Rounding Up

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.

Resources

Recording Student Thinking in a Mathematics Discussion

Transcripts

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.

Mike: Hmm.

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.

Mike: Yes.

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.

Nicole: Um-hm.

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…

Mike: Yes!

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…

Nicole: Um-hm.

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.

Nicole: Yeah.

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.

Mike: Um-hm.

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…

Mike: Right!

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.

Mike: Um-hm.

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.

Mike: Hmm.

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.

Mike: Okay.

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,'…

Mike: Um-hm.

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,…

Nicole: Um-hm.

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.

Mike: Um-hm.

Nicole: They have nicknames that they use at home,…

Mike: Yep.

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?

Mike: Um-hm.

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…

Mike: (laughs)

Nicole: …what had gone on, right?

Mike: Yes!

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,…

Mike: Um-hm.

Nicole: …similarities, or differences? Like, where they are in relation to each other, if you think about how we make sense of space, matters.

Mike: Yes.

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?…

Mike: Um-hm.

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,…

Mike: Um-hm.

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.

Mike: Um-hm.

Nicole: OK? If a student is counting up using their fingers,…

Mike: Um-hm.

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,…

Mike: Um-hm.

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.

Nicole: Yeah

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.

Nicole: Yeah.

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…

Mike: Um-hm.

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.

Mike: Gotcha.

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—

Mike: Um-hm.

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.

© 2022 The Math Learning Center | www.mathlearningcenter.org

Rounding Up: Season 1 | Episode 1

Educational theorist Charles Degarmo once said, "To question well is to teach well. In the skillful use of the question more than anything else lies the fine art of teaching." Today, Dr. DeAnn Huinker, author of Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5, talks about the art and the science of questioning and ways that teachers can maximize the impact of their questions on student learning.

Transcripts

Mike Wallus: Educational theorist Charles De Garmo once said, 'To question well is to teach well. In the skillful use of the question, more than anything else, lies the fine art of teaching.' Our guest today, DeAnn Huinker, is one of the co-authors of 'Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K–5.' We'll talk with DeAnn about the art and the science of questioning and the ways that teachers can maximize the impact of their questions on student learning. DeAnn, welcome to the podcast. It's great to have you.

DeAnn Huinker: I'm happy to be here, Mike. I'm looking forward to our conversation today.

Mike: So, I'd like to start by noting that NCTM (National Council of Teachers of Mathematics) has identified posing purposeful questions as a high-leverage practice in 'Principles to Actions,' and then again in 2017 with the publication of 'Taking Action.' And I'm wondering if you can make the case for why educators should see purposeful questions as a critical part of this practice.

DeAnn: Yeah, certainly. Let's just jump right in here. As we think about purposeful questions and why we as teachers need to be more intentional and strategic in the questions we use … I was honored to be a member of the writing team for 'Principals to Actions.' And in writing that document, we were really tasked with identifying a set of high-leverage teaching practices for mathematics. We reviewed the research from the previous 25 years (chuckles), and it was really clear: There's been a lot of research on teacher questioning. And what are the characteristics of effective questioning. So, as I think about making this case for purposeful questions, a couple things come to mind. First of all, researchers have estimated that teachers ask up to 400 questions each day in the classroom. I mean, that's more than one question every minute for the entire school day.

Mike: That's incredible (sniffs).

DeAnn: (chuckles) I know. That's a lot of questions. Also, if we think about it, it's not just how many questions we ask, but what questions. Because that depth of student learning is really dependent on the questions we ask them because our questions prompt them to consider and engage with the specific mathematical ideas that we're helping them to learn. The other thing I'd like to add to this, is that our questions also set the tone for what it means to learn and do mathematics.

Mike: Hmm.

DeAnn: Are we asking questions about getting answers or are we asking questions that let students know we value and respect their inquiries into mathematical ideas into problem solving, and that we really are about helping them make sense of mathematics? I think it's essential that we critically examine the types of questions we ask and how we can use them to best serve our students.

Mike: That's a really interesting way to think about it. That the questions we ask are really signaling to kids, 'What is mathematics?' In some ways we're informing their definition of mathematics via the questions that we ask.

DeAnn: Yeah, I absolutely agree with you.

Mike: Well, I think one of the most eye-opening things for me to think about lately has been just learning more about the different categories of questions and the different purposes that they can serve. So, I'm wondering if you can briefly sketch out some of the types of questions teachers could put to use in their classrooms.

DeAnn: So, in 'Principals to Actions,' we really looked at a lot of different frameworks that people have established over the years for questioning. And we kind of boiled it down to four specific types that are particularly important for mathematics teaching. One is to gather information. For example, can students remember the names for different types of triangles? Another is to probe student thinking. This is when we want them to further explain, elaborate or clarify their thinking. Uh, third type—which is my favorite category—are questions that make the mathematics visible. In other words, these are questions that prompt students to consider and explicitly discuss the underlying math concepts. Or that we want them to make connections among math ideas and relationships. Let me give you an example. If I were going to ask students to explain how to represent 3 × 5 with an array, they would have to consider more deeply the meaning of each of those numbers and that expression, and how that would connect to the representation. So, we're really getting at the mathematics there, not perhaps the problem or tasks that cause them to think about 3 × 5.

Mike: I see. I see.

DeAnn: Fourth category [is] questions that encourage students to reflect and justify. And I think of these as the why questions. Why does it work to solve 4 × 6 by adding 12 + 12? So, those are the four categories that we identified in 'Principles to Actions.' But since that time, in the 'Taking Action' book at the elementary level, my co-author and I decided to add a fifth category, because these questions really are emerging in classrooms often. So that fifth category is asking questions that encourage students to engage with the reasoning of other students. Many people refer to these as talk moves. For example, if we think about these talk moves that teachers use in their classrooms or that we need to use more often in our classrooms, an example would be, 'Who could add on to what Mateo just said?' Or another example would be, 'Could someone describe or put into their own words the strategy that Jasmine was just telling us about?' Those talk moves are the ones that really get students to listen to and start to have conversations with each other.

Mike: It's interesting because what comes to mind is, there are multiple reasons why that's such an important thing to do in the classroom. In addition to engaging with the reasoning, what it makes me think is it also gives the teacher the opportunity to position a child who may potentially have been marginalized as someone who has math knowledge or whose ideas are valuable.

DeAnn: As I was thinking about talking with you today, Mike, I got thinking about that same idea, which is how do we use questions to position students as capable and as having mathematical authority? And I think this is actually a new area in mathematics education that we need to explore in research further. Just by saying, 'Can you repeat what Jasmine just said?' I'm actually marking her idea as probably something we should all listen to and consider more deeply.

Mike: Absolutely.

DeAnn: I definitely agree that we can use questions to position students as capable in math classrooms, which is something that's greatly needed these days. And that also helps students develop a more positive math identity in themselves and even fosters their math agency and capability in the classroom.

Mike: So, for me at least, personally, my perspective on questions really changed after I read the '5 Practices for Orchestrating Productive Mathematics Discussions' that was written by Margaret Smith and Mary Kay Stein. And after I read that, I really found myself investing a lot more time in preplanning my questions. So, what are your thoughts about whether or how teachers should approach preplanning questions?

DeAnn: In the five practices model, the first practice is to anticipate. This involves anticipating student responses to the kind of key math task of the lesson, and also planning questions so that you, as the teacher, are ready to respond to your students.

Mike: Uh-hm.

DeAnn: Practice of anticipating is preplanning. And I would strongly suggest having those questions written down on a piece of paper so that we are ready to refer to them during the lesson, because it's going to keep us on track, and it's going to give us those tools to help press students to talk more about the mathematical ideas that we want to surface.

Mike: I think part of what really is illuminating for me is, we're anticipating how students might think, and then really, we're digging into what's a response that can help advance their thinking regardless of the angle that they're coming at the task from. So, it's, in some ways, what we're talking about preplanning questions, is really kind of a differentiation strategy to some degree.

DeAnn: Perhaps we want to talk a little bit more about the use of the math teaching practice talks about asking both assessing questions and advancing questions?

Mike: Yeah.

DeAnn: So, let's dig into that a little bit.

Mike: Yeah .

DeAnn: The teaching practice from NCTM says we should be using purposeful questions to both assess and advanced students' reasoning and their sense-making about important math ideas and relationships. So, assessing questions are those that really draw out students' current understanding and strategies. And then advancing questions are those that really move students forward in their thinking and understanding—and pushes or presses them or pulls them along towards those learning goals for the lesson. So, that has probably been one of the main things that has really evolved in my thinking in working on 'Principles to Actions,' is thinking more deeply about these assessing questions and the advancing questions that we need to be posing in our classrooms.

Mike: It strikes me that of the two—they're both important. But it may be that planning advancing questions is the more challenging task for an educator. Talk to me a little bit about preplanning or thinking in advance about advancing questions.

DeAnn: Certainly. So, first as we think about assessing questions, those tend to be more the recalling information, probing student thinking.

Mike: Uh-hm.

DeAnn: As teachers, I think we're pretty good at that. We can all say, 'Well, how did you think about that? How did you figure that out?' But the advancing questions are much more difficult because that means we, as teachers, have to know: Where are we going with this task and what's the math we want? So, in thinking about this … Or, for example, I was recently working with a group of teachers. And what they did is they worked in grade-level groups and even preplanned the questions they were going to use in an upcoming lesson. And it was really true that yes, assessing questions they had. But we took a lot of time to kind of unpack and think about these advancing lessons. So, I'm going to kind of share, like, three steps here to think about this.

DeAnn: One, you really need to know the math learning goals for the lesson because the advancing questions need to be about the mathematics students are learning. Two, it's helpful to work through the math task that the students are going to be doing in the lesson cause that's going to help you anticipate and approach the task and think about, 'OK, what might be happening in their work?' And then third, we can preplan those questions that should be specific to the task, to the anticipated student work, and most importantly to the mathematics.

Mike: Uh-hm.

DeAnn: If you can do that with someone, it's so invaluable to brainstorm and bounce ideas off each other.

Mike: I was thinking about what you were saying. And it's striking the difference between an advancing question that's, as you said, about the mathematics that we're trying to advance, versus a question that might move a child toward mimicking a strategy for a right answer right now, but that isn't actually in the long-term advancing the mathematics that we want. That really, for me, is jumping out as something that … it's a line that we want to help people see the difference between those two things, particularly in the moment. And I think that's why, as you were talking, DeAnn, the idea of, let's write some of these things down so that we have them on hand. Because in the moment it's often difficult to make those kinds of judgements when you're in a public space with a whole bunch of children in front of you. That's a superhuman task at some times (chuckles). So, there's certainly no stigma to writing it down. In fact, it's a strategy that makes a ton of sense for teachers.

DeAnn: Yeah, definitely agree. There's nothing wrong with having those questions on a piece of paper, on a clipboard, carrying that with you, pausing, taking a moment. 'What might be some questions I want to ask here?' I mean, asking questions is really a skill we develop as teachers, and we need to use tools and resources to, kind of, help us.

Mike: Well, I was going to say, the other thing that's really hitting me, DeAnn, is the connection between the learning goal and the question; how clearly we see the learning goal and the different levels of progression that kids will make as they're approaching the learning goal. And advancing means recognizing the meaning of a child's thinking at a given time and thinking about what's the next move, regardless of where they're at. Move children toward that deeper understanding.

DeAnn: Yeah. Perhaps it would be helpful if we share some examples.

Mike: Let's do that.

DeAnn: All right. So, assessing questions—as we were talking, it's like, 'Tell me about your thinking? Can you explain your picture to me? Can you tell me about the tape diagram you drew and used to solve this problem?' Just getting into that kid's thinking and where they're currently at. But then the advancing questions really move students' thinking forward. As you were saying, kind of along this continuum. So, we have to be ready to guide them, kind of step by step, to kind of scaffold that thinking, right? So, I might ask a question, 'What equation could you write for that problem?' Maybe they got the answer, but what would be an equation they could write? Because perhaps my learning goal is to help them make a connection between those different representations, the context and the equation. I might see that a child has written an equation, but then I might say, 'Could you label what each of those numbers means in your equation?' Because I really want to make sure they understand the mathematical meaning of each of those numbers.

Mike: Absolutely.

DeAnn: Just asking kids questions, like 'How are your strategies similar or different?' That's also going to make them think a little more [deeply]. So, all of these advancing questions, really the goal is sense-making and more depth of understanding.

Mike: That totally makes sense. I'm wondering if we can pivot a little bit and talk about the types of teacher moves that might accompany an assessing or an advancing question? What might I do after I ask an assessing question, as opposed to say, asking an advancing question?

DeAnn: So, with assessing questions, the goal of them is really to understand where the student is currently at. So, I would ask an assessing question, and as the teacher I would stay and listen. So, we could assume students are working individually or small groups.

Mike: OK.

DeAnn: So, I might ask an assessing question of a child or a small group and stay and listen because I'm trying to figure out, really to understand, what they did (chuckles) and why they did that. Whereas an advancing question, I would be more likely to pose the question to the individual child or small group and then walk away and say, 'I'll be back in a minute or two to see what you've done or what you're thinking about.' So, it's kind of like giving them time to pause and ponder and consider that question.

Mike: This is fascinating because I wonder if for a lot of people that might feel counterintuitive, that you would pose the advancing question and walk away. Tell us a little bit more about the why behind that choice.

DeAnn: Our goal really is to help students become independent math learners in the classroom. By asking the question and then saying, 'I'll be back in a couple minutes; think about that or show me what you've done,' we want them to be able to figure out how to proceed with a task on their own so that they don't become dependent upon us as teachers. But they really develop that agency in themselves to try things out, whether they're right or wrong, but at least that they're making some progress in the task.

Mike: You know what it makes me think, DeAnn, is that asking an advancing question and walking away might feel foreign to the educator, and it might at least initially feel kind of foreign to the child as well. But over time, it will start to feel like the culture of the classroom, and the child will actually get to a point where it's like, 'Oh, my teacher believes that I have the ability to think about this and come up with an idea.' And that's a real gift to a child. It does what you were talking about earlier, which is: Question sets the culture and helps children think about what is it to be learning about math.

DeAnn: Yeah. We've also talked about that other type of question to encourage students, to engage with the reasoning of each other. That also really helps with those advancing questions and that tone in the classroom. Cause you could ask a question as a teacher and then say, 'Why don't you talk with each other for a while about this?' Or ask one student to explain to another student some of their ideas. So, we can, again, use those talk moves when students are working in partners in small groups to learn from each other.

Mike: I'm struck by the idea that this conversation we're having about questioning is also really pretty tightly connected to, how do we support children when they need to engage with productive struggle? And I'm wondering if you could talk about the connection between high quality advancing or assessing questions, and helping kids manage and engage in productive struggle at the end.

DeAnn: Thinking back to 'Principles to Actions,' we identified eight high-leverage teaching practices for mathematics. And one of them is using purposeful questions. But another one is supporting productive struggle. So, the connection I think you're kind of alluding to here, Mike, really is they go hand in hand. We can use our questioning to encourage students to persevere in the mathematics that they're doing. But those questions, again, [mean] we are, first of all, trying to understand where the student is at by asking those assessing questions. And then we can encourage them to kind of, like, this bridge, right? With those advancing questions we're trying to get them to consider some of the mathematical ideas that might actually not even be on their horizon for them right now. So, if we say, 'How could you put this fraction on a number line?' Or 'How do you know this fraction is greater than or less than one?' We're asking a question to really make that math idea visible and to get them to consider it. And then we're pausing and giving them time and space to consider it and figure out how to proceed on their own. If I, as a teacher, tell them what to do next, that means I'm owning the math, I'm being the authority, and I'm not valuing struggle as part of the learning process.

Mike: Mm, yes. Yes, absolutely. Well, before we close, I want to dig into one more question type. And this is the one that I think really is just kind of transcendent. It transcends the task at hands and digs into students' understanding of big ideas. And it's the one that you would describe as making mathematics visible. Can you talk a little bit about the importance of these types of questions and perhaps some examples that would help people kind of envision what they look like in an elementary classroom?

DeAnn: So, you asked about these questions [that are] really making the math visible. As I think about that, what comes right to mind is a fascinating study conducted by Michelle Perry and her colleagues. They actually looked at the questions and examined very closely the questions teachers ask in a first-grade classroom for mathematics. And they compared the questioning of teachers in Japan, Taiwan, in the United States. Well, unfortunately they found that teachers in the U.S. ask significantly [fewer] questions that require high-level thinking than in Japan and Taiwan. In fact, teachers in those countries tend to ask questions that transcend the problem at hand. I love that phrase . The question goes beyond the surface of the task to really transcend that problem at hand, to get at the underlying math ideas, math concepts, and connections that we want students to make. And they found that teachers in Japan and Taiwan went beyond the surface to really make the math visible for students to consider. And really kept students engaged at higher cognitive levels of thinking.

Mike: That is fascinating. What it reminds me of is, I think it was Jim [James] Hiebert and [James] Stigler wrote about the idea of the mathematics classroom as a cultural activity, in that there's this kind of underlying script of what it means to be a student or a teacher in a mathematics classroom. And I think what we're really talking about in some ways is the role of questioning in building a different vision of what a mathematics classroom is or what it means to be an educator of mathematics.

DeAnn: Yeah. I think that ties right back to our earlier sharing about productive struggle. We think if students don't know the answer quickly that it's our job to step in and tell them how to do it.

Mike: Uh-hm. We're almost coming full circle though, in the sense that I think the promise of high-quality questioning—be it assessing or advancing—is that we're really, by considering the ways that students might think about a task and then considering the ways that you can assess that and advance their thinking from wherever they may be, we really are helping teachers see a different way. And I think that's the power of what you're describing when you talk about strong questioning, DeAnn.

DeAnn: Yeah.

Mike: So, we talked a little bit about what to do next. But I would love for you to take a moment to weigh in on the question of wait time. What are your thoughts about wait time and its value and how that can work in a classroom to support children?

DeAnn: So far today, we've been talking a lot about like the types of questions that teachers ask. But the implementation of those questions is also something we need to think a little bit more about. So, [there are] two types of wait time. Wait time is when I ask a question as a teacher, and how long do I wait until I call on a student? The research on wait time really shows that as teachers, we tend to wait less than a second.

Mike: That's incredible.

DeAnn: Yeah. We provide no processing time to our young learners to really formulate those ideas in their head and then be able to share back. So, just by reminding ourselves to pause for 3 seconds makes a huge difference in the learning that goes on in a classroom. Those 3 seconds, what happens is we find out that more students will respond to our questions. The length of students' responses increases. And those questions or those responses from students where they say, 'Oh, I don't know,' decrease. So merely waiting 3 seconds makes a huge difference. And as teachers, we just don't deal well with silence, and thinking time, and processing time. So, I think it's always a good reminder to just monitor the amount of time we give students to process ideas after we ask a question.

Mike: You know, as a person who works in math education, when I'm at a dinner party or in mixed company with people, and I ask them, 'Tell me about your memories of elementary school mathematics.' There are a few common things that always come up. One is typically, as I'm sure won't surprise you, the idea of memorizing my facts. And the other theme that kind of goes along with that is this sense that I wasn't good at math because I didn't know the answer right away. And the connection I'm making is, maybe that's because we didn't give you enough time to actually process and think. If we simply expand our time and give kids 3 seconds or 4 or 5, rather than 1, how different would that experience of mathematics be for children? How many folks would actually feel differently about mathematics and maybe, perhaps, not associate mathematics with just being the first and being the fastest to find the answer.

DeAnn: So, again, we're talking about using our questioning to kind of establish that tone and the expectations in the classroom about what it means to learn and do mathematics. And we shouldn't be in such a hurry (chuckles) for students to respond. We as adults need our processing time. Our young learners, [who] are first encountering many of these new ideas in mathematics, we need to give them time to think and to process and make connections before we expect them to respond to any of our questions.

Mike: The piece about 1 second is just so striking . It's odd because I suspect people imagine that by coming back that quickly, they're actually supporting the child. But you're actually doing the opposite (chuckles). You're teaching them: One, if you haven't had it in a second, what's wrong with you? And then two: You're also fostering dependency. It's fascinating how, what I think comes from a desire to help, is actually debilitating.

DeAnn: It really speaks to the need to reestablish not only norms for students in our classrooms, but really for ourselves as teachers.

Mike: Definitely. Well, I just wanted to say thank you so much for this conversation. It's really been a pleasure to have you join us and hopefully we'll have you back at some time in the future.

DeAnn: And thank you, Mike. I've really enjoyed talking about the importance of purposeful questions for teachers to consider more deeply in their classroom practice.

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.

© 2022 The Math Learning Center | www.mathlearningcenter.org

Rounding Up: Season 1 | Episode 1

There is a persistent myth in the world of education that mathematics is abstract and its teaching is not influenced by cultural contexts. This despite the fact that research and scholarship indicate that when students see how math applies to the world they recognize, they perform better. Today on the podcast Dr. Corey Drake , senior director of academic programs at MLC, talks about what it means to provide a culturally inclusive and relevant mathematics experience in the elementary classroom.

Recources

If you're interested in more on this topic, consider the following article for further reading:

Three Strategies for Opening Curriculum Spaces

Transcripts

Mike Wallus: There's a persistent myth in the world of education, that mathematics is abstract and its teaching is not influenced by cultural contexts. This, despite the fact that research and scholarship indicate when students see how math applies to a world that they recognize, they perform better. Today on the podcast, we'll talk with Dr. Corey Drake, senior director of academic programs at The Math Learning Center, about what it means to provide a culturally inclusive and relevant mathematics experience in the elementary classroom. This is a topic on everybody's mind, and we're excited to address it head on.

Mike: All right. Hello, everybody. Welcome to the podcast. We are excited today to have Dr. Corey Drake with us. And the topic of the day is culturally relevant practices in an elementary classroom. So, Corey, welcome. It's great to have you on the podcast.

Corey Drake: Thanks. Great to be here.

Mike: Fantastic. So I want to start this conversation and zoom way out as a beginning place. So one of the things that I'm thinking about is that lately it seems like you hear terms—like equity, culturally inclusive, culturally relevant—and those are being used across the education space almost as like kind of a catchall, to the point where it seems like in some cases they've almost lost their meaning. So I'm wondering if to begin the conversation, and really give these ideas the depth of discussion that they deserve, If you'd be willing to unpack … When you think about culturally inclusive and culturally relevant practices, help paint a picture of that for someone who's a listener.

Corey: Yeah. I think those terms do get used all the time in all kinds of different ways. And so I've been trying to think a lot about sorting them out and trying to think about a framework that makes sense for me, recognizing though that actually whatever the term is, I think the goals are the same, right? And so the goals of whether it be culturally inclusive, culturally relevant, culturally sustaining education, or to provide better experiences and more access to all students to high-level mathematics. So that's the underlying goal. And so I don't want to get too lost in the terms.

Mike: Thank you.

Corey: Having said that though, I think there are some important differences. I think if we think about things like culturally inclusive, we think about context representations that include all students so that every student can see themselves in curriculum so that students aren't excluded by the examples and representations they see in curriculum.

Corey: So I would think about that as more along the lines of culturally inclusive. When we start to get to culturally relevant, and then culturally responsive, culturally sustaining work, now we're really starting to think about who our students are, what their experiences have been, what their interests are, the kinds of activities our families and communities participate in, and how all of that can provide access and bridges into mathematics. And then if we would get all the way, kind of on what I think of as the far end of the continuum, we really would get to terms like anti-racist education. We're really there. We're talking about systemic racism, systemic oppression and privilege, and ways in which mathematics can disrupt those systemic issues of, not only who has access, but the kinds of outcomes and opportunities that students have based on various characteristics.

Mike: So let's unpack these a little bit.

Corey: Yeah.

Mike: I think one of the things that's really interesting is this idea of relevance and responsiveness …

Corey: Uh-hm.

Mike: … so, particularly because it made me think about the kids in my classroom when I was a classroom teacher, so it strikes me that a part of this work is like, as you said, like really getting to know your students. So paint a picture of what that might look like if I'm a classroom teacher, and I'm teaching fourth grade, what kind of process might I engage in? What does that look like as I'm getting ready to perhaps start a unit of study, or even as I'm just getting ready to start the year? Like, what might that actually look like for a person who's out in the field?

Corey: Yeah, that's a great question. And it brings up a really important point, which is that cultural responsiveness cannot sit just in a set of materials. And it can't sit just in the teacher's actions, right? Cultural responsiveness happens at that interaction of curriculum materials and the mathematics and the teacher and the students. And it's in those interactions that cultural responsiveness happens. And so for the teacher, what that means is really getting to know their students. But also—perhaps even more and importantly, and as a way to get to know their students—opening up those spaces for student voice in a classroom, right? Where do students have opportunities to share their ideas, to make sense of ideas, to bring in the connections that they're making? To the extent that it's all about the teacher, we're never going to get to that cultural responsiveness, where the students are allowed to bring themselves and bring their cultures into the classroom, and then be able to make sense of the math. With that in mind though, teachers can be thinking about looking at, say, a new unit of study or a task they're going to work on and think about, 'How do I open up the space within this task, within this unit, for that student voice to come in, for students to be able to make those connections?' So the teacher is really opening the space versus making the connections, right? They're opening the space so that the students can be making those connections.

Mike: So I love this idea of opening space. And I think I want to unpack this idea and just try it on. Is it fair to say that opening space, to some degree, is about two things? Part one is: How do I allow space for my students' lived experiences and their cultural background, and those pieces to kind of come into the, the work? And then part two is: How do I open space in a task that may actually funnel student thinking or constrict the opportunity for kids to share their thinking? Am I thinking about that appropriately, Corey?

Corey: Yeah, I think that's right. I think you open space for student voice. But you open space in ways … a main way that you would open space is by not overly directing, not overly restricting what that space is. So if I'm going to pose a task, I'm going to look for opportunities to bring in student voice and opportunities for students sense-making throughout that task. So I'm going to launch that task by asking students, 'What is this context about? What does this make you think about? Can you connect this to other things you know?' And then we're going to launch the task and we're going to get into the mathematics. And again, and I, as a teacher, am not going to be directing a particular way to solve a problem, a particular way to think about it. But again, opening up the space for students to make those connections, for students to make sense of the mathematics, and then providing opportunities for them to share and learn from each other. It's not a free for all though, right? It's not just bring in whatever you're thinking about, right? My goal as the teacher is to open that space and then facilitate those connections so that they really lead to the kind of sense-making that all students need.

Mike: Thanks for that. You know, I actually want to shift gears a little bit because it was interesting as you described the continuum … you also were kind of talking about the idea that we could consider, like, a series of steps that we deemed—or you deemed—anti-racist. And you talked about those in, in relation to, kind of, systems that exist. Can you say more about that? Just talk a little bit more about the types of systems that we might be talking about when we're talking about taking an anti-racist stance.

Corey: Yeah, absolutely. I think the two that come to mind right away are two that you mentioned, right? One is our around curriculum, and one is around assessment. And those are really tightly intertwined, right? So we have a curriculum that not only provides a set of standards, but provides a particular order and a particular path through which we think all students should reach the set of ideas that are represented in the standards. And in order to provide opportunities for all students, we need to think more broadly about that. We need to really think about, 'What are the big ideas? What are those goals? And how do we provide opportunities for all students to reach those goals?' … recognizing that what we know about student progressions and the way students get there have mostly been built on the progressions, honestly, of white, middle-class children.

Mike: Hm.

Corey: And so there's a lot we don't know, and it requires us to open up spaces. And I think assessment is probably the biggest.

Mike: Yeah. Talk about that please.

Corey: Yeah. So assessments are set up to label and categorize students, which is kind of inherently problematic. And I think even more problematic is that assessments and the assessment systems we have built tend to focus our attention on what students don't know, on what students can't do, right? So if we think about the various labels and categories we have for children, they're often around, 'Well, they can't do this yet' or 'They haven't learned that yet,' versus what is it that students can do? What do they understand? What are they bringing to the classroom? You know, I always tell pre-service teachers, like, something we know about learning is that new learning is connected to prior understandings. You don't learn new things in a vacuum. So if we don't know what students already understand, what they already can do, how are we going to help them learn new things? What are we going to connect it to? I can't connect new learning to the fact that you don't know X, Y, or Z. I can connect it to the idea that you do know this set of things, and I can help you build on that and learn the next set. And to me, that is a critical shift that we would need to make to really have a less racist, less oppressive education system.

Mike: Mm. Yeah. Can you just expand on that vision a little bit, Corey? I'm still really resonating with two things. One, we learn new things when we connect it to prior knowledge. And two, the whole design of the system—and really kind of the intent, for lack of a better word—this is really to kind of categorize what don't you know. And to label that very, very specifically.

Corey: Yeah.

Mike: As opposed to a different kind of intent, which is: What do you, in fact, understand?

Corey: Yes. And it starts with, we think about math tests we may have taken in the past, right? The focus was always on was the answer right or wrong? And when the answer was wrong, there was an assumption: 'You don't know this. You don't understand this.' And that's how you got grouped or labeled or categorized. And we still do that to students. Versus looking at a piece of student work. You don't want to forget whether the answer in the end is quote, unquote 'right' or 'wrong.' But what I really want to look at is how is a student thinking about a problem? How is a student making sense of this problem? What are the ideas and understandings they're bringing to this work so that I know what to build on next.

Mike: Absolutely.

Corey: And so focusing much less on right or wrong. And here's where I think curriculum and assessment are intertwined. Because when we set things up as here's the endpoint, here's the standard we're trying to reach, right, that leads us to saying, 'Yes, they got it' or 'No, they did not.' Versus what's the path, what's the pathway they're taking? What are the understandings they're building along the way?

Mike: So I'm imagining either a single teacher looking at their students' work, or perhaps a team of teachers who are looking at it … it's an entirely different kind of conversation, right? Like it's almost an entirely different process of, I've got—I'm thinking old school—I've got students' paper work in front of me …

Corey: Sure.

Mike: … I'm looking at it. I'm almost kind of thinking to myself, 'For someone who's new to this idea, what might that look like if you and I, and a couple colleagues were sitting together, looking at our students work?' What does that conversation sound like?

Corey: And how great would that be, right?

Mike: It'd be amazing.

Corey: We have these kind of data meetings and things like this in schools. But so often we're looking at printouts from standardized tests …

Mike: Right.

Corey: … that don't really to give us insight into the thing we would pay attention to if we sat around a table, looking at student work, is 'What do you think this student was thinking about? Oh, and where did they get that 10 from? Oh, I see they broke this number up this way. So that shows me they understand some things about place value. They understand something about the structure of numbers. I can see that here, they had a really interesting strategy, but they just miscounted at the end.' So I'm thinking, 'This show's really rich understanding.' And so we could have those kinds of conversations.

Mike: And those things are actionable, too, right?

Corey: Absolutely.

Mike: I, I mean, that's the challenge of having sat in so many data meetings is, like, what's actionable about what you're looking at?

Corey: Exactly.

Mike: It's really hard when you're actually trying to get into students' heads and think about their thinking. You, as a teacher, you have some agency, you can do something. So it's, it's like, wow, that's really powerful.

Corey: Yeah. It just lends itself to this next idea. OK, if I know that this is what this student is thinking about, and maybe this group of students is thinking about it this way, and this group is thinking that way, it supports also this idea of, like, teaching is inquiry, right? Because what we always want to do, we don't have the magic next step. But we could look at a piece of student work and say, 'Huh, I wonder what would happen if I posed this problem next? Or what if I changed the numbers in this problem? Would I still see this kind of thinking?' And that's what we want teachers to be doing to support student learning. To say, 'Here's what I see happening. Let me try this problem next and see what happens' And building that pathway for a student over time.

Mike: Which to me, actually, the connection I think I'm making is, that's actually almost like a generative path, right? In some ways that leads us right back to what you said at the beginning, which is, 'What's the role of the teacher when they're trying to provide a culturally relevant experience?'

Corey: Absolutely.

Mike: It's like, this is the pathway to get there.

Corey: Yes.

Mike: Yeah. That makes a ton of sense. Um, well, before we leave things, Corey, I guess the last question I wanted to ask is: If I'm a teacher who's new to this conversation or new to thinking about these ideas, do you have any references that you might share with folks? Things that would help them kind of continue to think about this, continue to think about how it shows up in their classroom? Is there anything you'd recommend?

Corey: Yeah, absolutely. There are so many great resources out there right now. I think the main problem is making sure we have time and space to be able to, to learn from the great work that's happening out there. I would say a book that's been really influential for me recently is actually in English language arts. But it's by Gholdy E. Muhammad and is called 'Cultivating Genius.' And she talks about what it would look like to build a historically and culturally relevant curriculum in ELA. And I think there are a lot of parallels with math. We've also been reading lately, 'Choosing to See,' by, um, Pam Seda and Kyndall Brown. And I think that has very actionable steps. It's really written in a way that teachers, either on their own or in a small group, could take it up and really think about some of these ideas shifting. It's these small shifts in curriculum and assessment, and just our orientation to children, that really makes such a big difference for the experiences of students.

Mike: Totally agreed. I read that and just felt like, 'If I'm a teacher, I can do something with this tomorrow.'

Corey: Yes, yes, absolutely.

Mike: Absolutely. Definitely. The other one that jumps out for me, and I'm wondering if you add some commentary, is just, 'Smarter Together,' which has been around for a while.

Corey: Yeah.

Mike: But has got some really powerful work inside it as well.

Corey: Absolutely. So 'Smarter Together' really helps us think about—within groups of students— thinking about status and privilege and how teachers can really bring to the forefront and, and hold up the different ways in which students are smart in mathematics. And I think that's a really important shift, which is that all students are brilliant, right? And it's taking that as a fundamental tenant and saying, 'The ways we've tended to think about what it means to be smart in math have been so narrow. They've been about being fast with your facts. Or being able to memorize things.' When really, the range of ways in which you can be and need to be smart in math are so much broader than that. And so, 'Smarter Together' really helps us think about, 'What are the range of skills and knowledge and interests that students would need to bring to really do well in mathematics?'

Mike: Sounds like we have another podcast on our hands.

Corey: Love it.

Mike: (laughs) Thanks so much, Corey.

Corey: Yep.

Mike: It was great to have you on the podcast.

Corey: Thank you.

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.

© 2022 The Math Learning Center | www.mathlearningcenter.org

Welcome to Rounding Up, the professional learning podcast brought to you by The Math Learning Center. Two things have always been true in education: Ongoing professional learning is essential, and teachers are extremely busy people. Rounding Up is a podcast designed to provide meaningful, bite-sized professional learning for busy educators and instructional leaders.

I'm Mike Wallus, vice president for educator support at The Math Learning Center and host of the show. In each episode, we'll explore topics important to teachers, instructional leaders, and anyone interested in elementary mathematics education. Topics such as posing purposeful questions, effectively recording student thinking, cultivating students' math identity, and designing asset-based instruction from multilingual learners. Don't miss out! Subscribe now wherever you get your podcasts. Each episode will also be published on the Bridges Educator Site.

We hope you'll give Rounding Up a try, and that the ideas we discuss have a positive impact on your teaching and your students' learning. © 2022 The Math Learning Center | www.mathlearningcenter.org