P.J. is right that we have to plan our questions, but then what happens when something occurs that we did not anticipate? It goes deeper than careful planning, at least in the usual sense of the phrase. We need to work hard to break our bad habits about questioning. We must develop habits of language and thought that enhance learning. We need to be clear about what our objectives are and pause before we ask--or answer--a question, and make sure that our response is consistent with the outcomes we desire.
With regard to PJ’s surprise solution, I think the moment in class when a student connected two ideas from totally different places and solved the problem ought to stand as one of the high points of any teacher’s career. At that moment, PJ’s student achieved a higher level of doing mathematics than merely understanding a well-known theorem. PJ’s student demonstrated, in the presence of the entire class, what it means to do mathematics. The student’s insight is more important than the theorem that was meant to be taught that day. A meta goal was reached. Exciting moments like that are rare and can only happen when students are encouraged to approach problems using their own intellect and intuition. If we could make those events happen every day, mathematics would be the most exciting class for every student in every school. It is important that PJ knew to set aside the lesson at hand and celebrate the insight of one of his students. But how does a teacher ensure that this sort of exciting moment happens?
I think we can establish an atmosphere in our classroom in which students will be willing to take chances and are not afraid to be wrong. I think we can structure our classes so students will try to think of clever solutions and will try to make connections. How can we do this? By asking problems that are rich enough that students can get started on them but will not necessarily see the end for a while, by celebrating many different ways to solve a problem, and by asking questions that encourage students to think for themselves. The first thing is to give them time to work and encourage collaboration. In other words: instead of asking if everyone understood, how about asking: Did anyone do the problem a different way? Does this solution remind anyone of another problem we have done? What made that student think of using the Power theorem?
Asking questions should be a means to stimulate further thought. If we want to know what students know, we can ask them to do an interesting problem, and then we can walk around and see if they can do it. We can observe their errors and let them sort out a solution among themselves. If a question does not take us further along in the investigation at hand, we don’t need to ask it. We can ask students: what relevant questions arise after this problem has been solved? What is the next question we might ask? Would this solution have worked if the coefficients had been irrational numbers? What if the point had not been on the circle? Did you need everything that was given to solve the problem?
And often, there should be no question—just a situation; the students’ first task is to determine what questions can be asked and answered.