In this recent podcast, Anne Watson and John Mason describe the importance of asking questions that promote mathematical thinking (among other things). In the podcast, they refer to their book, Questions and Prompts for Mathematical Thinking, which I re-read after listening to it. I also re-read Janet Ainley's article Telling Questions (from MT118), which they refer to. In light of this, I decided to record yesterday's (triple) lesson and try to analyse my questioning.
Listening to the audio, I identified a few loose categories that my questions seemed to fall into:
- It would help me if you could speak a bit more about what you are doing/experiencing. These might fit into what Janet Ainley calls 'genuine' questions, in which I want to know what the learner is thinking. An example might be: "Could you talk me through what you are doing?"
- I think it might help you to speak a bit more about what you are experiencing. This may be used if I feel something useful is at the edge of a learner's awareness. The question might be the same as above, but the intention is different, in that I suspect I have more of an idea what the learner might be thinking.
- May I suggest a way of thinking about this, or something you could do, that might be useful? This might be where I am hoping to promote some mathematical awareness, similar to what Ainley calls a 'provoking' question, for example: "Will you expect this answer to come out negative?"
- Does it help if...? This is a bit more 'direct' than above, perhaps more like providing hints, and might happen when a learner is stuck. See below for more about this.
- When is... ? Why did...? Can you remember...? This might be what Ainley calls 'testing'. An example might be: "When do you get a negative answer for an integral?"
- Too vague questions, such as "What else do you notice?"
- Rhetorical questions, such as "This is the essence of the whole thing, isn't it?" Ainley calls these pseudo-questions.
- Checking questions, such as "Is that the answer that you've got?"
Most of the questions I asked invited learners to articulate their thinking, either for their benefit or for mine. Less frequently, I suggest a way of thinking, give help, or ask testing questions.
Sometimes it is hard to decide which category a question falls into. Some questions fall into more than one of these categories. It is often not the question itself, but the situation and intention behind the question that determines its nature. Often it only became clear what type of question I had asked by listening to the response of the learner.
A quick note on the Ainley's use of the word 'genuine' to describes question to which you (the teacher) don't know the answer. I'm not sure this is the most useful expression, as it might suggest that asking questions to which you know the answer is not genuine, and to be avoided. However, I think there are instances in which it might be useful as a teacher to ask questions or pose problems for which you know the answer.
I think it becomes more problematic when the answer only resides in the teacher's head, or when the teacher embarks on a series of questions (funnelling) in order to elicit an answer that they already knew, and could have more easily just expressed themselves. Listening back to the transcript, there were examples in which I recognised myself doing this. They are often noticeable by my response to a student answer of "Er, not really" or "Yes!"
So you’re not recognising what’s happening here, you can’t see it?
[I’m trying to get where A is, to help him from his low place. He shakes his head]
So let’s get a blank sheet of paper… Does this picture help you?
[I'm trying to set out the significant parts of the problem more clearly, but he’s still not seeing what is required, he is not speaking, he is visibly frustrated - there are lots of crossings-out on his paper. He seems tired. I know I must proceed with caution here. A long pause… Student B asks a question, so I turn to work with him, which has the possible benefit of allowing A some time to think on his own. When I turn back to A, he doesn't seem to have got much further.]
What are your thoughts? You’re not seeing it?
[Shakes head… I try a more direct explanation of what might be required, but he now seems to have a ‘block’, he is rubbing his head with his hands. I rearrange the expression so that it more closely matches what I hoped he might notice. I have suspected for a while that what is 'blocking' him from solving this problem - at least initially - is as simple as not noticing this rearrangement, and think I could have shown him this rearrangement sooner.]
Does that help?
Does that help you work out…?
If I want to integrate that, how does this help?
[I'm more or less showing him the answer by pointing, but he isn’t seeing anything now, the fog has descended]
Do you see what’s going on here?
[A sort of blank look… not helped, I don’t think, by his awareness that B solved this problem a while ago... Then I realise I can create a set of simpler examples that will lead him up to be able to solve the problem we are working on, through use of variation. Why didn't I think of this earlier?]
The set of questions I created proved to be useful; he was then able to solve the original problem.
Questions which are derived from ways of thinking about mathematics... are likely to be useful because they reflect mathematical thinking.
I became aware that a problem with my questioning is that it was not often specifically mathematical, perhaps drawing attention to more general psychological elements of learning, such as articulating, looking/recognising, making associations, and so on - perhaps reflecting my personal interests.
Watson and Mason suggest the creation of a small set of questions that are used repeatedly (but not too repeatedly) for a while, in order for them to become internalised by the learner, before fading them out. It is all very good the teacher asking questions, but what is required is that these actions come to mind for the learner in similar situations in the future. As they suggest (p.36):
Becoming a good questioner is not a matter of using someone else's list of questions (even our list!), but of developing questioning as a personal activity with mathematics. Asking oneself: "What can I ask of my students?" rather than "How can I get them to say what I sam seeing?
So the quality of questioning is not necessarily about each individual question, but rather about the thinking that repeated use of these questions promulgate in the learner.
My attention returned to the episode above. It is certainly the case in this episode that I was trying, and struggling, to find ways to get A to see what I could see. What questions could I have asked?
I think one approach is to look at what I did to 'help'. What did I do ? And what do I generally find useful when solving problems? I invited A to read/articulate the question, in the hope of him recognising what it was about. I invited him to compare the two expressions. As I started to help more directly, I set out more clearly what he knew, and what he wanted. I then created a set of (simpler) questions that allowed him to see how to solve the original problem.
These actions suggest to me the following three questions:
- Of what is this question an example? It would be useful when faced with this question to recognise that it is about (reversing) the chain rule. If a question is not recognisable, it will be useful later to identify of what it is an example, and explore this more fully (see this follow-up work).
- Articulate and connect: What do I know now? What is the question asking me to do? How are they connected? For example: "I know that the derivative of sin^3(x) is 3.sin^2(x).cos(x), and I am being asked to integrate sin^3(x). These are connected because..."
- Can you create a simpler problem, or series of problems, that will allow you to solve this problem? For example: "What if that cubed wasn't there? Could I form/solve that problem? Then, what if it was a squared?"
What is interesting is that we have already worked extensively on these things this year, but I had not formulated a set of questions in order to promote this thinking. In the next post, I will describe what happened in the following lessons.