Nicole Garcia, Recording Student Thinking During a Mathematics Discussion
Rounding Up: Season 1 | Episode 3
Learning to record students’ mathematical thinking might best be described as “on the job training” with a great deal of trial and error and a lot of practice. Today Dr. Nicole Garcia from the University of Michigan talks about the practice of recording student thinking and offers insight on this challenging but crucial practice.
Recording Student Thinking in a Mathematics Discussion
Mike Wallus: If you're anything like me, learning to record students’ mathematical thinking might best be described as on-the-job training, which meant trial and error, and a lot of practice. Our guest on today's podcast is Nicole Garcia, the co-author of an article, published in Mathematics Teacher, that explores the practice of recording student thinking, and offers insights and some principles for making them as productive as possible. Welcome to the podcast, Nicole.
Nicole Garcia: Thank you for having me.
Mike: So you and your co-authors start the article by acknowledging that representing and recording student thinking—when you're in the moment, in a public space, with students—it's challenging, even for veteran teachers. And I suspect that most teachers would agree and appreciate the recognition that this is a skill that takes time and it takes practice. What makes this work challenging and why is it worth investing time to get better at it?
Nicole: Well, so I think you said a lot in your question that points to why this is really difficult work, right? First of all, it's in the moment. We can't predict what students are going to say. We can do some anticipatory work. We might have guesses. And as we move along in our careers, we might have gathered some really good guesses about what students might have to say, but you never can tell in the moment. So unexpected things come up. Students’ phrasing can be really different from time to time, even if we're familiar with an idea. And we're also standing in front of a room full of children, and we're trying to manage a lot in the moment—while we're listening, while we're interpreting those ideas. And then we're trying to figure out: What do we even write down from this mass of ideas that was shared with us? So that's a lot to coordinate, to manage, to think about in the moment. But it's really critical work because part of our goal as mathematics teachers is to build collective knowledge, to support children in being able to listen to, make sense of, interpret one another's ideas, to learn from each other, and to build on one another. And so if we want to make that happen, we need to support making students’ ideas accessible to everyone in the room.
Nicole: And listening is only one part of that, right? If you think about what it takes to make sense of ideas, it takes multiple representations—those are things that we're working on in math. So we need the kids in classrooms to have access to the words that children are speaking. We need them to have access to visual representations of the ideas that are being shared. We need them to have access to the ways that we typically record those things in mathematics—the symbolic notation that we typically use. And we need that to happen all at once if we want kids to be able to unpack, make sense of, and work with others’ ideas. So it's really important work. And I think it's worth investing the time in to get better at this because of the power of having children learn from one another and feel the value of their mathematical ideas.
Mike: You know, as you were speaking, part of what I was doing is making a mental checklist from principles to actions. And I felt like, check one: asking purposeful questions. Check two: connecting mathematical representations. I mean, as you describe this, so much of what we see as really productive practice is wrapped up in this event that takes place when teachers get together and listen to students and try to capture those ideas.
Nicole: And that capturing is really important if we want those ideas to stay with us, right? Like, I think about the number of times that I've been in a discussion with a group of people—it may have been in a class, it may have been in another space—and the whole thing happens. And when I leave, sometimes I wonder, ‘What just happened? What did we think about together? What ideas did we engage in?’ And I can't hold onto them. And recording on the board in the public space offers an opportunity for those ideas to stay with us, for us to hold onto them, for us to revisit and come back to them. So it's critical for continued learning and mathematical growth.
Mike: Absolutely. So this particular part of the article that you wrote—as I was reading it, and you were describing the challenge of recording student thinking during a discussion—this particular statement really struck me, and I'm just going to read it as it was in the article. ‘The thinking being recorded is not the teacher's own, requiring the teacher to set aside their own strategies and interpretations of the math work, to focus on representing student thinking.’ I would love if you could talk about why you felt like it was so important to explicitly call this out in the article.
Nicole: Yeah. So I think that there are a couple of things here that are important. One is that, as a teacher, you're thinking always about the trajectory of your lesson, the trajectory of student learning, where you want to be and steer. And so a lot of times, when we're listening, we're listening for something in particular, right? We have a plan in mind, we have an idea, we know where we want go, and we're listening really carefully for a catchphrase, a vocabulary word—something that we recognize, that we can pick up and pull into the discussion and move forward, right?…and march on, and accomplish our lesson. And a lot of times that kind of natural way of listening is not aligned with what students are actually trying to communicate, because the ways that children express themselves—in particular around mathematics—are really different than the ways that adults, who know math well, express their ideas about mathematics. So there's a lot to hear in the language that they're using, in the trajectory of their talk, that's both difficult to follow and difficult to figure out what the big idea is that they're communicating. And when we're listening for our own understanding, our own ways of working, our own strategies, we often miss what children are actually bringing to the discussion, to the conversation. We miss their thinking. I think about the number of times where I've been a student in class and I've said something and the teacher rephrases it in the way that they really wish that I would have said the thing.
Nicole: And it's not, like, it's not even my idea anymore, but you kind of nod and you go along with it. And so I think, you know, as a teacher, you get those cues that, yes, you did just rephrase what the kid said. They just said, ‘OK.’ And you record that thing and you move on. And so I think reflection— checking back in with children about whether or not you heard their idea, whether or not the representation that you're putting on the board actually matches what they were thinking about—is really, really critical. Because it isn't your thinking. It's the child's thinking and we want to make sure that that's what we're representing.
Mike: Yeah. I read this and I will confess that a part of me thought back to the points in time when I was teaching kindergarten and first grade. And I suspect anyone who's taught and tried to record students’ thinking has been in a spot where you have kind of a pathway that you're thinking the learning will follow. You have an idea of how the big ideas might roll themselves out.
Mike: And I think what I found myself thinking is, there are certainly many, many times where I felt like I was true to student's ideas, but I was really conscious that there were definitely points where, what I heard and what I represented differed, probably because I was thinking to myself, ‘Gosh, I really want this model to kind of come forward.’ And the truth was, the kids weren't taking me there and I was trying to force it. I guess what I'm saying is, it really caused me to think back on my own practice and really kind of reconsider—even when I'm doing professional learning with other adults and children— the need to listen, as opposed to kind of have the path sketched out in my own mind.
Nicole: Well, it's really difficult to do, because sometimes as a teacher, you really do need the lesson to go in a particular direction. There are all kinds of constraints around teaching. And I think what's important is knowing that you've made that decision. (laughs) Right? Because sometimes you might. You might…
Nicole: …rephrase it a particular way because that's the move that you need to make in that moment. And I think that sometimes that can be OK. We need to give ourselves permission as teachers to make the best choices for our whole class and the students whose ideas are being shared in the moment. But I think knowing that that's what you're doing is really important,
Mike: Right. Like, it's a conscious decision to say, ‘I've heard that. I'm going to take this in a different direction.’ Rather than just imagining, ‘I've heard that. I'm going to represent it.’ And not kind of questioning whether what's being represented is the student's thinking or your own thinking.
Nicole: Right. Or even better, making the decision that, ‘I heard, what that child said. And I'm going to say back to them,’ for example, ‘so I think what I heard you say is…bop, bop, bop. Can I try an idea out?’, and actually sharing the idea that you have on tap. Or saying something like, ‘You know, I've heard some of my students in the past say something really similar. Can I share that idea with you? And let's see what's similar or different.’ So thinking about how can you get that idea out there, that you really wanted to record, that the student didn't say, in a way that isn't totally disingenuous—pretending you heard something that you didn't hear.
Mike: Right. You're kind of acknowledging that they said something and you're…. It's powerful; the language you used is really subtle. But it's essentially saying, ‘I've got something that I'd like to contribute that your idea made me think about,’ or…
Mike: …that you want to also put out there. And I think that subtlety is important. Because as you were describing that feeling of, ‘I said something. Teacher revoiced it in a way that was totally different,’ and kind of the bad aftertaste that that left.
Mike: You know, that subtle ask—of the child—for permission, really kind of shifts that dynamic.
Nicole: It's saying, ‘I value your idea and let's consider this other idea.’ It's OK for teachers to put ideas out in the space.
Nicole: But acknowledging that that isn't what you heard and you're going to record this other thing, or maybe you record both of them…
Nicole: …and talk about the similarities and differences.
Mike: So I'd love to shift just a little bit and talk about the role that recording can play in developing students’ mathematical vocabulary. And I'm wondering if you could talk about the ways that recording can help students make connections between their informal language and the more formal mathematical vocabulary that we want them to start to be able to use. Can you talk a little bit about what that might look like?
Nicole: Yeah. So I think that there are a couple of ideas to be thinking about. One is that we actually know a lot about how children develop vocabulary. We know that that's a progression and that students need opportunities to play around with ideas, to have something to hang that vocabulary word on.
Nicole: Once they have the kind of core idea and they have some informal language—some way to describe that idea—that's the prime place to be able to introduce the formal mathematical vocabulary. They're able to make connections to that big picture, that core idea that they've come up with. They have some informal language to go around with it. And now they have a real name for it—the formal mathematical name for it. We also know that one of the ways that students remember and are able to recall—and use appropriately—vocabulary is by having a visual representation that goes along with that mathematical vocabulary.
Nicole: So one way that representations and recordings can support students in learning that vocabulary is first, by having them build some representations that go with that vocabulary word, but then also having those labels on the representations that make their way onto our boards.
Mike: Ah, yep.
Nicole: In addition, you know, when we do things like dual labeling, um, where maybe in our classroom space, we've named something with someone's name, right? As we're beginning to talk about an idea, we might call it Diego's idea, Diego's strategy. Then when it makes our way onto the board, we can label it with ‘Diego's strategy’ and the formal mathematical name for it so students are able to connect the of things. But even if it's not a student's name as the name of the strategy, there's lots of informal language that students bring to mathematical ideas. They have to have a way to talk about things. And so we can dual label those ideas on our board to help students make that connection and to let them walk between using their informal language and using that formal mathematical language, and being OK with that.
Mike: So just to go back… Describe dual labeling again, because I think I've got an idea of it, but I want to make sure in my own mind I've captured that correctly. How does that work?
Nicole: Let's imagine that we have a strategy—a student has shared a subtraction strategy in our discussion, and I've represented that strategy on the board, say, using a number line.
Nicole: And, say, the kids are calling it scooting—they're scooting the numbers to make this subtraction problem. So I might actually write on my board, like, on the left hand side of the strategy ‘scooting,’…
Nicole: …and then on the right hand side, label it ‘shifting the numbers’ or whatever our formal mathematical language is going to be for our classroom. So we have both of those things labeled on top of the strategy. And I might even draw a double sided arrow between the two to help…
Mike: Oh! OK.
Nicole: …[undecipherable] that the strategy that's there has these two names and I can use those names interchangeably. But over time, we get to a place where we're calling it by its formal name. And kids also have the idea that, ‘oh, that's the one that's the scooting strategy.’ They have their own name that they gave that idea
Mike: That is really helpful. And I think the example you shared really kind of shows how dual labeling kind of progresses and there's almost kind of a fade out at a certain point. Not that you're purposely not permitting kids to use ‘scooting,’ but that a certain point you're kind of fading and you're starting to use the more formal name. They can use it,…
Mike: …but that you're really kind of trying to help them make a transition to the formal vocabulary.
Nicole: Um-hm. And if you think about, you know, kids are really used to using multiple names for things.
Nicole: They have nicknames that they use at home,…
Nicole: …they have their home name, they have their school name, they have their friend name. There are lots of different labels on the same kind of thing. So that's a natural progression of language for them. And it doesn't cause complications to have, like, these multiple names for this idea. And we can shift toward using the formal language once everybody has that tied up.
Mike: Yep. So as I was preparing for this interview, and even as I was reading the article, I found myself thinking about my life as an elementary school teacher. And I think what I found myself thinking was, is that I learned how to facilitate and record math discussions—like a lot of folks—trial and error and a heck of a lot of practice. And I think what I really appreciated about what you and your co-authors put together is that you actually laid out some principles for recording that support mathematical understanding. And I'm wondering if you could just unpack some of the principles that you think are important, Nicole.
Nicole: Yeah. So as we… as we were working on these principles, we were trying to think about, like, what are the big ideas of what gets recorded, right?, and how we record in a classroom. What are the big things that we want to make sure get attention in that work? And so we kind of organized under three big umbrellas of principles, one being around advancing mathematical ideas. Because the goal of discussion in mathematics is to build ideas together and to move the mathematics forward using student ideas. So when we think about what gets recorded, we want to record in ways that are helping us build those mathematical ideas together. So in that area, we'd really be thinking about recording the core ideas, deciding, like: What's important enough to get on the board? What do I want to make sure gets up there that's going to help push people's thinking forward? And then at the same time, thinking about: What's the right level of detail?
Nicole: Sometimes you look at a board recording… If you walked out of the room and you came back in and you looked at it, you would have no idea what happened…
Nicole: …what had gone on, right?
Nicole: Like, there's not enough there to really, like, get a sense of what happened. But sometimes there's so much there that it's a jumble and you can't discern, like: What's important here? So that ‘just right’ space of managing the detail—so there's enough that you can make sense of it when you come back the next day, you get what happened; it's enough to prompt your memory, but it's not overwhelming—um, is really important because we want kids to be building on those ideas over time. So we want those recordings to be in that kind of level of detail. And then thinking about that arrangement. Where am I going to put things so that I can help students make connections between the ideas that have been shared? Right? Do I want kids’ strategies to be next to each other? Are there particular strategies that, if I stack them on top of each other, kids are going to be able to see different kinds of connections,…
Nicole: …similarities, or differences? Like, where they are in relation to each other, if you think about how we make sense of space, matters.
Nicole: So that was… that's one kind of bucket. A second bucket is really respecting students as sense makers. And this comes back to what we were talking about earlier, with really paying attention to: What were students trying to communicate? So, ‘Did I actually record what the student said it or did I write down what I wish they had said?’ But trying to stay true to: What was the core of that student's idea? And am I representing that correctly? But then also adding enough detail so that the other students in the class can figure out what that student's idea was about. And we can do that through questioning, but part of that has to come out in the recording as well, because we want that record to be like the full representation of the ideas that students are communicating. And then labeling those ideas so that we're able to talk about them easier, right?…
Nicole: …that we're not just like pointing to a general space, but we have some language, we have some vocabulary, we have some kind of label to be able to talk easily across those ideas.
Mike: I had a follow up that I wanted to ask you. So, again, I'm paraphrasing, but one of the things that really stood out for me in the way that you unpacked the principles was: Our recording should show the thinking behind the idea rather than the steps in the solution alone. I would love for you to expand on that a bit.
Nicole: Yeah. So the thing that we're trying to get out when students are sharing strategies in class, when they're sharing the ideas in class, is in some ways the generalizability—to use my big math vocabulary. We want to get to what is the core of the idea that they're sharing that can be used across multiple kinds of problems in lots of different ways. And so recording just the steps that get followed, may show—or it may not—the steps that somebody followed for that particular problem, but doesn't show the thinking that could be used to solve other similar or different kinds of problems. Right? So we will want to be able to record in a way that gets to the heart of the thinking. So if you think about a student, for example, using counting up to solve a subtraction problem,…
Nicole: …then I might think about what's important are the steps that a student is taking to count up. So they're either thinking about it on a number line and they're hopping along the number line to count from one number to another. And so on the board, I would actually want to record those hops because that's the underlying idea—is that we're looking at the repeated unit distance between those two numbers.
Nicole: OK? If a student is counting up using their fingers,…
Nicole: …then I might want to actually record a hand on the board and the count that the student is doing, so that other students in the class are able to try out that strategy, use that strategy, and think about when it's useful. But if all I've recorded on the board, are the words ‘counting up’ and then the problem that they solved, that doesn't necessarily support other people in being able to try out that strategy or that idea, or even think about when would it be useful or not.
Mike: That's super helpful. I love the idea of generalizability. If I've done recording well, allows other kids to have access to the strategy that's being highlighted, rather than simply putting together the steps that showed how a person came to this individual answer, at this particular task, at this particular time. That's a really helpful clarification, I think—in my mind.
Nicole: If you even think about things like annotation and the power that annotation on a recording can have. And we think about the U.S. standard algorithm for addition,…
Nicole: …where students are… they're adding and when they get a number that's greater than nine, they're making groups and carrying that group, right?, to the next place value. If we're actually annotating that process with what each of the numbers means as we're doing that work together, that can really support students in continuing to make meaning. I think that one of the things that often happens is, we make meaning when we're introducing the algorithm, we do some work together. Students are really in a place where they're understanding place value, they're understanding making groups, they get what that recording means. And then we kind of say, ‘Great, then we're just going to record this way from now moving forward.’ And we continue to do that recording without the kind of reinforcement about, again, what are… what are we saying these numbers mean? What are we actually doing here? And so we move from meaning toward this recording without meaning?
Mike: Sure. That absolutely makes sense.
Nicole: Very quickly for children. And then, you know, too… I know that, for example, my fifth grade teachers would say that oftentimes their kids come to them and… and can't explain what's happening when kids do that addition. They do the work—they know how to do the work—but they can't say what it is that they're doing. Right? And so annotation can really support that, that remembering of what have we…? What kind of collective understanding have we come to?
Mike: Sure. That totally makes sense. So I wanted to ask you a bit about guidance that you'd offer to teachers. I suspect there's a fair number of people who are listening, who are really thinking about their own practice and are wondering: What steps might I take as a teacher—or maybe within the team of folks that I work with—to really try to attend to the principals and the practices that we've talked about? What's your sense of how teachers can support one another in, kind of, practicing the principles that that we've unpacked today?
Nicole: So I think there are lots of options for what it might look like to focus on and practice this work together in a teaching community. I think one way that we talked about in the article—and it's not the only way—is using video. There are lots of videos that are available on YouTube, on TeacherTube, etc., of classrooms where people are leading discussions, are recording student thinking. There are lots of videos of student thinking out there where—in a pretty short amount of time—I could, with my peers, watch this video and practice recording—either on a board, on a chart paper, on paper in front of me— recording what I'm hearing from students. And then afterwards comparing our recordings together and talking across them. What are the features that each of us has picked up on? In what ways were we in line with what the student was sharing? Where are there differences in how we interpreted what a student was sharing? And that's a pretty quick activity. I can find a five-minute video. We can do that work together, talk about it in, like, tops 20 minutes, really, to do that kind of activity together. We can also do work where we’re visiting each other's classrooms.
Mike: That's what you had me thinking, Nicole.
Mike: Yeah, absolutely!
Nicole: I can go to somebody's classroom. I can—on my lap—have my piece of paper where I'm trying to record as students are talking. And after that lesson, debrief with a teacher that I'm observing, about, ‘What was it that you decided to record? How did you make that decision? Here's what I had.’ And really talk across those ideas because it's small changes in practice over time. This is an overwhelming set of work, this recording work. And it's going to get better by increments, but it's going to take practice, talking with colleagues, and really coming back to these principles and thinking about: Am I adhering to these things? Where is it that I really want to work and I improve my practice? Because I would encourage people to pick one—to start with—that you really want to get better at and focus on that one.
Mike: Yeah. I think what's powerful about this too, is that I would imagine you could certainly do some of the things that you described if you were the only teacher at a grade level.
Mike: But gosh, when you put other people together and think about the ability to help one another raise your consciousness about why you made a particular decision or why you chose to go in a certain direction with a representation… That's kind of that intricacy where teachers can really help one another. I mean, we are keen observers of behavior. That's… (laughs) that's kind of the bread and butter of a lot of what we're doing when we're talking about differentiation. It's really powerful to think that teachers could help one another build their craft around this.
Nicole: Um-hm. Well, and it's… it's a really interesting practice, I think, in that there isn't one right way. (laughs) Right? There isn't a right way to represent a particular idea. Um, there are lots of really good features of different kinds of recordings, and so there's lots to discuss and… and a lot to learn from each other. And your… your comment about the being alone had me thinking about the work that you can do just by studying student work…
Nicole: …and thinking about: How are students inclined to represent their particular ideas and how might I translate that into how I represent things for the class on the board? Because students do a lot of their own translation of their thinking into representations on their homework. We can pull student work sets. You know, if we look at Inside Mathematics, there are lots of student work site, sets up there on that site that you can pull and study and look at how children are inclined to show their thinking.
Mike: So I'm going to back up and just ask if you can identify and source that resource that you just shared about Inside Mathematics. Would you… would you mind—for people who might not be familiar—just unpacking what that is and where folks can find it?
Nicole: Yeah. So, Inside Mathematics is a really great resource for teachers. It came out of a project funded by the Noyce Foundation. The website is insidemathematics.org, and it's currently housed at the Dana Center at The University of Texas at Austin.
Nicole: Great resources for teachers. There are videos of lessons. There are problems. There are assessments. There are lots of resources up there, but one of my favorite resources is that, with each of the problems, they have student work samples. And so you can really see a lot of student thinking inside of those.
Mike: That's fantastic. You really answered my last question, which was going to be: For folks who, again, are listening to this conversation and thinking about steps, they might take… resources that you would recommend to someone who's really wanting to think more deeply about representation and the practice of representing student thinking.
Nicole: So I think the big three are ones that we've covered and that would be visiting your colleagues classrooms—
Nicole: …whether in person or via video—depending on what the setup of your school is; visiting sites of video, right?, so going to YouTube, TeacherTube—seeing how people are representing that work and then comparing how you might choose to represent that work; and really digging into student representations of their own thinking.
Mike: That's fantastic. Nicole, thank you so much for joining us today. It has absolutely been a pleasure to talk to you.
Nicole: Thank you so much for having me. It's been really fun.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.