article by JJ Sylvester on a property of concurrent lines in a triangle. He pointed out that if you had a triangle and located a point in its plane (I begin this with students by placing the point in the interior of the triangle) then the three cevians (lines from a vertex cutting the opposite side, perhaps extended) through the point will lie on a circle. He went on with some more detail, but ended the article by saying something about it being a good classroom exercise because it raises many good questions.

It reminded me of all the quotes I have about the importance of "questioning" to being a good mathematician.

"In mathematics, the art of asking questions is more valuable than solving problems." Georg Cantor

I wonder if math teachers in general agree, and if you could tell they did by the way they run their classes?

"The scientist is not a person who gives the right answers,
he's the one who asks the right questions." Claude Levi-Strauss

So if the art of asking questions is more important, should we be spending more time getting students to ask, rather than answer questions.

"Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs. " Felix Klein

I don't think I teach them to ask quesitons very well. My kids are good at asking, "How do I do that?", but not at the kind of questions that develop and reinforce intuitive development. I think I model asking questions well... "what might happen if we changed this? What does this remind you of???", etc...but I don't have any activities that actually are designed to help them learn to ask good mathematical questions.

So do you do that? And how do you do it?

"It is better to solve one problem five different ways, than to solve five problems one way." George Polya