Pretending Closed Questions Are Open

I was in Avery Pickford’s session at CMC-South when he put up this image and polled his participants for questions that interested them.

They asked about the slope of the diagonal. They asked about its length. Avery then constrained their questions. “What things could we count?” he asked.

His participants responded with “the perimeter” and “the number of squares.” At that point, Avery just asked the question that interested him:

How many squares does the diagonal pass through?

His session ended on that problem but 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.

Open And Closed Questions

If you have a question you’d like your students to answer, ask it. But before you ask it, consider creating a visual – something short and sweet, one photo or one minute of video – that orients your students to the context of your question and makes that question seem like a natural one to ask. Like:

If you’d like your students to pose some questions of their own, ask them what questions they have. But questions about what? Give them something to ask questions about. Consider creating a visual – something short and sweet, one photo or one minute of video – that lends itself to different perplexing questions. Like:

What Happens On Twitter Stays On Twitter

How can you tell in advance if students will be perplexed by your closed question or if they’ll have open questions about your photo or video? You pilot it. There’s no right way to pilot curricula, only optimizations for different variables that are often in competition with one another. Like:

  1. Are you piloting with students or with some proxy for students?
  2. How easy is it for your participants to give you feedback?
  3. How many participants are giving you feedback?
  4. How helpful is that feedback to your development process?
  5. How far into the development process are you waiting to get feedback?

Here’s one optimization: show teachers your photo or video on Twitter and ask them what questions they have about it.

This means (1) you aren’t piloting with students, which is unfortunate, though no students are harmed if your idea is a dud, (2) it’s easy for your participants to give you feedback, (3) the number of people giving you feedback is proportional to your followers on Twitter, (4) that feedback is often useful – if you plan to ask a closed question, the feedback will let you know if that question is interesting; if you plan to ask for open questions, the feedback will let you know what questions to expect.

Or you might pilot your curriculum on the same day you’re teaching it, making modifications for your afternoon class based on feedback from your morning class. You can evaluate the variables for yourself on that one.

Make it work for you. Twitter, #anyqs, your classroom, your faculty lounge, whatever. Make it make you a better teacher. Just understand that when you’re using curriculum in the classroom, you’re optimizing for an entirely different set of variables than when you pilot that curriculum somewhere else.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. 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!

  2. Dan, I’ve had the chance to tell you in person, and this post reminds me to say it again! What you’re talking about are narrative and literacy skills for teachers. You’re asking math teachers to pay attention to narrative and context, and make sure the math they WRITE has a story worth telling!
    And by consequence, you’re asking students to READ a context, extract a mathematical idea, formulate a problem, represent it, and solve it. These are E-ticket ideas, and for those of you who’ve read Adding It Up, strategic competence.

  3. 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?

  4. I think the premise is flawed.

    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.

  5. @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.

  6. The problem itself and how it is used for instruction are separate, but related topics. While it’s not always possible to try to put lipstick on a pig, it might be too difficult for teachers just beginning this type of teaching to focus simultaneously on creating perplexing problems and facilitating a discussion around the problem. Of course the problem is the pig and the teacher is the lipstick applier :).

    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.

  7. 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.

  8. Some formatting did not come through – there would be questions after 1. and 2.

    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.