I’ve been reflecting on-and-off on the way I use questions in my classroom. If I could, I would use questions all the time – I would try to ‘tell’ the students as little as I could. In fact, if there’s one pedagogy article that I have completely and wholeheartedly internalized, it’s Never Say Anything a Kid Can Say (If you haven’t read this, I highly recommend it). My whole strategy for instruction/lesson design/unit design is asking the question ‘How can I get my students to come to this conclusion rather than have me tell it to them?’. Part of this, in my opinion, is providing the right motivation – the right problem to solve or question to answer. After they’ve started to come up with ideas and stumble onto the correct procedures or answers, my job then becomes to help them organize the information and give them the vocabulary that they’re probably missing. Jason Buell explained it so much better than I think I could in his post about Layering – essentially, in my own ideal classroom, my lessons follow a similar pattern as his: pose the question, investigate, summarize/synthesize/organize/give new vocabulary (which is a big part of teaching Geometry), practice, extend.

In another example of ‘The blogotwittersphere works in mysterious ways’, Kate Nowak just shared her Formula for a Math Lesson, which is incredibly similar to my own version of an ‘ideal lesson’. I highly recommend reading it.

So: I want my classroom to be one where learning is framed around solving a problem. This means introducing students to new scenarios they may not have seen before, as well as introducing them to a new way of thinking they may not have seen (or been conscious of) before. My job is to guide them through the process of applying previous knowledge to a new situation, or making a discovered pattern into something explicit. This means modeling my thinking, which is a big part of how I plan certain lessons and scaffold certain activities.

But here’s something you should know about me: I think in questions. Which means when I’ve posed a new type of problem to the class and am modeling how I would try and solve it once they’ve struggled with it a little bit, I’m asking questions. But I’m not directing them at the class – I’m asking them of myself, using them to guide my own internal dialogue. I’m asking rhetorical questions (which, I’ve been told, are the worst kind of questions – and, based on the reflection to follow, I somewhat agree). They’re big general questions that are sometimes hard to answer, like “where should I start?”, “do I know any theorems or definitions that can help me out?”, “have I solved a similar problem to this?”, “what was I given and how can I use it?”, “What happens if I…”, etc etc. I’m essentially walking through Polya’s process for problem solving right at the front of the room for all my students to see. And as I answer them, I step through how I break down a complex problem into manageable pieces, or how I relate a new situation to familiar content. I’m modeling my own problem-solving strategies (which I also have listed on a poster in my room) and showing my students its okay to not know the answer right away, but at least we have some strategies on where to start.

But I have a problem: I’ve accidentally blurred the lines the questions intended to model my own thinking, and the questions that come later meant to peek into what my students are thinking.

These same types of questions – “Where would you start?”, “Have you solved a problem like this before?”, “What do you think about the information you were given?” – are meant to help my students start problems, so it’s important that they be comfortable answering them. From there, I can get a little peek into any misconceptions that may be brewing or where there may have been a gap in my explanation. I find that when I ask my students these questions, meant to formally assess how they’re thinking, I’m usually met with silence. It takes a while for them to take a risk with this, if they take that risk at all. For a long time I thought this was just an issue with my students comfort level with mathematics and taking risks in the classroom, but I overheard a twitter conversation between Dan Meyer and David Cox that changed my opinion. I now think what I’ve actually been doing is blurring the line between the closed questions I pose to myself and the open questions I pose to the class. This terminology comes from Dan Meyer’s post on Pretending Closed Questions are Open – in fact, here’s a quote from that same post describing how the general question ‘What question do you have?’ eventually became a single specific question:

“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.”

Here’s what I think happened: sometimes when I model my own thinking and am asking myself those rhetorical questions at the beginning of class – “What should I do next?”, “How should I start this problem?” – an excited student will raise their hand to answer the question. And then I get excited because I’m hoping that this student is really thinking the same thing I am and will take over this conversation (because, remember, I never want to say anything a kid can say). So I call on this student to answer my rhetorical question that I’m using to model my own thinking… and the student won’t be thinking the same thing as me at all. They’ll confidently announce an incorrect answer or misconception, or want to go off in a direction that proves irrelevant and I, in the interest of time and not creating any other misconceptions, will need to curtail that discussion. But really what I’m doing is: I allowed one of my closed questions, intended to have exactly one right answer that I was thinking of, become an open question – and in doing so, I allowed a student to take a risk and get shut down. As a result, when I go to ask this question later – “what should you do next?” – after we’ve done several problems together hopefully building up their confidence with these types of problems, I’m met with blank stares. Or, they sit an try to wait me out, since earlier when I asked this same question, I gave the answer. So maybe they’re thinking that if they wait long enough, I’ll give them the answer again. Only the ‘apple polishers’ are willing to volunteer an answer even though I’ve checked in with many of my students and know they now know how to do these problems.

I guess what I’m realizing is I need to be acutely aware of the points of my lesson when I’m modeling my own thinking with questions, and I need to make sure my students are aware of it too – “Let me tell you how I would handle this problem if I were in your shoes and you can see if your thinking is similar to mine”, or maybe “If you’re not sure what to do, you would probably ask me for help. And if you did, the first thing I would ask you is…”. I need to not give in to that temptation of calling on a student when I’ve asked a vague, general question but have a specific answer in mind. I need to prepare the class for when I’m expecting an answer out of them: “Alright – we’ve talked through a few as a class – let’s see if we can start this next one on our own”. Maybe at this point they turn to their partner and talk through it with them before I poll the class full-group (which is always a good idea – check with a neighbor before sharing full-group). I need to be careful about when I choose to poll the class with one of these Polya-esque questions (“Anyone have any ideas of where to start?”) – if I’m doing it as I’m introducing a new type of problem to my students, this is probably a mistake – I usually have a specific way to start this problem in mind and it’s not fair to make my students ‘guess’ what that specific strategy is. Doing it as we’re going over homework seems like a better idea – this is where I want to see what my students are thinking and, hopefully, some students will be able to explain the answer to the class.

I guess this idea of blurring the lines between an Open and Closed question has really manifested itself in my classroom with how I model problem-solving and my enthusiasm and desire to have my students tell me the answer, rather than me tell them it. If I want to keep modeling my thinking by asking questions, I need to accept that these are just very general closed questions and it’s only fair that I’m the one who answers them – otherwise, I’m setting my students up for failure by giving them too broad of a question with too specific of an answer. And if they get enough of those wrong, they stop wanting to take risks, which is the opposite of my ideal classroom.

Update 1/2: I was looking through some old teaching resources and found this – Developing Mathematical Thinking with Effective Questions. There are some goooood question starters in there.