Today on the Heinemann Podcast, how do we have productive conversations that help surface a student’s mathematical thinking?
We confer often with our readers and writers, but these conversations are just as needed in mathematics. What does it look like to talk with students about their mathematical thinking so that their thinking grows? How do we know when to step in and when to nudge? In Jen Munson’s new book, “In the Moment,” she offers educators clear guidance for conferring with your students in math, responding to their thinking, and helping them grow their mathematical ideas.
Our conversation begins with learning to embrace opportunities for what Jen calls “productive struggle”…
Below is a full transcript of the conversation...
Brett: You opened the book up talking about how you struggled on how not to say too much or when to step in. You write that, when you first started teaching you really needed to lean on your students to teach you, on when to step in and when not to say too much. What was that process like for you? Talk me through your thinking there?
Jen: When I first started teaching, I was lucky enough to have some materials that asked students to solve problems together. Which gave me a good start in the right direction. But as I was watching them solve those problems, I knew that they were struggling. And I didn't quite know how to interpret that struggle. I hadn't been introduced to the idea of productive struggle yet. I had an intuitive sense that it was a good thing. But knowing when to step into that struggle, to push them, and how to do it, so that I didn't do all the work for them, was really, really tricky. I found that I had to read the kids to decide when I had done the right thing. I would know from what they said back to me, if I had made the right move. But I learned by reading their responses.
When I would ask a question and then ask it again, and ask it again, and I kept getting blank stares, or sort of hesitant responses, I knew I was just on the wrong path. When I was asking a question and all of the sudden their eyes would light up, or they would start to talk, and even if it wasn't what I expected that they were going to say, but I could her their voices, I knew I was on to something. And I didn't always know what I was on to, but at least when I was hearing their voices, and I was hearing their thinking, I knew I was at least onto their thinking rather than my own.
Brett: Mm-hmm (affirmative). So you've taken that and you learned quite a bit, you've taught at a number of different schools across the country. From your perspective, what should a productive math conversation look or sound like?
Jen: Well when you step into students thinking in the midst of them working, we know that they're not there yet. They're in the middle of their process. And what it should sound like, is it should sound a little messy. It should sound like, thinking that is evolving. The opportunity for teachers to find out what are you thinking right now? Even though it's not fully formed, you haven't perhaps found an answer yet, and you may even be doubting your own thinking. It should first look like, getting a sense, and getting your mind around what students are thinking right now. And then, there should be some move that we can make, to extend that thinking. To push it just a little bit further, in a direction that's productive, that's going to get students a little bit further on the pathway that they want to be on.
They are thinking about solving an aspect of a probably, and they're trying to figure out how to put in on paper. And we want to figure out how we can push them a little bit farther, into making a plan for them. And so these conversations aren't meant to be full instructional moments, from start to finish. They're just an intervention in the middle of that process. Just a little bit to help us learn about student thinking, and push it just a little bit farther.
Brett: When you talk about math conferring, how is it different from other kinds of conferring?
Jen: So when you confer with readers and writers, in my experience as an elementary teacher, was that I often would make a schedule for wanting to confer with my readers and writers so that I got to everybody over the course of say, a week, or maybe a little bit longer, depending on how many students I had, and how much time I had for my readers and writers workshop. But in mathematics, I want to get to as many kids as I possibly can every day. And I'm helped to do that by the fact that students are collaborating with each other. So I can sit down with two, or three, or four students at a time, and really dig into their collective thinking, and then move on to another group.
So I really could probably talk to at least half of my class in a particular day. I might even be able to talk to the same kids more than once, so that I could revisit the process and see how it's unfolding. The conversations tend to be a little bit quicker, because I'm not intending this to be the one time I get to talk with you this week. It's just a snapshot of where you are in today's process. So I get to talk with more kids, more frequently. And I'm not trying to model my own thinking. I'm trying to figure out where students are in their own thinking, and just to nudge it just a little bit farther, along their own pathway. Which is just slightly different than what we do in reading and writing.
Brett: There's two things that write about in the book, and I want to ask you about both of them. You write a lot about nudging. You give us a tremendous amount of advice of when to nudge, how far to nudge, strategies on nudging. I want to ask you a little bit more about that. But I also want to kind of marry that to, you write about surfacing a student's thinking. So how do we best nudge, and how do we best surface the student's thinking?
Jen: Well surfacing students' thinking, it has to come first. And it's the most important job that we're going to do in that conversation. We can't figure out how to push a students thinking, if we don't know what it is. And oftentimes students' thinking is, like I said, still in process. But also it's communicated in ways that are difficult to interpret. Kids don't always communicate their thinking in ways that are linear, step by step. They often start sentences and then restart them, and they may talk in circular ways. Their language may be imprecise because they're still learning the language of mathematics.
So all of these things make just understanding what students are thinking, and giving them the opportunity to grapple with describing it, really, really challenging, but fundamentally important. So the first thing we have to do is ask as many questions of students as we possibly can to find out what are you thinking. What's important about that is that first it's a chance for formative assessment. Like I get to learn exactly what you're thinking right now. I get to learn where you are in your learning in this very instant. It's the most immediate form of formative assessment I can imagine.
But also, the students themselves get a tremendous benefit from the chance to articulate that thinking. Putting their thinking in words is hard, and it's an act of learning itself, to describe your thinking. Anybody who's ever been asked to try to describe a thought that's just beginning to come together in their mind, like putting that into words is hard. And you learn a little something about what you're thinking as you try to put words to it. And research shows that this is what students are doing when you elicit their thinking and you probe their reasoning. That they're not just reporting out to you something that's already in their minds, they're actually putting the pieces together as they're talking.
So those two things are happening at once, and it's incredibly important. Once students have a sense of what they're thinking, and you have a clear sense of what they're thinking, this is our opportunity to nudge that thinking forward. So nudging is about asking, usually a provocative question that can take where they are, and push it just a little bit forward. Whether that's about how they represent their thinking mathematically, about how they're communicating their thinking, about their conceptual understanding of the mathematics involved in the problem, the strategies that they're choosing, or even how they're working together with partners, which is no small effort in and of itself.
Brett: On of the things I love about this book, is how when you write about something like that, or you write about those conversations, you have anticipated the reader's thinking about the question that comes next. Certainly in my mind as I was reading, my first question when I read that was, but what if the student gives me the no answer or the one word answer? Or what if it's always the same student talking all the time? And as I read it, both of those questions immediately came up in the book. So, I'm going to ask you, based on what you just said, what if it's the brief one word answer from a student, and or what if it's the student that's always answering, and we can't get the other students to speak up?
Jen: Yeah, well I'm not new to this work, so the wrong question, more times than I can count. And I have encountered all of those responses, and I still do. There's no opportunity to get this perfect every time. Because every conversation is new. And even when you know your students well, that helps a tremendous amount, but today's problem is a new problem. And so every conversation can be a challenge. And that's part of what I love about this work. It never gets to be routine and old hat. But when students give me the blank stare or when they look at me and they give me a single word answer if I ask them like, is there another way that you could try that? And they just say no, which happens. What it tells me is that I probably haven't asked the right question, or perhaps I haven't asked it in a way that the students really understand what I'm asking them to do. And so I need to step back and think where have I gotten off track. And usually what that means is, me asking them again to tell me about their thinking. I can step back to eliciting their thinking, and I can get more. Because if I've misinterpreted what they need, my only tool is to get a little more information.
And usually when kids are describing their thinking, I can then kind of pinpoint something. I can say, so well you just said this, now, is there a way that you could say it a different way? Or is there a way you could show it on your paper. If I can kind of tie my question really tightly to what children are saying, I'm far more likely to get a rich response from kids. And the no, or the one word answer or a blank.
Now the issue of having a dominate kid in a group is a different one. Because what it means is that you get a lot of information about one kid and next to nothing about the rest. What it can mean, is that one kid is dominating the group, and the others are in one way or another marginalized. Which is not at all what we want from collaborative mathematics. It kind of defies the promise of working together, that we can do more together, than we can separately. And so to me that's a big red flag, and I try to pursue that by deliberately structuring my conversation so I ask questions of all of the kids. And if I hear one voice a lot, I might even just say that out loud. Like I've heard a lot from you, what do you all think and turn to the rest of the group. And deliberately and pointedly ask them questions about how they are understanding, or what their process is.
And I want to hear it from them in their own words. Kids very rarely all say the same thing in the same set of words. If they really are understanding it, they probably say it and understand it in slightly different shades. And so I want to hear that, and if I do then I know that the group is actually all working together, and on a pathway together. But oftentimes you don't. Oftentimes you hear one student. When one student is talking a lot, the other students aren't with them and have in some ways been marginalized in that group, and haven't had access to working on the problem. And to me that doesn't reflect that one kid is learning and the other three aren't, it means that one kid is dominating and taking away the opportunity from the other three kids to learn. And so that requires me to intervene on how the group is working together.
The great news on that front is that, kids meet our expectations. And if we talk with them directly about what we expect of them, about how to work together, and we teach them how to work together, they will. I've seen that again, and again in classrooms from kindergarten through middle school.
Brett: Well the book is beautifully designed. I absolutely love it. It's incredibly thoughtful. It's just absolutely brilliant. Explain to people your best thinking for how you'd like readers to use the book?
Jen: Well I think first as with all of teaching mathematics, if we're striving to teach math in a way that is meaningful for kids, we're probably trying to do it in a way that was different than the way we were taught. Again and again as I go across the country, and talk with teachers from the last like 15 or 20 years, people have had the same experience I had, and there's very little difference. And that experience typically wasn't rich with talk. So the first step I think is building a vision of what this kind of math teaching looks like. And I hope that the book is laid out in a way that allows readers and teachers to think about what would it look like to confer with my students during math? What would this sound like? And how might it fit into my classroom? There are written examples and videos that hopefully can establish that kind of vision. And I think that's really the first step, is trying to figure out what it looks like.
Then second, knowing what it looks like, isn't enough to be able to make it happen in your classroom. And I know as a teacher that I have often seen some really gorgeous teaching and then gone back to try to implement it in my classroom and realized like how hard the mechanics of that work actually are. So what I hope to do is to kind of take apart the practice of conferring, into some pieces that make them learnable. And that we can practice in classrooms. And so that teachers can use both the text in the book, and also the video examples to think about how to practice with different parts of what it means to confer.
Like how do we surface student thinking and figure out where kids are? What does it mean to nudge like this? What does it mean to nudge like that? How do I deal with the different problems that come up, because they will? And to start to practice those in pieces so we can kind of put the whole practice of conferring back together, into something that really feels comfortable for you and your students as a daily way of living in the math classroom.
Brett: There's a lot of videos with this book. Can you just walk people through a little bit of what they can expect in the videos?
Jen: So I collaborated with two teachers in a school that serves a predominantly Latinx and Pacific Islander population, in a Title I school in northern California. They served some of the students that people often say can't do the rich math work, that you will see them doing in these videos. And they are able to make conferring and rich math work in their classrooms. So one thing that you'll see in the videos, is a fourth grade teacher Mary, and a second grade teacher, Faith, talking with their students about the mathematics that they were doing on the days that we came to film. And it's just the math that they were doing that day and just the conversations that they were having with kids on those days. And those conversations include everything the particular kids in front of them happen to need in those moments. So what you'll get to see is how they elicit their students thinking, how they think about what students are doing, the questions that they ask, and how they pursue those ideas to try to push them forward.
You'll also see some examples of struggling with those conversations. Because that happens with real teachers every day. And there are some great videos of both Mary and Faith reflecting on their practice after those lessons, that allow us to get a view of how they were thinking about those conversations, the decisions they were making, and how those conversations influence what they are going to do the next day. The big goal of all of this work is that, each little conversation impacts the kids that are in front of us, but also, they collectively impact our whole instruction. They are collectively formative assessment, and together they tell us, what kids need tomorrow. And they can allow us to make really important instructional decisions about the whole group as well as the individual kids. So we get to see a little bit about how Mary and Faith reflect on those kinds of decisions, and think about what they're going to do next.
Jen Munson is a postdoctoral fellow in learning sciences at Northwestern University, a former classroom teacher, and a professional developer who works with teachers and school leaders across the U.S. to develop responsive, equitable mathematics instruction. She is coauthor of the Mindset Mathematics curriculum series.