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Introduction

Listening is one of the most important aspects – no, scratch that – the most important aspect of my work in the Maths Learning Centre.

It is not obvious to people starting out tutoring in the MLC that this should be the case. To a beginning tutor, it seems that it's their job to explain things to the students, and to show them how to do stuff. But even if the actual goal was to explain, you can be much surer which explanation to give the student if you first listen to their ³¦³Ü°ù°ù±ð²Ô³ÙÌýunderstanding. More importantly, you can never improve as a teacher unless at some point you listen to the students to see how well your explanation has gone.

But  how do you go about doing the business of listening? This blog post is about my interpretation of a framework that describes different levels of listening for the purposes of teaching, which I read about in two papers:

1. Davis, B (1997) Listening for Differences: An Evolving Conception of Mathematics Teaching, Journal of Research in Mathematics Education, 28, 255-376
2. Yackel, E, Stephan, M, Rasmussen C and Underwood, D (2003) Didactising: Continuing the work of Leen Streefland, Educational Studies in Mathematics, 54, 101-126

I spoke at a conference about this framework some years ago, and I have been meaning to write about it ever since. I am finally actually writing about it now (and you are reading it). My thinking has evolved a little since then, so you get the updated and extended version.

The papers

Davis 1997

In the  first paper, Davis tells us about how he and schoolteacher Wendy reflected on the types of listening Wendy did in her classroom, and how they were related to her beliefs about what mathematics is and what the teacher's role is in helping students learn it. It is a truly fascinating and powerful paper and I recommend everyone read it.

Davis notes from previous research that "the quality of student articulations seemed to be as closely related to teachers' modes of attending as to their teaching styles", which is a very deep observation. Before, I said that at the very least a teacher needs to listen in order to figure out what to do next, but this says even more that the way you listen may change the very things the students say. Davis goes on to give three vignettes from Wendy's classroom to display three types of listening.

1. Evaluative listening

When a teacher is listening evaluatively, their reason for listening is to evaluate the correctness of what the student is saying. Ultimately, they are "listening ´Ú´Ç°ùÌýsomething in particular, rather than listening to the speaker". The vignette describes a whole-class discussion where student responses are dismissed until the exact right one was finally accepted. Even right responses that were perceived to be in the wrong form were dismissed. This reminded me of so many times when I had been a frustrated student in such class discussions (and several times I had been a teacher leading one).

I had never quite put my finger on why this felt frustrating until reading this quote from the paper: "No one is attending to the answer in a way that will make a difference to the course of subsequent events". The teacher in such discussions is waiting for the right response in order to continue on their pre-planned course. As Davis says of Wendy's vignette, it was "a teaching sequence that seemed impervious to student input". It sounds harsh, but Davis was more forgiving than that. He noted that Wendy was indeed seeking information from the students. She could see that the students were or were not able to give the responses she hoped for, and she could see that her lesson was more or less successful based on how quickly students could use her explanations to produce the right answers. The listening was doing exactly what she wanted it to do: evaluating.

2. Interpretive Listening

When a teacher is listening interpretively, their reason for listening is to interpret what ideas are actually happening inside the student's minds. They are still usually seeking to bring students to the understanding they perceive as the correct one, but now they do it through figuring out how to talk about ideas in shared ways that move students forwards in their thinking.

Davis notes that teaching sequences with an interpretive listening stance need to have materials that "serve as a commonplace for learners to talk about ideas, enabling the process of re-presentation and revision". For example, in the vignette, Wendy used two-coloured chips to help her students talk about adding and subtracting negative numbers. In my own teaching in the MLC, drawings or play dough often play this role.

3. Hermeneutic Listening

When a teacher is listening hermeneutically, they are listening not only to interpret what their students are thinking, but also to understand how their own thinking relates to that, and how the group as a whole understands. This is my description of it, anyway. Davis has several long paragraphs discussing philosophical and theoretical standpoints, which are a bit heavy (though his style makes it much lighter than I'm sure it could have been). The two main takeaways for me are that understanding isn't only something that lives in one person, but lives in the shared communication of many, and that teachers listen not just to help students grow their understanding but also to change the teacher's own understanding. A relevant quote: "Instead of seeking to prod learners toward particular predetermined understandings, Wendy seems to have engaged, along with her students, in the process of revising her own knowledge of mathematics."

Davis notes that this type of listening seems to go hand in hand with a teacher's conception of what mathematics itself is. You have to be prepared to believe that mathematics concepts have multiple valid ways to understand and describe them, and that mathematics is at least in part a construction of a community, all of whom (including novices) have a part to play in the construction. Otherwise you won't be ready to listen in this way.

A final comment on the terminology... The word "hermeneutic", no matter how often I look up definitions, still remains more-or-less meaningless to me. It seems to refer to a type of inquiry that in itself seems difficult to describe and has different meanings in different disciplines, so I can't borrow meaning from whatever it means elsewhere to make it meaningful in this context, like Davis seems to have done for himself. This makes it hard for me to hold onto the framework.

Yackel et al 2003

In the second paper, the authors are thinking about how teachers structure and restructure their instruction, a process they call "didactising" after Leen Streefland. The reason the paper is here in a post about listening is that the way that you get information about what needs reworking in your instruction is to listen to the students.

I have used the word instruction, as opposed to teaching, because it's the word the authors used. And they really do seem to be thinking about instruction, in the sense of a sequence of explanations and activities you do with students. The main theme of the paper is about how listening to students helps design these instructional sequences, which I do not question the importance of. It's just that the overall feeling I get is that students aren't quite real people but sources of data, and that making good instructional sequences is a good in an of itself, as opposed to being something for the students. I've been a bit too dramatic there, and it's not really as bad as I've made it sound, but still my feeling is that it dances a little too far from viewing the students as people.

Anyway, the most useful thing in the paper for me was a new terminology for what Davis called hermeneutic listening; these authors call it "generative listening".

3. Generative listening

These authors decide to use the word generative rather than hermeneutic because it's easier to process for their purposes. They say, "Listening in this way can ²µ±ð²Ô±ð°ù²¹³Ù±ðÌýor transform one's own mathematical understandings and it can generate a new space of instructional activities."  While Davis was more focused on the way that hermeneutic listening changes the listener and the community's understandings, these authors are more focused on the way generative listening generates new instructional activities. I'm happy to have both in my life. I think it's important to recognise that teachers still have to decide what to do each day and that listening can help them make those decisions!

I'll finish off with three questions the authors list to help people focus on generative listening, which really do bring it back to the students as people at the last moment: "How does student thinking suggest alternative ways of thinking about particular mathematical ideas? How does student thinking suggest what mathematical ideas are experientially real for them? How can the instructional sequence be redesigned to capitalize on the fresh points of view that students offer?"

Some thoughts

From these two papers, we have a framework with three levels of listening: evaluative, interpretive, and generative. The authors of those papers focused a lot on the mindset of the teacher, and how this makes a difference to how you attend to what the students are doing and saying. Davis talks a bit about the kinds of questions people ask when they have those mindsets. But it occurred to me that even if you have a particular mindset, if you ask the wrong questions, you still won't get the information you need. So yes the kind of question you ask is evidence for the kind of listening you hope to do, but also the kind of question you ask can also dictate the kind of listening you have to do, because you will only get certain kinds of responses.

For example, if you ask a yes-or-no question (eg Is this a subspace or not?) or a direct question about factual information (eg What is the definition of subspace?), you are unlikely to get much information about what a student is thinking, even if that's what you hoped for. You will have no choice but to simply evaluate their response.

And there is one question that is famous for giving you no choice for what to listen to: "Does that make sense?" If you ask this of a whole class, students will usually give no response at all. If you ask it of a single student, they'll say "yeah ok". So basically it tells you nothing at all: to ask this question is to give no opportunity for you to listen. So actually there is another lower level of listening: not listening.

And if we're talking about not listening, then there is something worse than asking "Does that make sense?", which at least shows you think things ought to make sense, and theoretically has a chance of a student saying "no" and so giving you some information to work with. What's worse is asking no questions of any kind. It is amazing how often a maths teacher, even one-on-one, will speak continuously for half an hour with no opportunity for the student to say anything. I always feel such a sense of shock and shame when I realise I've done this and that I have absolutely no idea how the student is going.

I think sometimes the impulse to talk continuously comes from a belief that it's your main job to provide explanations, and sometimes it comes from believing in the power of an explanation you've worked hard to perfect. However, even if maths teaching were transmission, that process can't possibly be perfect, and so you really do need to check in every so often! As Davis says, "Implicit in the act of questioning is a certain lack of faith in the transmission process." I think everyone needs to have that certain lack of faith.

So it's good not to have total faith in the power of a single explanation. But what ²õ³ó´Ç³Ü±ô»åÌýyou have faith in? I think you need to have faith that students actually are thinking. Implicit in the interpretive listening stance is the assumption that there is something to listen to. You have to believe that students have ideas if you seek to interpret them. If you don't believe they do have ideas already, then of course you don't seek to listen to them. For me, this is a huge part of working in the MLC that changes the whole approach. The next level above this is to believe that students have ideas that can change your own, which is where generative listening lives.

My version of the framework

So, finally, this is my interpretation of the listening framework of Davis (with the third level renamed by Yackel et al). There are a lot more aspects to this, such as the nature of the teacher's role, but this version helps me think about what I am doing with students on the fly. You can download a if you want.

Level 0: Not listening

Goal:

• Tell what the teacher thinks is important
• Give clear explanations

Types of questions:

• Not asking questions
• "Does that make sense?"

Beliefs:

• Faith in the power of the teacher's explanation
• Students are waiting for your ideas

Level 1: Evaluative listening

Goal:

• Judge student responses against a standard
• Get a specific response so you can continue the plan

Types of questions:

• Yes/no questions
• Direct questions about raw information
• Results of calculations

Beliefs:

• The teacher's explanation is not perfect
• Students are waiting for your ideas

Level 2: Interpretive listening

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• Decipher the sense that students are making
• Understand student thinking
• Create a shared language to describe thinking

Types of questions:

• Open-ended questions about thinking or process

Beliefs:

• Students are reasoning
• Student ideas are worth listening to

Level 3: Generative listening

Goal:

• Jointly explore ideas
• Discover new ways to think about or to learn concepts

Types of questions:

• Open-ended questions about thinking or process
• What-if questions and I-wonder questions

Beliefs:

• Students are reasoning
• Student ideas are worth listening to
• Student ideas can change yours

Final thoughts

I have deliberately numbered the types of listening and called them levels, because I wanted to explicitly say to myself that some are higher than others. However, I don't want to say that you should never seek to provide clear explanations and never listen evaluatively. Of course you should explain things when you need to, and of course there are times when you need to know students can do things in a standard way. And I also don't want to say you should spend all your time listening generatively. That would be exhausting for everyone. It's just that the types of listening definitely do progress in how student ideas shape what happens, and it is definitely a good thing for students to feel that what they think and do makes a difference to the outcome.

What I want is to always be open to the opportunity of finding out how students think and possibly having it change the way I think. I also know that while beliefs definitely guide actions, it also works the other way too. If I spend all my time talking, I may come to believe implicitly that the students have nothing to say. If I spend all my time evaluating against a standard, I may come to believe implicitly that the students have nothing wonderful to say. I need to actively work in opportunities to listen at the higher levels, so that I never go too long without them.

In daily work, where I spend most of my time one-on-one with students, this is even more important. Because when you're right there next to the student, what a waste it would be to never hear the wonderful things they have to share, or to never make something wonderful together.