So, the point I am making is that this sequence of questions – from open to particular – can serve a purpose. Namely, it can prepare the ground for a rich question.

1. What questions would you ask about this diagram?

2. Thank you! If we had an extra hour or five, students could solve these fine questions and more. Hopefully, forming the questions helped you to get to know the diagram more personally. The next task: ask just questions about things you can count.

3. Thank you! So, again, we could pursue these questions with students. I prepared a question of this type ahead of time to demonstrate the next stage of the process. I like it because it recently inspired a good discussion in math teacher blogs, and has some history in Olympiads.

I would like to offer a couple of strategies for presenting a problem a teacher wants to pursue in a particular way while still eliciting student interest in other aspects of the problem. This is based on the premise that the teacher has chosen to use a particular problem based on a specific learning goal for the lesson in which that problem appears.

1. Present the problem at the end of the previous day’s class and have the students submit questions on slips of paper. The teacher can peruse these to get the lay of the land for the next day’s lesson. The questions might fall into categories or form a hierarchy of sophistication. (Wouldn’t it be cool to track the complexity of student questions over the school year?) On problem day, the teacher simply says “There were several interesting questions submitted. Today we’re going to talk about this one.”

2. The teacher can record or collect student questions and assure students that these will be addressed in some way either individually or in the future. The teacher might even write a reply to some students to encourage individual pursuit of particular questions.

3. Revisit the recorded questions during the lesson to see if any get addressed during the discussion the teacher had planned to have.

Dismissing student questions is not good, but there are other ways to encourage curiosity while directing focus on a particular learning goal.

Couldn’t agree more. I’ve been teaching in Michigan for 11 years now, and can attest to the “fear” of veering off the path. We feel that we are so constrained to cover so many concepts that to “lose” a day to student curiousity is just not allowable. I’ve just started to take some risks, and just let things play out. I had a great day yesterday with a lesson on rate of change with my Algebra 1 kids. I still find myself falling back into my safe traditional approach, but I am slowly testing the waters to create a more student driven atmosphere. I have a long way to go, but the first step was definitely the hardest one to take.

Before Dan’s blog, I wouldn’t have thought twice about the question that was asked in this problem. My question would have been, what concepts are going to apply to the solution to this problem? That is what we were trained to do, drive towards covering the GLCE, HSCE, CCS, etc.

Giving students the opportunity to take in the information and ask the question themselves doesn’t necessarily mean that the teacher is pretending closed questions are open. On the other hand, I don’t mind just asking the question, either. Having a clear, guiding question is an important first step in any problem solving situation; just as important as developing a plan, carrying out that plan, abstraction, verifying results and, finally, answering the question. We’d like students to be proficient in carrying out all these steps as independently as possible.

The situation isn’t all that different from making a decision about whether to pursue an unexpected answer/solution/ strategy/idea offered by a student. Look at the classic situation in Deborah Ball’s 3rd grade classroom in 1989-90 when a student declares that “some numbers are both odd and even.” Ball was faced with making a choice: pursue the student’s idea (which wasn’t even well-formulated yet in his mind, it turns out), deflect it, delay pursuit, ignore it, “correct” it, or some other path? Fortunately, in this case, she chose to allow students to follow the opening and it led to some really rich conversation and thinking. The mathematics education community is richer for her decision.

What likely led her to the choice she made was her pedagogical content knowledge: she had enough understanding of the potential for something valuable to emerge from the student’s speculation to risk straying from her lesson plan. I suspect, too, that had things proved unproductive, she would have gotten back to the plan, and she trusted herself to be able to switch gears and paths in a reasonably timely manner.

Unfortunately, not many mathematics teachers (and very few elementary teachers in this country) are inclined to take such risks. Further, Ball was in the happy circumstance of being “outside the system.” She was guest-teaching mathematics in an elementary school classroom for a year, working in conjunction with the regular teacher for part of each day as part of a research grant. She had the permission and freedom to do what made sense to her rather than follow a pacing chart and district-determined curriculum plan for the year.

As a mathematics coach in Detroit, I’ve worked with teachers who were fearful of the repercussions of going off the reservation, grounded in experience with district officials who slapped down those who weren’t “on pace.” Other teachers with whom I worked were less fearful, perhaps, of consequences from on high, but lacked the sort of pedagogical content knowledge needed to recognize and pursue profitably opportunities that presented themselves, let alone teach the sorts of lessons that engender such moments.

To return to Dan’s example, I think there’s an insidiousness in pretend-openness in that it buries the truth about how school in general and math class in particular works under a veneer of intellectual freedom. There truly is nothing wrong with asking pertinent, rich, specific questions. Many of the most wonderful lessons I’ve observed came precisely from such questions, particularly when they occurred in lessons that built thoughtfully to just that question. Expecting students to play mind-reading games and then disrespecting their ideas when they fail to come up with exactly the “correct” question/answer seems like an error to me. Why fool students (who as Dan correctly suggests, won’t be fooled for long) into thinking that you’re actually interested in what they think or wonder about, when it’s obvious that you really want to get to what’s on YOUR mind? Why do we think our students are so blind? Why are we so blind to our own transparent moves? And why are we so disinclined to find out what happens next?