Dan,

Thanks so much for this insightful post!

“I’m extremely curious what would have happened had he presented a new image and asked his participants for new questions. I can’t be sure but I suspect they would have held out. They’d know from their last experience that Avery had a question in mind and everyone but the apple-polishers would have waited him out.”

Point well made! It happens all the time. Teachers with the best of intentions, shut down their students and usually don’t even realize they (yes, I’ve been guilty too!) did it. Then, the worst part, teachers get frustrated that students don’t want to take risks. The frustration leads the teacher to fall back into their comfort zone of spoon feeding students.

So, one take away for me is to always remember to be aware and reflective about what we’re doing in our classes or presentations that might cause students/participants to shut down and loose their desire to learn.

You also reminded me of a key element in the questioning/problem solving process that I’ve been forgetting to share with teachers. After providing the catalyst, gathering student generated questions and choosing a question to answer, teachers need to have students determine what information is needed in order to answer the question! — This reminder came from Act 2 of your Three Acts links.

Thanks again for sharing your insights and making me reflect on my own practices when working with teachers and students!

Part of the problem, of course, is grounded in teachers’ fear of what would happen if a truly open question led to something that exposes teacher ignorance. It takes intellectual courage and integrity to risk revealing oneself to a class filled with students who’ve been “schooled” to pounce on each other’s errors and weaknesses, and hence prone to smell blood in the water when teachers make so much as a calculation error or simple mistake in writing a problem (something I’m very prone to do if I try to talk and write at the same time).

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?

@Michael

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.

I think it would come across better for you if Avery was more explicit about the presentation structure. I don’t want to second-guess Avery’s reasons, but I sometimes start with a very open question that generates a large example space, then narrow it down – just to give students some familiarity and ownership of the context. But you have to tell people, explicitly, what it is you are doing. Here is how I would plan it:

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.