Rounding Up

We often ask students to share their strategies. But what does it look like to uncover and highlight the reasoning that informs that strategy? Today on the podcast, we'll talk with Nancy Anderson , a classroom teacher and professional learning developer about strategies to elicit the reasoning at the heart of a student's thinking.

Mike Wallace: We often ask students to share their strategies. But what does it look like to uncover and highlight the reasoning that informs that strategy? Today on the podcast, we'll talk with Nancy Anderson, a classroom teacher and professional learning developer, about strategies to elicit the reasoning at the heart of the student's thinking.

Welcome to the podcast, Nancy. I am so excited to talk to you today.

Nancy Anderson: Thank you. Likewise, Mike.

Mike: I'd like to begin with a quote from your article, " Keep Calm and Press for Reasoning ." In it, you state: "Mathematical reasoning describes the process and tools that we use to determine which ideas are true and which are false." And then you go on to say that "in the context of a class discussion, reasoning includes addressing the strategy's most important ideas and highlighting how those ideas are related." So, what I'm wondering is, can you talk a little bit about how eliciting a strategy and eliciting reasoning may or may not be different from one another?

Nancy: So, when we elicit a strategy, we're largely focused on what the student did to solve the problem. For example, what operations and equations they might have used, what were the steps, and even what tools they might have used. For example, might they have used concrete tools or a number line? Whereas eliciting reasoning focuses on the why behind what they did. Why did they choose a particular strategy or equation? What was it in the problem that signaled that particular equation or that particular operation made sense? And if the strategy included several steps, what told them to go from one step to the next? How did they know that? And then similarly for the tools, what is it in the problem that suggested to them a number line might be an effective strategy to use? And lastly, eliciting reasoning sort of focuses on putting all those different pieces together so that you talk about those different elements and the rationale behind them in such a way that the people listening are convinced that the strategy is sound.

Mike: That's actually really helpful. I found myself thinking about two scenarios that used to play out when I was teaching first grade. One was I had a group of children who were really engaging with the number line to help them think about difference unknown problems. And what it's making me think is, the focus of the conversation wasn't necessarily that they used the number line. And it's like, "Why did this particular jump that you're articulating via number line—what is it about the number line that helped you model this big idea or can help make this idea clearer for the other students in the class?"

Nancy: Exactly, yes. So, when I think about reasoning, I think about different pieces coming together to form a cohesive explanation that also serves as a bridge to using a particular strategy for one particular problem, [and] as a tool for solving something similar in the future.

Mike: So, I have a follow-up question. When teachers are pressing students for their reasoning, what counts as reasoning? What should teachers be listening for?

Nancy: Broadly, mathematical reasoning describes the processes and tools that we use to determine which ideas are true and which are false. Because mathematics is based upon logic and reasoning—not a matter of who says it or how loudly they say it or how convincingly they say it—but rather, what are the mathematical truths that undergird what they're saying? That's sort of a broad definition of mathematical reasoning, which I think certainly has its merits. But then I think about the work of teaching, particularly at the elementary level. I think it's helpful to get much more specific. So, when we think about elementary arithmetic, reasoning really focuses on connecting computational strategies to the operations and the principles that lie underneath. So, in the context of a class discussion, when we have a student explain their reasoning, we're really trying to highlight a particular strategy's most important ideas and how those ideas are related, but in such a way that others can listen and say, "Oh, I get it. If I were to try the problem again, I do believe that's going to lead to the correct answer." Or if it was this problem, which is similar, "I think I can see how it might make sense for me to use this approach here with these slight adjustments." So, do you want to take an example?

Mike: Yeah, I'd love to.

Nancy: So, for example, in a first grade class, there might be a class discussion about different strategies for adding seven plus eight. And I think in a lot of classes at one point, the teacher would likely want to highlight the fact that you can find that sum using doubles plus one. So, in this particular instance, if a student were to talk about their reasoning, we'd want to encourage that student and certainly help that student talk about the following ideas: the connection between seven plus eight and seven plus seven, and the connection between their answers, namely because the second addend has changed from seven to eight, and noting the connections between the second addend and the answers, namely, if the second addend increases by one, so, too does the sum. And finally, we'd want to emphasize what it is we're doing here. Namely, we are using sums that we know to find sums we don't know.

Nancy: So, that's an effective example of what reasoning sounds like in the elementary grades. It's very specific. So even though reasoning is the thing that allows us to move from specific examples to generalizations in elementary mathematics, it's oftentimes by really focusing on what's going on with specific examples…

Mike: Uh-hm.

Nancy: …that students can begin to make those leaps forward. Some of my thinking lately about what I do in the classroom comes from the book Make It Stick , which talks a lot about learning processes and principles in general. And one of the points that the authors make in the book is that effective learners see important connections, for whatever reasons, sometimes more readily or more quickly than others.

So, what I try to do with my teaching then is to say, "OK, well, how can I help all learners see those relevant and important connections as well?"

Mike: Absolutely. So, it really does strike me that there are planning practices that educators could use that might make a press for reasoning more effective. I'm wondering if you could talk about how might an educator plan for pressing for reasoning?

Nancy: One thing that I think teachers can do is anticipate, in a very literal sense, what is it that they want students to say as a result of participating in the lesson? So, I think oftentimes we, as classroom teachers, focus on what we want students to learn (i.e., the lesson objective or the essential aim). But that can be a big jump from thinking about that to thinking about the words we literally want to hear come out of students' mouths. So, I think that that's one shift teachers can make to thinking not just about the lesson objective as you'd write on the board, but literally what you want students to say, such that when you walk around and you sort of listen in on small groups, those moments where you say like, "Oh yeah, they're on the right track." And then I think another key shift is thinking more towards specific examples rather than generalizations.

Nancy: So, as an example, suppose that in a third or fourth or fifth grade classroom, students were talking about fraction comparison strategies, and the teacher had planned for a lesson where the objective was to determine if a fraction was more or less than a half by using the generalization about all fractions equal to a half. Namely, that the numerator is always half of the denominator. So, that certainly could be something that we might see in a, you know, teacher's guide or perhaps in a teacher's planning book. But that's different than what we'd want to hear from students as the lesson progressed. For example, I think the first thing that we'd want to hear as the students were talking, is a lot of examples, right? The kinds of examples that are going to lead to that key generalization. Like if a student was talking about nine-sixteenths, I think we'd want to hear that student reason that nine-sixteenths is more than half because half of 16 is 8, and nine-sixteenths is a little bit more than eight-sixteenths.

Nancy: And so, what's effective about that kind of planning is that it alerts you to those ideas when you hear them in the room. And it can then help you think about, "What are the pieces of the explanation that you want to press on." So, in this case, the key ideas are finding half of the denominator, connecting that value to the fraction that is equivalent to one-half, and then comparing that fraction to the actual fraction we're looking at so that we can bring those key ideas to the fore, and the ideas become a strategy for students to use moving forward.

Mike: You're making me think about two things kind of simultaneously. The first is, I'm reflecting back on my own practice as a teacher. And at that time, my grade-level team and I, we tried to really enact the whole idea of anticipating student strategies that comes out in the Five Practices book. But what you're making me wonder about is, we went through, and we said, "Here are some of the ways that children might solve this. This is some of the strategies." The step we didn't take is to say, "We know that there are multiple ways that children could attack this or could think about this, but what's the nugget of reasoning?" What would we want them to say in conjunction with the strategy that they had so that we were really clear on if a student is counting on to solve this problem, what's the nugget of reasoning that we want to either press on or encourage. If the're direct modeling, again, what's the nugget of reasoning that we want to press on. If they're decomposing numbers? Same thing. So, really it makes me think that it's helpful to anticipate what kids might do. But the place that really, like, supercharges that is that thing that you're talking about, is—what's the thing that we want them to say that will let us know that they're onto the reasoning behind it?

Nancy: Exactly. And I think the conversations you're having or have had with your colleagues reflects where we are with the field generally. I think that the field of mathematics education is at a place where, for the most part, we're on board with the use of discussion as a pedagogy. I don't think that it's a tough sell to convince a lot of folks that students should be spending some amount of time talking. But I don't think that we as a field are nearly as clear on what to do next. And again, as you alluded to with the Five Practices book, and while I would certainly agree that all of these are important aspects of classroom talk, I think that they skip over this essential idea of pressing for reasoning. Namely, staying with the student beyond just their initial explanation so that their ideas become clear, not just to others, but also clear to them.

Mike: I love that. I want to go in a direction that you started to allude to, but you really got to in, in your article. This idea that there's a certain number of questions for follow-up that can really have a tremendous impact on kids. I'm wondering if you could talk a little bit about that.

Nancy: My article and more broadly, my interest in press for reasoning, is motivated in large parts by my professional interest in figuring out, you know, what it is about discussion that makes it such a powerful tool for learning. So, although we have enough empirical evidence to support discussion as an effective pedagogy in math class, we as a field are much less clear in knowing which of the aspects of discussion are most efficacious for learning. What are the mechanisms of student talk that help students learn math more deeply? I had the good fortune many years ago to find some compelling research by Megan Franke and Noreen Webb and their colleagues at UCLA who did some digging into press for reasoning. And through their studies, they have shown that follow-up questions, questions that press students to clarify and strengthen their initial explanation, are associated with students giving more robust and more accurate explanations.

Nancy: What their research revealed is that it takes two to three specific follow-up questions in order to either have the student say, more math and more accurate mathematics. So, I think about that so often in my work in the classroom because so often I'll ask a student to explain their reasoning and because they're learning, the explanation comes out either partially correct or partially complete, and I need them to say more. And I might ask them the first follow-up question and either they or I suddenly start to worry. The student might think, "Am I saying something wrong? Am I totally off track here? Uh, I'm not really sure why I did what I did." And then I, of course, as the teacher, I'm so worried about, "Am I putting the student on the spot? Am I losing the rest of the class?" And in those moments, I hear myself say, "Two to three follow-up questions, two to three follow-up questions," as a way to remind myself to stay with the student. That if we really do believe that students learn by talking, then it only makes sense that we should expect them to need more than just one turn to get their ideas out in such a way that are clear and accurate to them as well as to the listeners.

Mike: So, that's fascinating, Nancy. I think there's two things that stood out from what you said. One is, as a classroom teacher, I appreciate the fact that you acknowledge that feeling of, "Am I losing the class?" [It] is something that always exists when you're trying to question and support. But I think the thing that really jumps out is, we have research that says that this actually does have a tremendous impact on kiddos. So even though it might feel counterintuitive, staying with the press for those two to three questions really does have a tremendous impact. I'm wondering what it might sound like to take a student's initial response and then follow up in a way that presses for reasoning.

Nancy: So, suppose a fourth grade class is working on strategies for multi-digit multiplication, and one particular strategy that the teacher would like to emphasize, or showcase, is compensation. Namely, how we can change one or both factors in a multiplication to create an easier computation and then make an adjustment accordingly. For example, we can multiply 19 times 40 by thinking about 20 times 40, and then subtracting 40. Let's suppose that students are working in groups and—on this computation—and the teacher overhears a student talking to their partner about how they use this exact strategy, and briefly checks in with the student and asks, you know, if they'd be willing to share their strategy with the whole class. And the student agrees. So, the teacher calls on the student to tell us, "How did you compute 19 times 40?" And the student says, "Well, I did 20 times 40 minus 40, and I did that because 20 times 40 is easier."

Nancy: Great. So, we've got some ideas on the table, and so now let's unpack. So, maybe the first question to ask the student is for them to interpret 19 times 40. What does that mean? Literally, it says 19 times 40, but can they give a context? Can they provide an interpretation of that expression with the hope of getting the idea out that we can think of 19 times 40 as 19 groups of 40. And similarly, 20 times 40 as 20 groups of 40. So, once we have the idea of groups of a number out there, can the student tell again why it made sense for them to think of 20 times 40? Why is that easier? Then another follow-up question to ask is, "Well, what's the connection between changing that first factor to 20 and subtracting 40?" Because if you think about it, if you're a listener who's unfamiliar with compensation, that's a pretty big leap to go from changing the first factor by one to a second step of subtracting 40. Huh?

Mike: It sure is.

Nancy: (laughs) Right? Like, how does changing it by one mean you subtract 40. And so, here the students can talk about the fact that we found 20 groups of 40, which is one too many groups. So, we compensate by subtracting 40. So, those are some follow-up questions that I think we'd want to ask.

Mike: This example just makes so many connections. I'm struck by the fact that, simultaneously, that press for reasoning is helping the child who came up with the idea really build a stronger vocabulary and a justification, and at the same time, it's actually providing access to that strategy for kids who didn't come up with it, who maybe kind of wondering, "What? Where did that come from?" So, really it's beneficial for the child who brought the reasoning to the table and to everybody else. The other thing that jumped out is, even in that question where you said, "Can you offer this in context?" That's kind of connecting representations, right? Like the child was articulating something that might show up in equation form and asking them to articulate that in a contextual form. [That] is actually a way of challenging their thinking as well.

Nancy: Exactly, yes. For many students—and, unfortunately, many more adults—symbols are just that, their symbols. Yet, we who engage in mathematics know that many times symbols are linked to not just one representation, but several , that there's certainly a literal interpretation of any kind of symbol string or numeric expression. But then we can interpret what those expressions mean by connecting back to the different meanings of the operation. So yeah, like you said, Mike, there's two things going on here at least: Helping the other students learn about this particular approach and trusting that it works, but also helping the original speakers see what it takes to convince others. And in this case, part of that includes the fact that, "Oh, when I talk about multiplication, it's helpful to remind people that multiplication refers to putting groups together. Or that it's helpful to think about multiplication in terms of putting equal groups together."

Mike: Well, before we close the podcast, Nancy, I typically ask a question about resources because I suspect for some folks, this conversation is one that they've been thinking about for a while. And for other folks, this idea of thinking past strategies toward a reasoning might be a new idea. So, I'm wondering if you'd be willing to share resources that you think would help support people maybe taking this conversation we've had and deepening it.

Nancy: Sure. So, my work in this field rests upon the shoulders of many brilliant mathematics educators and some of whom are people I admire from afar, like Megan Franke and Noreen Webb and their team at UCLA. And still others who I've had the honor to work directly with and learn from over the past 20 years. And two educators, in particular, are Suzanne Chapin and Cathy O'Connor of Boston University, who are a mathematics educator and applied linguist, respectively.

Mike: I adore their work. I'm just going to cut in and say, I'm excited for the resource you're going to share because I've read some of their stuff, and it's phenomenal.

Nancy: They were kind enough and generous enough when I was very new in the field to invite me to collaborate with them on a book called Talk Moves , which is essentially a teacher's guide to facilitating productive math talk. Many years ago, Cathy, Suzanne and I worked together on a research project where we were using discussion in elementary math classes in the city of Chelsea, Massachusetts, and we realized that there really wasn't a how-to guide out there for doing this kind of thing. So, from our work together came the book Talk Moves, which is now in its third edition and includes written vignettes in the book showing composite examples of teachers and students using talk moves to learn more mathematics, but also includes a set of video clips that were filmed in actual math classes with real-life teachers and real-life students using productive talk moves—including press for reasoning—to help students talk about their reasoning and respond to the reasoning of others. It's a very user-friendly guide for people who want to dig more deeply and see what this thing called productive math talk looks like in action.

Mike: So, I'll add to your plug. I read that back when I was teaching kindergarten and first grade, and it actually had a huge impact on my practice and just understanding at a granular level what this could look like. Nancy, thank you so much for joining us. It really has been a pleasure talking with you today.

Nancy: Oh, it's been a real pleasure for me too, Mike. Thank you so much for having me.

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.

© 2023 The Math Learning Center | www.mathlearningcenter.org

Today we talk with Kim Markworth, director of content development at The Math Learning Center about using and designing rich tasks to support student learning.

Mike Wallus: Lately, terms like "rich tasks," "multiple entry points," and "low floor, high ceiling" are being used so often in the world of mathematics education that many educators are confused about their meaning. Today we talk with Kim Markworth, director of content development at The Math Learning Center, about what these terms look like in practice and how they support student learning. Welcome to the podcast, Kim. It's great to have you.

Kim Markworth: Thank you, Mike. I'm really honored and excited to be here.

Mike: I would love to start this conversation by talking about what it means for a task to be open-ended and have a low floor and a high ceiling. So, is there a way that you think about these terms that might help educators clarify their meaning?

Kim: That's a great question, Mike. In truth, when we think about these terms, they're really all interconnected. And I don't know that anyone has really settled on meanings. And lately there's been a bit of a transition from thinking about low floor and high ceiling to low floor and no ceiling at all. And so, when we think about the variations across continua, this really contributes to how we look at tasks and what we appreciate about tasks. And so, when I think about open-ended, this might correspond with high ceiling or no ceiling. We could keep going with the task. There's not really a defined ending point, uh, but instead many pathways where we could take this task further. And when I think about multiple entry points , this might correspond with low floor. And there [are] so many different ways to approach or enter a task. And rich task, probably my favorite term of them all, corresponds to what we've maybe called a problem historically, where we don't have an anticipated solution path or maybe we haven't solved something like this before. We might have multiple solutions to the task. We might have multiple solution paths or viable strategies. We could have opportunities for extension or generalization. And I love tasks that have an unexpected twist or a need to think about something in a unique way, like where your first inclination, your intuition, might be wrong. And all of a sudden you're like, "Ooh, this is really something that I wasn't expecting."

Mike: Can you share an example or a few examples with folks who are listening? There's so much that you said just there about the nature of these things. Are you OK telling us a little bit about one or two of them?

Kim: Yeah. One task I'm really excited about is a new third grade task that we've been working into our new curriculum. And this task is positioned at the beginning of a unit on multiplication. And so, for third graders, this is really an introduction-to-multiplication task. And I'm going to apologize to the audience here because this really does require some mental visualization, but I'm going to describe this image that third graders look at, and it involves a pet store. And so, if you can imagine a pet store and going into a pet store, then we have a dog bone display, and these dog bones are hung on three hooks, and Annie took, there are two packages—so that the back package isn't visible because it's behind the front package. So, three hooks, two packages per hook, and in each package there are eight dog bones. And those eight dog bones are arranged in a four-by-two array.

Kim: And finally, one additional detail here. Each package is labeled $12. And this is a problem-posing situation for kids where we put this image in front of kids and ask them to think about the different mathematical questions that they might ask about this particular image. So ultimately, the question that we ask them to explore is, "How many dog bones are on display?" Once you have that image in your head, I want you to be thinking about that final question. And the numbers right now are kind of irrelevant for this discussion, but it's highly unlikely that it's something that kids will already know.

Mike: Hmm.

Kim: So, this is a problem-posing situation for the third graders. It's asking them what mathematical questions could you ask about this particular display? And ultimately, we're going to direct them to how many dog bones are on display, but they could explore other questions with additional time. So, how much for all of these dog bones? How much per dog bone? If you had a given number of dogs, how many dog bones would each get? And so, this to me is a task that is open-ended. We could go multiple ways with this, although we are going to focus in the classroom on a particular question. But it's also rich in that kids are going to connect with it, especially if they've been into shops like this. It has multiple entry points. And so, in a lot of ways it really connects to these different terms that you've brought up earlier.

Mike: It's interesting because as you describe it, particularly the fact that there's a visual component to this, it really comes clear how there are multiple ways that a child could think about the question or attack the question that you asked. Is there any role for number choice in thinking about how to design a rich task?

Kim: Yeah, the numbers are really important. And it's fun to play with different numbers and see how they pan out. And so, in thinking about the dog bone task, we want to keep the numbers accessible. So, when I think about a single peg or a single hook for the dog bone packages, I can think about eight plus eight. And from where kids are coming from, eight plus eight should be accessible. When I think about what I'm seeing with the dog bones, I'm seeing two groups of four in each of the packages. And so, that's a nice way looking at doubles that students might find useful, but I'm also looking at three packages of eight. And so, I could add eight plus eight plus eight, which could bring the teacher very easily to a three-times-eight multiplication expression. One that is manageable and a great way to introduce multiplication, the times symbol where we're going with all this, but one that is also still accessible for kids to be thinking about as repeated addition.

Kim: Ultimately, the numbers get to a final answer of 48 dog bones, and you could think about this in terms of six times eight. But kids aren't going to know six times eight, at least not very commonly in this point of an early introduction to multiplication. But there's various ways that they could get to the eight. We ultimately landed on these numbers: 2, 4, 8, 16, but also this additional number 3, which as a separate prime number really throws some additional mathematical thinking into the mix that elevates the task itself. There's things that are critical to be thinking about as you imagine this task and how it might play out instructionally. It's really important to think about how this stands in the curriculum sequence. So, it is introductory, it's using numbers that the kids are probably not going to know off the top of their heads—related multiplication fact and an answer—but the numbers themselves might elicit different strategies, and the visual might elicit different strategies. And all of this connects to the commutative and associated properties for multiplication and how they might play out with student thinking.

Kim: And so, it's all connected. All these pieces really fit together into what I think is a really interesting and engaging task for students. It's challenging enough, but it's also very accessible simply by counting, kids could count what they see. And the context is engaging for kids as well , because they might be thinking about displays that they've seen and how this corresponds to trips to the store that they've had.

Mike: Part of what you've got me thinking, Kim, is there's the design of the task and then there's the element of how a teacher might go about facilitating it. And I think, I want to come back to that, too, because I heard you not only describe the task—the way that it was designed—but you also described some of the ways that a teacher might introduce it, some of the things that they might pose to kids. I wonder if you'd be willing to talk a little bit about facilitation and some of the things about facilitation that can bring a task to life?

Kim: So, one of the things that I like to think about in implementing tasks, Mike, is the very intentional letting go—that kids need time to think. They need time to explore. I think all too often as teachers, we have this desire to go in and help and direct. And sometimes we just need to back off and let them think about the different ways that they might approach that, those different entry points, and let them explore and let them take the time to do that. And so, one of the things that I always used to describe to pre-service teachers was the walk away . That I would go up and talk to a group or talk to some partners working together and listen to what they were doing and maybe pose an additional question and then I'd walk away. I didn't want to hear their answer right away. I wanted them to talk about that amongst themselves. But if I stood there, they would start talking to me. And so, I would walk away, move on to a different group, come back later and hear what their thinking was. But it creates that space for letting kids explore, think about, and also not feel the pressure to be getting to a particular answer in a particular time frame. And I think that's really important for kids to have that freedom.

Mike: Yeah. It also strikes me that you're reframing your role for kids, too, in the sense that by walking away, you're sending the signal that, "I actually have confidence that you and your partners can think about this and reason about this."

Kim: Absolutely. It is putting some power, some agency, with the students themselves. "You are capable of doing this; you're capable of thinking about this. You do not need me here to be your sounding board or the mathematical authority."

Mike: Kim, can you talk a little bit about the idea of entry points? I'm wondering for teachers in the field, how would you actually define an entry point? What does that look like?

Kim: I'm not sure I have a good definition for it, but I do have an analogy, and I would compare it to on-ramps for highways. And when I think about on-ramps, we can all get on the same highway, but we might do it at different places, and we might make choices for where we get on based on our current location or what we know about the on-ramp. But as long as I have a workable vehicle, I can do it . And maybe that's our prior knowledge. But unfortunately, often kids, they don't think that they have a workable vehicle or teachers might even underestimate the child's vehicle that they have. And so, I've probably gone far enough with this analogy, but kids come onto that mathematical highway at different places with particular problems. And I think making sure that we as teachers, as educators, as curriculum designers, that we're thinking about all those different possibilities for getting into a problem and knowing that we can all go to the same place regardless of where we've gotten on.

Mike: That's really helpful. So, if I'm an educator and I'm designing a task, or even if I'm facilitating a task that comes with my curriculum, what guidance would you offer to folks to ensure that there are entry points for kids?

Kim: I think it really depends on the mathematics and what you're trying to accomplish. And so, with this one in particular—the dog bones visual—I might be asking myself, "Can I do it without multiplication since this is an entry point for multiplication. Can I do it without that, or could I do it with basic counting skills?" And so, are those viable entry paths open for kids if they don't have where we're going with the task already in their toolkit? When I think about ensuring that there's entry points, I like to think about stripping away the expectations for where you want to go with the task, really allowing kids to have that freedom for exploration. And it's the variety of entry points that leads to the multiple strategies. And when you have multiple strategies, you can make connections between and among those representations. And then you've got something really robust. Or I might go back to that term rich task . And so, it's about can they do it without where you're going, that mathematical goal, and however they encounter that or engage with it, does it still connect to other strategies that will bring them to your mathematical goal?

Mike: That is really helpful . What that has me thinking is we have heard in the field about the idea of the five practices and anticipating. But this is a little bit of a twist on that in the sense that you're evaluating the task and saying, "What's possible for a kid to get into this task?" I love the example of multiplication. So, for example, if I only have partial or emergent understanding of multiplication, can I still work my way toward an answer to that? And if the answer is no, then what?

Kim: Right? But you mentioned partial or emergent understanding, whereas I think this task, actually, you can get in with no understanding of multiplication. I can look at three sets of eight dog bones, add eight plus eight plus eight to get to 24, and then the teacher has that to latch onto to say, "We have another way of writing this. I can write three times eight to represent eight being added three times." And so even that visual structure leads us to something that we can hook on to, to bring forth the connection to multiplication.

Mike: I think that's helpful because it means that there's a value and there's a utility to having ways of doing this that you can ultimately connect to the place where you want to go, right?

Kim: Yeah. And I think there's opportunities for that kind of reasoning or openness throughout math education; imagining what we can do with tasks to really not just ensure that there's entry points, but value those entry points as really important connections to all students' prior knowledge and where we're going mathematically.

Mike: That totally makes sense.

Kim: If we can't imagine that our kids are capable of problem-solving and engaging in challenging tasks, then they won't be able to imagine that themselves. And so, in a way, we have to pass along the agency by sometimes just believing ourselves that, 'Yeah, maybe they can do this,' and giving them that time and seeing what happens.

Mike: Before we close the conversation, I'm wondering if you have any resources that you think would help someone listening to this conversation deepen their understanding of designing or implementing rich tasks?

Kim: I think the best way to really think about task design and build your own facility is to do some rich tasks and just engage with them as a learner. So, one of the resources that I frequently turn to is the NCTM journals, both old and new. They have really good problems in there. And you can sometimes take one of those and change it in a way that is making it more challenging or making it a more generalizable situation. There's various problem-solving publications. I could certainly plug my own books, Problem Solving in All Seasons ( PreK–2, 3–5 ).

Mike: I have read it, Kim.

Kim: (laughs)

Mike: I would absolutely recommend it.

Kim: Another resource that I love is NRICH. It's a website—NRICH—which is nrich.maths.org . Those are just incredible tasks that really get you thinking in various ways about, [some] good problem-solving experiences. So, what I would recommend for teachers or other people who are interested in this, is to really do that mathematics and then reflect on what made it interesting for you . What surprised you? Were there twists? Were there things, stuck points where you had to get past? And then also to extend the thinking by asking yourself something like, "So does this always work? Or when does this work?" And how could you apply the mathematics to more broad situations? And finally, I think it's really important for teachers to put themselves in the minds of their students and sense what might excite them or challenge them.

Mike: Thank you so much for joining us, Kim. It's really been a pleasure talking to you.

Kim: Thank you, Mike.

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.

© 2023 The Math Learning Center | www.mathlearningcenter.org

Adopting a new curriculum can be challenging. Beyond the materials themselves, a curriculum adoption may represent changes to long-standing practices, beliefs and classroom culture. On today's podcast, Dana Nathanson, an elementary math coordinator in Leander, TX, talks about how leaders can effectively design, manage and sustain a successful curriculum adoption.

Mike Wallus: Adopting a new curriculum is not for the faint of heart. What makes this challenging? Well, beyond the materials themselves, a curriculum adoption may represent many things: changes to long-standing practices, beliefs and classroom culture. On today's podcast, we'll talk with Dana Nathanson, the elementary math coordinator in Leander, TX, about how leaders can effectively design, manage and sustain a successful curriculum adoption. Welcome to the podcast, Dana. I'm thrilled to have you and be able to talk with you a little bit about the work that goes into adopting and supporting the implementation of a new curriculum.

Dana Nathanson: I'm excited to be here. Thank you for the opportunity.

Mike: Absolutely. So, in your case, we're talking about the work that you did in Leander, TX, when you supported the adoption of Bridges in Mathematics . I'd love to start by talking about something that feels really critical when a school or a district adopts a new curriculum: the idea of buy-in. How did you think about building buy-in for teachers when you adopted Bridges in Mathematics in your district?

Dana: I think that's an interesting question, because we do hear a lot about, "How do you get people to buy in?" And in our district when we think about buy-in, I think about, "That's my idea. And so how am I going to get people on board with my idea?" And so, really, we want to kind of flip the script on that and think about ownership. And so, when we think about, "How do I get people to kind of own this idea with me?" Then that is really where we see true empowerment. And so, we really approach this with that kind of lens to be thinking about, "How do I get people to own this, um, process and own what good math instruction looks like with me?" So that when we do adopt that we are adopting something that aligns with our vision for mathematics and what we want to see students participating in and being a part of in the classroom.

Mike: That really feels different even just to hear you talk about it. Ownership kind of conveys this idea that there's a shared responsibility as opposed to buy-in, which is, can I convince you to do a thing?

Dana: Right, right. And so, to get that ownership, we were at a time in the state of Texas where we were adopting new standards. And so, it was kind of, like, the perfect timing to think about, "How are we going to really get a clear picture of what we want math instruction to look like?" So, we did a lot of work with our teachers up front prior to adoption on what are those standards going to look like and how are we…or what do we feel like is the best way to teach math, really, in the younger grades? And so, we did a lot of learning together, a lot of reading. We really grounded ourselves with some of the work of Cathy Seeley, who is a former NCTM (National Council of Teachers of Mathematics) president. She wrote a book, called Faster Isn't Smarter . And so, we kind of looked at that as a good starting point for, "We want all students to have opportunities to make sense of math, do the math and use the math." And that kind of became our foundation. It's not just about procedural fluency, but conceptual understanding and then ultimately, transfer. And so, we grounded our work in that and tried to bring people along as far as owning that vision. And then from there, we really looked at what teachers wanted from a resource. And thinking about the use of continuous improvement tools, we used feedback loops, consent-o-grams to—all along the way—so that we could really feel like everybody was owning. They wanted a parent component. They wanted more technology. They wanted practice opportunities through, through games. And so, when we established a rubric together with teachers and administrators, then that really helped us when we came to adoption because we were looking for something that checked all of these boxes.

Mike: Yeah. The story that I make up as I hear you talk about that, is that you had a level of consensus around what you were looking for, which made it a lot easier to make a decision that you felt good about, that you felt like people could own.

Dana: Right. Exactly.

Mike: So, I think anyone listening to this podcast knows that schools and districts have limited resources. So, the thing that I'm wondering about is, what were some of the supports that you prioritized during the first year of your implementation of Bridges?

Dana: So, I'm fortunate to work in a team of—there's three of us at the district level, to support all of our campuses. We have over, well, we have 28 elementaries, and we're about to open 29, and so over a thousand elementary teachers that we support. But we knew that the three of us could not do it alone. And so, we are also fortunate that we have an instructional coach at each campus. Now this instructional coach is not specific to math. They support all content areas, but we had to bring them along. We had to get them to own it, and we had to have them feeling comfortable. And then we also created a teacher-leader system where we had a lead teacher from each campus. And we really focused on the instructional coach and the lead teacher as our early adopters or our campus champions to really help us rally—rally everyone around, um, owning this vision for mathematics and also the implementation of a new resource. And what a great opportunity along with the implementation of our new standards. And so, we did pay our teacher leads a stipend for that year. And having the instructional coaches in place was critical because it's those two groups that we would be able to lead and then they would take back to their campuses. Another thing that was also critical in that first year was administrator support. And I know that we're going to talk a little bit more about that, but I just want to highlight the fact that our campus principals were really great about giving teachers time in that first year of implementation to work as a professional learning community together, to have half days to plan and support the new adoption that we had.

Mike: There's a lot that you shared there…

Dana: (chuckles)

Mike: (chuckles) …that I'd love to dig into a little bit. I think what strikes me about what you said though, particularly at the last part first, is the way that you worked with and supported administrators in really designing a year one where teachers had space and time to actually really devote mental space to thinking about a new curriculum: how it's designed, giving space to plan. That feels like it was an intentional priority that you worked with your administrative team to create.

Dana: Yes, that was very intentional. And it was evident when we began our first Getting Started trainings that summer. And we also trained our ICS (In-Class Support) and our lead teachers first, so that they could kind of get the buzz going for summer professional learning. And I thought it was also great that we were able to have the resources available. If you attended the training, you left with your resources. And teachers were so excited to get all of the great resources that are provided with Bridges. So, that was kind of a draw for them. But then once they had their resources and you start to dig through everything, there's another level of support that is needed. And so, we actually had what we called open houses prior to school starting so that teachers could go around to different teachers' classrooms in the district to see, "How did you set up your Number Corner? How did you provide space or how are you structuring space in your classroom for Work Places?" And so, we had a lot of teachers [who] would go around to other teachers' classrooms at other campuses and kind of explore to see and get ideas from each other, which was really powerful. And we created the space up front for that prior to the school year so that they would have that opportunity. And I also want to say at this time, seven years ago, we had a pretty good Twitter presence during this, so that we could also have people online. And I know Twitter's kind of blown up since then, but we were on Twitter a lot, and just being able to share that way, as well.

Mike: So, I love this idea of giving teachers space and time to get their materials and get set up. And the open house idea feels really supportive. One of the things that I sometimes think about is an adoption and an implementation might be a pedagogical shift. There might be a different understanding of the mathematics. But the truth is for a lot of people, the very first thing is, "How am I going to find a home for all of these things? What will my classroom look like?" You're kind of attending to that really important need that people have to have met even before they're trying to grapple with the curriculum itself.

Dana: Right. And so, to give that time for them up front to kind of get settled in—with what's this going to look like and how do I make it work—I think was key. And I talked a little bit earlier about the principals being able to provide some half-day plannings for teams throughout the year. But we also offered what we would term "power hours" after school. And we would host these in teachers' classrooms. And so, this month we're going to talk about the Work Places because we thought it was so critical that the teachers played all the Work Places so that they would know. And that's how you kind of get their ownership of that, too, as well. And so, we would have these power hours after school, where they would come and play the Work Places, or maybe the next month we're going to do a Math Forum together. That's coming up. And then the next month we're going to go through all of the Number Corner. Now you guys have all these great videos, but this was before you had those for Number Corner. And so, we were just really trying to get teachers in each other's classrooms sharing and making it easier. And we would all make the charts together so that they would have them ready for the next month. And we would see a lot of people on Twitter posting: "Here I am at my son's baseball game with my binder, learning." (laughs) But I mean, that's just part of the process, too, right?

Mike: Well, you've really started to address the next thing that I wanted to bring up, which is, when I think about having been an elementary teacher for 17 years, what strikes me is that in education, we sometimes give ourselves really short windows of time to do a complete "implementation" quote unquote. I can't tell you how many times I've heard, "This year is literacy. Next year is math."

Dana: Right.

Mike: I think what you're starting to address, but that I wanted to ask you directly is, as an instructional leader, how have you really tried to maintain the integrity of your implementation over time? Maybe just talk a little bit about how you've thought about that process of maintaining and sustaining.

Dana: So again, we leaned heavily, and we still continue to lean heavily, on our instructional coaches at campuses. So, each nine weeks, especially in the first three years of implementation—but even now—we'll dive into what does that curriculum look like for the upcoming nine weeks? And we'll give them ideas and point out specific things that are coming up so that they know how to share or how to kind of pull these things out when they're planning with the different grade levels. And so, we would continue to meet with them, but we always start with that unit introduction.

Mike: Hmm.

Dana: And if teachers can just take the time to read this, and this was another big sell from our department for Bridges, was the built-in PD (professional development). If you read those introductions, just, like, how much learning that the teachers can have. So, those first years we really wrapped ourselves around those introductions and the learning together as teams. But we also took, at the time you guys had an Implementation Guide…

Mike: We still do.

Dana: Then I will plug the Implementation Guide. Now it's expanded a lot more. But we took that and we had teachers really pick what's a strength for you on here so that other teachers could come see that modeled for them. And then, what's your area of growth for this nine weeks or for this year? Are you going to focus just on Number Corner, but what parts of Number Corner? Or you want to work on the Work Places, but you're not really implementing the sentence frames correctly. So, whatever that goal is for you, and then the instructional coach and the campus administrator would know what that is, and they're able to support you or come give you feedback on that. And that has really helped us because that gave also administrators, kind of the look-fors that they should see when they walk into classrooms. And our department is fortunate to be able to walk with administrators and our instructional coaches so that we could all kind of participate in this coaching together around what we want it to look like, and then where it's going well. And we bring teachers across campuses and classrooms to see where it's going well, and really having them focus on some goals that they want to set to improve.

Mike: So, I suspect unless Leander is a magical school district that's different from everywhere else, you don't have exactly the same staff that you did…

Dana: (chuckles)

Mike: …seven years ago when you started your process. So, you probably know where I'm going, which is…

Dana: Yes.

Mike: …how do you account for the fact that teachers, like everyone else, have lives? And sometimes they move on from the grade level that they're teaching or their families move somewhere else. You have new administrators and educators coming in. How do you account for, kind of, that turnover that's just natural in education?

Dana: Right. So, we have the natural turnover. But also we are one of the fastest-growing school districts in Texas. And we continue to open about one school at least , sometimes two a year. So, we know that training and learning together is so important. And so, we have sent our curriculum specialists—have participated in many of the Bridges trainers of trainers, trainers of leaders, and for Getting Started. And so, we still offer a two-day for that every summer and also in the fall. And we offer that special session for our new administrators, and we even have turnover in our cabinet. So, we offer that training, and I sit down with superintendents and our area superintendents, because we all have to own, own this. And so that is just a yearly thing that we do. But then also continuing to use our campus champions. We have continued that teacher-leader program. They support our new-to-district teachers as well, and then our instructional coaches. So, it is an ongoing cycle. And I will tell you, at first we kind of say, like, "If you can get Number Corner, your Problems & Investigations, and your Work Places down," then we kind of introduced then the assessment piece the next year and then the intervention piece. So, we have layered it in that way so that it's not so overwhelming for our teachers. And then it just becomes part of your practice.

Mike: Thank you so much for that, Dana. The next piece that I wanted to go to, and you've alluded to it throughout this, is the role that instructional leaders—be they administrators or instructional coaches—play… I was reading a bit from The Wallace Foundation about how critically important principals are. Anthony Mohammad talks about how administrators are the ceiling on where a building can go. Can you talk in a little bit more detail about the kind of work that you did to bring your instructional leaders, particularly your principals, into the process of owning the adoption and the implementation?

Dana: This is still a journey. And so, I want to make sure that I plug that, that even though we are seven years into this adoption, we're still on a journey. Everybody's on a journey. We're not at the end of the race when we think about best practices and instruction in mathematics. But to bring our administrators along, we are fortunate to have instructional leadership meetings every month. And so, we really focus on curriculum with them. We focus on best practices and really, we bring learning to them. And we use a lot of the resources that The Math Learning Center provides. We will learn through some of the blog posts together, reading those together. But really what we wanted up front before adoption and through the adoption process was for our principals to really own the fact that all students, each and every student, can learn math, and making that accessible to all of our learners. And so that is a mindset. We did a lot of work around the mindset work with Jo Boaler and Carol Dweck. And so, thinking about how then, we wanted—we're not a district that just throws out the direct instruction piece either. We still value that direct instruction. But we want to see that blended with investigating and exploration for our students. And then also having that small group time where they're able to reinforce through Work Places. And so, we really wanted our principals to be firm in the components so that they would know what to see in the classroom, but also firm in the fact that we want to see visual models. What do our standards say? What are the best practices for mathematics say? And the use of manipulatives. And that our Number Corner is meant to be a routine and why we value that for practice for pre-teaching and reinforcing. And what's the value of playing the games in Work Places? So that they would understand these components and really own that they want to see these in the classroom because that's what we know is best practices in mathematics.

Mike: When you think about Bridges, in particular, as a curriculum that you've adopted, were there features of the way Bridges is structured or organized that you really felt like it was important to help people understand going into it? And what I mean by that is, in some ways, Bridges is a departure from a traditional curriculum. And I'm wondering what were the things that you identified that's like, gosh, I've just got to make sure people understand this about how it's designed to work?

Dana: Again, it's kind of the three components that I already alluded to, but really that Number Corner piece. Really thinking about Number Corner as an opportunity for the whole class. And we even kind of connect it to a read-aloud. This is an opportunity for the whole class to come together and to, either it's going to pre-teach some things or it's going to reteach some things. And so how are you making sure that those routines are in place and making sure that we have secured small group time for the Work Places to happen? And that's what we call our small group time, is Work Place time. Because we're talking about how the teacher is floating about the Work Places and observing how they're communicating and playing the game and how they are talking about the math with each other. So, I would say, the Work Places and the Number Corner are really, kind of, the areas that were a little bit harder to bring people along.

Mike: What strikes me about what you said is that you describe the function of those two pieces of the curriculum, Number Corner as a tool to have consistent, long-term opportunities to either reengage with big ideas or pre-engage with big ideas that are coming up. And then the idea that Work Places are an opportunity to practice. But they're so much richer of an opportunity to practice than the worksheets that I remember as a kid, where there were 25 naked number problems and two story problems at the bottom (chuckles). They function in the same way in the sense that they're the opportunity for long-term practice.

Dana: Right.

Mike: And the added bonuses, as you said, when the teacher's moving about the classroom, they can formatively assess and listen to what kids are saying. But they can also jump in and do some mini conferring with children in the moment.

Dana: Right.

Mike: To help guide them or move them or advance their thinking.

Dana: Exactly. And just thinking about that Work Place time and when teachers are thinking about, "Oh, I have to plan something different for this small group." Well, bring that group together to engage in the Work Place with them so that you are right there observing and having, like you said, that conferring time or that mini-lesson over the Work Place.

Mike: Well, before we close, one of the default questions that I ask anyone who's a guest is, if someone was listening to this podcast and they were charged with leading an adoption or an implementation of a curriculum, what are some of the resources you would recommend for someone who is looking for guidance on how to do this work?

Dana: Well, now I would definitely use the blue (laughs) Principles to Actions NCTM book, because I think this sets the great stage for, what are those teaching practices that we want? But also it talks about the elements. One of the essential elements is specific to curriculum. I didn't mention this earlier, but we also had parents give us feedback along the way. And I think that that is also critical, as well as students. Let your students have some hands-on experiences with the resources so that they're able to even advocate and say, "This is how we want to learn math." There's no denying when you see that students are feeling successful, but also when they are loving what they're doing in the math classroom.

Mike: Well, I was just going to say, everything that you talked about today, I think that the word that comes to mind in addition to ownership is investment . As I've listened to you, I keep thinking, you invested time and energy to make the things that you were looking for come to fruition…

Dana: Uh-hm.

Mike: …to continue the journey, as you said. And without investing in those really important things, the outcome might look really different at this point in time.

Dana: Right.

Mike: Well, thank you so much for joining us, Dana. I've learned a lot from the conversation. It's been a pleasure talking to you.

Dana: Thanks, Mike. I appreciate it.

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.

© 2023 The Math Learning Center | www.mathlearningcenter.org

Take a moment to think about the students in your most recent class. What assets do each of them bring to your classroom, and how might those assets provide a foundation for their learning? Today we're talking with Dr. Jessica Hunt about asset-based learning environments. We'll talk about how educators can build an asset-based learning environments in their classrooms, schools and school districts.

Mike Wallus: Take a moment to think about the students in your most recent class. What assets do each of them bring to your classroom, and how might those assets provide a foundation for their learning? Today we're talking with Dr. Jessica Hunt about asset-based learning environments. We'll talk about how educators can build an asset-based learning environment in their classrooms, schools and school districts. Welcome to the podcast, Jessica. Thanks for joining us.

Jessica Hunt: Thank you . I'm so excited to be here today.

Mike: Well, I would love to start our conversation asking you to help define some language that we're going to use throughout the course of the podcast.

Jessica: Sure.

Mike: I'm wondering if you can just describe the difference between an asset-based and a deficit-focused learning environment.

Jessica: I think historically what we see a lot of is deficit-based thinking. And deficit-based thinking focuses on perceived weaknesses of students—or even a group of students. And it focuses on students as the problem. And as a result, we tend to use instruction in an attempt to fix students or to fix their thinking. So, an asset-based learning environment means focusing on and beginning with strengths as opposed to what we think kids need or how to fix them. So, this means viewing kids as able and recognizing that the diversity of their thoughts, their culture, their experiences—all of these things are valuable and can actually strengthen and add meaning to classrooms and to instruction. I think asset-based learning environments involve a shift in our own mindset as teachers. And, of course, what we hope results from that is a shift in our practice. We talk a lot about growth mindsets for kids. I think I am referring to growth mindsets that teachers have about kids. We can ask, "What do students know and how can I use that? Or how can I build upon that through my teaching?" I've never met a kid that didn't bring something to instruction. Every student that I've met [has] had strengths that they bring to mathematics classrooms and to communities to expand their thinking and also that of their peers.

Mike: It's fascinating listening to your description. I find myself thinking about how deficit-based many of the systems and structures…

Jessica: Yeah.

Mike: …and practices are, even though we do these things with positive intent.

Jessica: Yeah.

Mike: Can you just say more about that? How do you see deficit thinking filtering into some of the systems and then impacting the learning environments in our kids?

Jessica: Sure. I think two ways that I see deficit thinking filtering into, driving—and driving systems in classrooms—involve things like time and priorities. Time and how it's used in classrooms and schools is one area that deficit thinking can impact in a big way. How are systems recommending that teachers actually spend their time with students in the context of a particular day or a week or even a unit of instruction? And I ask that question because I think that it's one thing to state that we have an asset-based approach. Yet it's quite another to consider the need to develop meaningful habits within classroom spaces that can really promote student strengths.

Mike: So, one of the things that you just said really struck me, which is this idea of habits in the classroom. I'm excited to hear what you're going to say about that.

Jessica: I think one of the key habits that we have in asset-based learning environments is this idea of listening to kids. I've never met a student that didn't have viable and valuable ideas about mathematics. The key for me is having the time and space to uncover and understand what those are. So, we've got to have a way to listen to students' thinking. When we do that, when we understand the reasoning and the strengths that they're bringing, that supports us in selecting instructional tools and strategies that leverage both their individual strengths and those that they bring to the group in order to promote learning.

Mike: Let's pick up on that a little bit. This idea of listening to kids and understanding their thinking and understanding of what it means about the assets that they bring. For a person who might be listening, help them form an image of what that might look like in an elementary classroom. Talk to me a little bit about on a day-to-day basis, how might this idea of listening to kids or attending to kids' thinking—and really considering the assets—how might that show up?

Jessica: One way it shows up is this focus on learning. And before I go on with that, I want to talk a little bit about how learning and a focus on it is a little different than focusing on performance. So, focusing on performance as opposed to learning, risks looking at change as something that's fast and quick as opposed to something that grows and endures. So, part of focusing on learning means that we're looking more at the process as opposed to only examining quick outcomes or products of what students are experiencing in classrooms. It's actually interesting to think about that in terms of educational equity because there's some research that actually suggests that performance gains don't necessarily equate to learning gains.

Mike: I think that's fascinating. You're making me think of two things. One, and I'm going to reference this for people who are listening, is Taking Action , which is NCTM's work. Really trying to say what do some of the really critical principles of high-quality education look like in grades pre-K through 5? And they have a really specific focus on attending to what do we want kids to learn versus simply what's the performance.

Jessica: Yes, absolutely.

Mike: I also just wanted to key in on something you said, which is that performance can be short-lived, but learning endures.

Jessica: It sure does. If we want to focus on learning, it means that we have to be intentional in our classroom practices. And I also think that links to a lot of things. Like you brought up NCTM, and a lot of the things that they advocate for. I think there are some natural linkages there as well. So, for me, being intentional , one key part of that is ensuring that students are doing the thinking so that teachers can listen to and promote that thinking. So, we want the placement of the learning and the thinking on the students for a good percentage of the instructional time. We want to ensure that we're immersing students in content rather than simply presenting it all the time. And I think another part of that listening involves positioning students and the ideas that they're bringing forward as competent. So, I think, together, what all of this means is that we're supporting students to make meaning for themselves, yet definitely not by themselves.

Jessica: Teachers have an intentional, key role. And part of that intentionality involves things like slowing down and thinking carefully about how to structure learning experiences. And taking more time and planning and ensuring that students have access to multiple ways to engage in and represent and express their thinking with respect to those tasks and activities that they're using and drawing upon to learn. And I think that asset-based learning environments allow for that intentionality. It allows for that time and space and planning. And in teaching, it allows for that immersion and thinking and listening and positioning of students as the sense-makers, as the doers and thinkers of mathematics.

Mike: I think the connection that I'm making is this idea that there are some shifts that have to happen in order to enable asset-based listening and intentionality. One of the things that comes to mind is it really starts with even how you structure or imagine the task itself. If you're posing a problem, that problem isn't accompanied by a, "Let me show you how to find the answer." That actually allows kids to think about it. And there might be some divergent thinking, and that's actually a good thing. We want to understand how kids are thinking so we can respond to their thinking.

Jessica Absolutely.

Mike: That's a big contrast to saying, "Let me show you a task; let me show you how to do the task." It's pretty difficult to imagine listening in that kind of context because really what you're asking them to do isn't thinking about how to solve it. Does that make sense?

Jessica: It sure does. And I think for me, or a hunch that I would have, is that that also goes back to this whole idea of teaching and listening and maybe even assessing, if you will, for what we think kids need versus what they're bringing us versus their strengths. I see some connections there in what you're seeing.

Mike: Let's talk about that a little bit.

Jessica: Sure.

Mike: Particularly assessment, I think when I was getting ready for this episode, that was the first thing that came to mind. I found myself thinking about previous PLC meetings or data meetings that I've had where even if we were looking at student work, I have to confess that I found myself thinking about the fact that we were looking at what kids didn't understand versus what they did understand. And I tried to kind of imagine how those conversations would've looked from an asset perspective. What would it look like to look at student work and to compare student work and think about assets versus thinking about, "What do I need to remediate in the type of thinking that I'm seeing?"

Jessica: Uh-hm. I hear you there. I think it speaks to something that if we really want to build asset-based learning environments, we need to make some shifts. And I think one of those shifts is how we look at and use data and assessment. Primarily, I think we need to assess strengths and not needs. I heard that a lot as you were talking. How can we focus on assessing strengths and not needs? I say that to a lot of people, and they're like, "What's the difference?" (laughs) Or, "That seems so small." (laughs) But I think it winds up being a really big deal. If you think about it, trying to uncover needs perpetuates this idea that we should focus on what we see as the problem, which as I mentioned earlier, usually becomes the students or a particular group of students. And I think it's very problematic because it sets us up as teachers to keep viewing students and their ideas as something that needs to be fixed as opposed to assets that we can build from or learn from in the classroom.

Mike: Yeah. One of the other ideas that we've talked about on this podcast in different episodes is the idea of relevancy and engagement. And it strikes me that these ideas about listening to kids for assets are pretty connected to those ideas about relevancy and engagement.

Jessica: Yeah, most definitely. I think, again, figuring out, we sometimes call this prior knowledge, but I look at it as when kids come to school, they bring with them their entire experience. So, what are those experiences and what from their eyes are things that are relevant and engaging and things in which they are passionate about themselves? And what do they know about those things? And how might they connect to what others in the classroom know about those things? And how can we, to borrow a term, how can we "mathematize" those things (laughs) in ways that are beneficial for individual kids and for the community of learners in our classroom? Like, how can we make those connections? I don't think we can answer those types of questions when we use assessment from this place of, "What don't students know?" Or, "How can I get them to this particular place?" If that makes sense.

Mike: It does.

Jessica: I think we can ask those questions from a strengths-based lens that is curious about and passionate about really getting at, again, this whole experience that kids are bringing with them to school. And how we can use that to not only better students' learning, but better the classroom community and maybe even better the mathematics that kids are learning in that community.

Mike: Absolutely.

Jessica: That's, that's interesting to think about.

Mike: So, you started to address one of the questions that I was going to ask, which is, I'm imagining that there are folks who are listening to the podcast, and they're just starting to think about what are some of the small steps or the small moves that I might make? What small steps would you advise folks to think about if they're trying to cultivate an asset-focused learning environment?

Jessica: It's an interesting question, and I would suggest putting into practice some of the bigger ideas that we're getting at in asset-based learning environments themselves. And the first is, look at your own strengths. And when I say who I'm referencing there, it can be a teacher, it can be a school, it can be a district. If you look at your own strengths first, look at how your practices, your structures, your priorities are uncovering and using strengths. And if they're not, why not? Kind of looking at what's there, what capacities do we currently have that we can build on toward asset-based learning environments? And I think I would pair that with just a commitment to, to action, if you will. You know, start small, but start now. If you're a classroom teacher for instance—I tend to go to that (laughs), that grade size a lot 'cause I still very much, uh, identify as a teacher—start with one task or one day, or part of a day, where you can slow down and use your instructional time to listen for kids' strengths.

Jessica: What brilliance and valuable ways of reasoning are they sharing with you? And what kinds of activity or task or environment did you need to put in place to uncover that? What did you learn about it? What did you learn about yourself in this process? So, we learn about kids, and then we learn about ourselves. It becomes sort of this beautiful back and forth between students and teachers where we're all learning about ourselves and about each other. And I think that learning piece is the third thing that I would suggest. Again, going back to, let's focus on learning. Let's celebrate our own learning as teachers and schools and districts and et cetera. Reframing your practices and structures will take time. That's OK. But learn to celebrate the steps that you and your communities are taking toward this asset-based model of instruction. And know that, again, you know, when we work to do that, we enable kids as mathematical thinkers and doers. So, we take that problem off kids, and we place it as a challenge in our instructional design, in our experiences and our interactions between teachers and students. So, I think for me, I would really invite folks to take those small steps, uncover your own strengths, learn to listen, and celebrate your own learning.

Mike: Before we conclude the episode, I'm wondering if you can recommend any resources for someone who wants to continue learning about an asset-based approach to elementary mathematics?

Jessica: Yeah. There [are] so many good examples of this. I think about my own learning as a teacher and a teacher of teachers, (laughs) and a researcher. And I think about things like cognitively guided instruction or the work of the The Dream Project in early childhood or even TODOS , where I know they provide a lot of wonderful examples of asset-oriented resources. I'll also do a shameless plug (laughs) for my, for my own book, you know, myself…

Mike: Plug away!

Jessica: … (laughs) and Jenny Ainslie put together, called, Designing Effective Math Interventions: An Educator's Guide to Learner-Driven Instruction . And that book came off of a project that I did with, uh, National Science Foundation support, where we looked at kids' thinking over time and designed some tasks and activities to support conceptual understanding of fractions. But there are those. And, and so, so many more. But those are the ones that come to mind immediately.

Mike: That's fantastic. And we'll share links to those things with the podcast.

Jessica: Great .

Mike: I want to thank you so much for joining us, Jessica; it's really been a pleasure talking to you.

Jessica: Oh, thank you. It's been an immense pleasure talking with you as well. And thank you for inviting me. I really appreciate it.

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.

© 2023 The Math Learning Center | www.mathlearningcenter.org

Today on the Podcast, a team of educational leaders from TNTP , an organization dedicated to great teaching, talks about myths surrounding multilingual learners and introduces specific strategies educators can use to leverage their assets and support meaningful understanding of mathematics.

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

How to Support Multilingual Learners With Higher Expectations

Mike: As a young educator, I was often unsure how to support the multilingual learners in my classroom. And my well-intended attempts didn't always have the impact that I hoped they would. Today we're returning to a topic we've discussed before on the podcast: support for multilingual learners in the mathematics classroom. We'll talk about some of the myths surrounding multilingual learners and dig into specific strategies educators can use to leverage their assets and support meaningful understanding of mathematics. Today we're joined by Shannon Lindstedt, Jean Harvey and Christa Beebe from TNTP (The New Teacher Project). We're going to talk with them about a set of tools and practices they've developed to support educators who serve multilingual learners.

Mike: Welcome, Shannon, Jean and Christa. Great to have you with us today.

Jean: Thanks for having us.

Shannon: Yeah, happy to be here.

Mike: So, Jean, I'd like to start with a question for you. I'm wondering if you could talk a bit about the misconceptions that we have in the education community involving multilingual learners. What is it that we've misunderstood about multilingual learners and how to support them in a mathematics classroom?

Jean: So, one of the most prominent misconceptions is that multilingual learners—MLLs as we call them—cannot engage in grade-level math because they do not yet have the language to understand the task. In MLL Good to Great, we take teachers through a planning protocol that has them assess English language demands in a task. They consider what mathematical academic language a student needs to know to answer a problem. We ask teachers to also analyze what language in a problem may be new to students, and then they think through what visuals and additional supports could help students to understand the language and the problem. We also think through what language students will need to use to express their understanding. This step is so important because it empowers MLLs to be part of the conversation, and they can grow their language at the same time. When teachers first implement the supports, they're always so delighted how well their MLLs were able to participate in class that day. When the language is supported and MLLs can fully engage in the task, teachers see how capable they are and how eager they are to dig into the rigorous learning.

Jean: The supports also help to dispel another common myth, which is that MLLs might lack the confidence or the ability to engage in class discussions. Sometimes teachers avoid calling on MLLs because they fear embarrassing students. However, when our teachers provide the language supports that help students to understand the task and to produce the language needed to express their understanding, they become part of the conversation. MLLs need that access to critical language, and they'll need some independent think time to craft a response. But they're fully capable of engaging in grade-level math and expressing their understanding. By offering both receptive and productive language supports, MLLs are able to unlock content and demonstrate their incredible learning. We know that actively engaging in class discussions is important for all students, but it's absolutely essential for MLLs.

Mike: There was a particular piece that you mentioned. You talked about the need for individual think time. I'm wondering if you can just say a little bit more about that, particularly with respect to MLL students?

Jean: Absolutely. So, one thing that we learned early on was that it's not always instinctive to give kids the think time that they need to gather their thoughts because they're not just processing the math in a given problem, they're also assembling the language that they need to use. In many cases, they're translating from their native language into English and trying to create…figure out how they're going to express their understanding in English. So, giving them that independent think time is incredibly important for MLLs.

Mike: Well, I will say that is most certainly something that is a shift in practice for folks. That level of comfort with what feels like silence—but for the learner is actually think time. That makes a ton of sense to me. Jean, I'm wondering if you could talk in a little bit more depth about the work that you did around vocabulary. And particularly, like, I taught kindergarten and first grade for quite a long time, so this actually feels really relevant to some of the things that I remember thinking about when I had children who may not have been familiar with language, let alone not having the language we were working in be their first language. Can you just talk a little bit about what that process was like for educators as you took them through it?

Jean: We would ask teachers to first think about what's the mathematical academic language that students need to know to access this problem? And so, if it was a problem on ratios, we'd think of, "What are the terms they might need to use to discuss this problem?" They might not be terms that are specifically listed in the problem, but it's the mathematical academic language that might come up. Then we look at the problem itself, and we wouldn't just focus on vocabulary. There might be phrases in there that are really unfamiliar. We were working with one problem that was about students running a ticket booth and what they were charging for different blocks of tickets. And just the phrase "running the ticket booth" was really different because running has multiple meanings. And students know what it means to run, um, you know, using their feet. But running the ticket booth was very different. And so, we supported that with some illustrations and put a sentence by it so that students could make that connection. Sometimes teachers will make some connection to native language supports as well. So, using Spanish or whatever the student's native language is as a bridge to accessing some of the new language and making sure they have that connection as well. And then finally, we'd think about what language the students are going to produce. So, what do they need to say to express their understanding and how can we support them in forming the language to express that understanding?

Mike: That's fascinating. What strikes me is how often the work that you're describing stops with the mathematical vocabulary and doesn't actually do that next piece, which feels really important. Like this idea: What is it about the vocabulary that we're using that we assume people understand, but that, like, "running the booth," that's (chuckles)—as you say it, and actually think and contemplate it—that's confusing.

Jean: Yeah, it's very confusing. And once teachers realize that that's what it takes to support language, you don't have to have an advanced degree in linguistics. It doesn't have to be deeply complicated. You're just really planning for what students might need to know to understand the mathematics in that task.

Mike: What are some of the moves that educators can make when they discover this language that we take for granted as everyone understanding? Would you be willing to talk a little bit about, what are the adaptations or the steps that folks take to help unpack that for children?

Jean: Yeah, absolutely. I think once you've identified different terms within just that day's lesson versus your academic language, you're going to want to have some consistent supports in your classroom. So, a lot of teachers will create a word wall. But a word wall isn't really effective unless students are using it. So, terms, definitions, and I'd also say having an illustrated word wall can be a game changer for some of the common vocabulary you're going to see within a unit—having that up so students can continually reference it and understand what it means. When we looked at the vocabulary and the phrases within the problem, we also connected it to visuals so we can explain what it means. We can provide students a written definition, but when you're still learning a language, the visuals are so essential to actually understanding what the term means or understanding it in context.

Mike: So, one of the things I'm curious about is, what are some of the understandings, the ahas, and the practices that you saw emerging as teachers engaged in this cycle of PL (professional learning)?

Shannon: I can respond to this one. We work with teachers to implement specific instructional strategies during their math classes, such as those mathematical language routines or the five practices. So, by using the variety of language supports incorporated in the program, we have definitely seen teachers develop a more nuanced understanding of what makes an appropriate scaffold and how to differentiate support for students based on their levels of English proficiency. It's not uncommon for teachers and the program to voice concerns that the tasks that we're using or how we're asking students to participate is too hard. And we know that this is coming from a good place. Teachers want their students to feel supported and be successful. So, we talk a lot about productive struggle and the role that it plays in students' meaning making and development in math class, and how critical it is that multilingual learners also get those opportunities to grapple with deep math concepts.

Mike: I think you're hinting at my next question, too, which is: Can you talk a little bit about the impacts that you observed on student identity and their learning as a result of this work?

Christa: Yeah, I'll take this one. This is my most favorite thing to talk about, cause I think this is where we saw the biggest impact, um, in the work that we were doing. And when we think about student identities, we almost had to take a step back and think about teacher identities. Especially when we think about mathematics and the role that that plays. We know that there's been a big emphasis on mindset and, and how important it is when we're learning mathematics to have this growth mindset and recognize that mistakes are OK and good, and that's how we learn. But we also know that math classes historically haven't been set up that way, right? We focus on a right or a wrong answer. So, there's not a lot of opportunity for kids in a traditional math class setting to experience the joy of making a mistake and working through it.

Christa: The hard thing about that is, we want teachers to create that type of math class for kids, but they may not have experienced that type of math class as a learner. So, in Good to Great, we give teachers the opportunity to reflect on who they are as math learners, who they were as math learners, and what their experiences were. And it's not surprising that many of our stories were the same, right? Like, we didn't see ourselves as math people, math is not our favorite subject, you know, on and on. And when we started to reflect on, "Well, how does that come through in our teaching?" Some things kind of bubble to the surface. Some teachers would look at that and say, "Math is hard for me, so I want to make it easier for my kids." They want to make this a more positive experience, trying to make it easier for them to, to solve the problem.

Christa: So inadvertently, they're kind of taking away that power, making that mistake, and learning through it. And so, teachers had the opportunity to pause and think about, "Who did I position as mathematically capable today?" Really what that means is, "Who did we give the opportunity to be seen as a mathematical thinker, who got to answer the questions, who got to share their thinking?" And when teachers were reflecting on that, some of them started to realize that, "No, I may not be giving my multilingual learners the same opportunities as my native English speakers." And once we had those discussions, we pulled in those tools that support that productive and receptive language, and we challenged teachers to call on their multilingual learners the next day. And let's see what happens. They did the supports in class, called on those kids, and what we noticed in those debriefs that came after that: The teachers were starting to share, "Once I gave them those tools, they ran with it." We heard things like, "My kids enjoy math class; they want to participate. They're raising their hands." All of this from providing the right supports, digging in deeper to some of these mindset issues that we may have ourselves as math learners. And then how do we shift that experience for students so that they can develop their mathematical identities in this?

Mike: The psychology of all of this is fascinating because you're making me think about the idea of intent versus impact, right? So, the intentions of an educator who might be making some of the choices that you're talking about are positive, right? Like they're genuinely in a spot where it's like, "I don't want to make a child feel embarrassed." On the other hand, the child doesn't know that. They just know that they're not getting called on, and they're making up their own story about why that's true. And that's also true for all the other kids in the class who are noticing that as well. And I think the thing that I'm coming around to is, it really does come back to the practices. You all gave them a set of tools to allow them to feel comfortable calling on those kids because they felt they could support them in the moment, and that produced a massive shift.

Christa: Yeah, absolutely. Once they had the tools, they were able to see what their kids had in them all along.

Mike: You know, one of the things that jumps out for me is, there are a lot of demands on teachers' time. But what you described, I can imagine this happening in a grade-level team. I can imagine it happening at a PLC, and really investing in the types of practices that you all just described feels like the payoff is pretty solid. So, I wanted to ask you all for educators or instructional leaders who are interested in learning more about the Good to Great professional learning that you all have built, designed and implemented, where can they go to actually learn more?

Jean: Sure. Thanks for asking that. So, we recently published a free toolkit that contains many of our MLL Good to Great resources, including the planning and reflection tools that we've been talking about today, as well as videos and exemplars. So, if someone just wants to learn a little bit more, they can go to the toolkit and see what some of the tools look like. The toolkit is called More Than Right Answers: Math Instruction for Multilingual Learners , and it's available on tntp.org. So, the toolkit also includes links to contact us at TNTP with any additional questions. And anyone interested in learning more could also email me directly. It's [email protected].

Mike: Thank you all so much for this conversation. I've learned a lot, and it was a pleasure talking to y'all.

Jean: Thank you so much for having us.

Shannon: Thanks, Mike. It was great to be here.

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.

© 2023 The Math Learning Center | www.mathlearningcenter.org

Today on the podcast, Pamela Seda and Kyndall Brown, authors of Choosing to See: A Framework for Equity in the Math Classroom, talk about what culturally relevant mathematics instruction looks like and identify practical steps educators can take to start this important work in their classrooms.

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

Choosing to See: A Framework for Equity in the Math Classroom

Mike Wallus: What does it mean to offer our students a culturally relevant experience in mathematics? This is a question on the minds of many, particularly elementary mathematics educators. Today we're talking with Pamela Seda and Kyndall Brown, authors of Choosing to See: A Framework for Equity in the Math Classroom . We'll talk with our guests about what culturally relevant mathematics instruction looks like and identify practical steps educators can take to start this important work in their classrooms.

Mike: So, hello, Pam and Kyndall. Welcome to the podcast. We're so glad to have you with us. I'm wondering if both of you would be willing to take a turn and just talk a little bit about what brought you to writing the book.

Pamela Seda: OK, well I'll start. This book really started with my dissertation research. And when I started my PhD program, I was very well aware of the achievement gap and the lack of opportunities for so many students, and I just wasn't satisfied that there was a gap. I had to find answers. And so, my PhD program was my quest to find answers. In the process of finding answers, I created this framework that came out of my study, and I had the opportunity to think about how to support teachers. Firstly, implement it in my own classroom and then figure out how to help teachers implement this. And it was just one of those things that I knew that there were a lot of people who wanted to do better for their kids, but they weren't quite sure how to do it. And so, therefore, this book was really kind of a nuts-and bolts place to start.

Mike: And, Kyndall, if you can pick up the story, how did the two of you start collaborating around the book?

Kyndall Brown: So, I met Pam at the National Council of Supervisors of Mathematics Conference in Boston in 2015. I was doing a keynote presentation focused on equity and mathematics, and Pam was in the audience. And at the end of the presentation, she approached me and suggested that we start doing presentations together. So ever since then, we were collaborating to do presentations at national conferences. I had been approached by a publisher about writing a book focused on equity in mathematics. So often when those of us who've been doing this equity work over the years, what we hear from math teachers in particular is, "What does it look like in the math classroom?" In language arts, you can read the literature that's reflective of your student population. And [in] a social studies class, you can study the cultures of the student populations in your classroom. But math teachers were always wondering, "What does equity look like in a math classroom?" And so, one of the first things Pam did when we met was, she introduced me to her ICUCARE framework. It just made perfect sense to use her framework. I asked if she would like to collaborate. She said yes, and this is what we did during the pandemic.

Mike: Well, I'm wondering if the two of you could just start and unpack the premise of the book and describe the framework that you all have proposed for people who may not have read it yet.

Pamela: Well, ICUCARE is the acronym. The first part is, "I include others as experts; C, be critically conscious; U, understand your students well; and then the second C is, use culturally relevant curricula." Kyndall, you want to take it from there? (laughs)

Kyndall: (laughs) Sure. The next principle is, "Assess, activate, and build on prior knowledge; then comes release control; and the final principle is, expect more."

Mike: You know, we could do a podcast episode for every component of the ICUCARE framework, but today we're really focused on using culturally relevant curricula. I suspect there are many educators listening who are kind of in the shoes that Kyndall was describing earlier, this idea that they're interested in the work, but they're not sure how to start, particularly in the math classroom. So, I'm wondering if you all could just spend a little bit of time talking about the guidance you would offer folks when it comes to culturally relevant curricula in a math classroom.

Kyndall: Well, first of all, in order to make a task or your curriculum culturally relevant, you have to know who it is that you're teaching, right? You can't make assumptions and assume that you know who they are based upon some physical characteristic or some other information that you might have with your students. The first thing you have to do is get to know who they are, what their interests are, what their concerns are, and then you can begin to start making the curriculum culturally relevant.

Mike: Hmm.

Pamela: I always say, if we're talking about a task, let's start with something that is cognitively demanding; something that is accessible but also cognitively demanding. And so, oftentimes we describe that as a low-floor, high-ceiling task. And it's real important that students have that opportunity to be able to have cognitively demanding tasks. I say that's a good place to start. We can use textbook problems, we can go to websites—things like Jo Boaler and Achieve the Core and Bridges—those kinds of things. And that's a good place to start. And so, then you might say, "OK, well how do I know that's culturally relevant?" Well, that's what we start with, the good task, and then we're going to take that and make it culturally relevant. And one way I say to take a baby step is, take that task and then just change the names. Put some names in there that are meaningful to your students.

Pamela: And I say, put your students' names in there. Rather than just trying to come up with some ethnic-sounding names, put your students' names so that they can see themselves in there. Put your school's names. Put the other teachers' names. The key is students need to be able to see, "I am a part of mathematics. Mathematics is a part of who I am, a part of who we are." And so, I think that's a very good baby step to take is just put meaningful names in there. I know that it was very effective. My students really enjoyed it. I could tell, like, even I purposely oftentimes would do that on tests to help reduce the anxiety level of taking a test. And my students, you would see them kind of smile and look around for the persons that they saw whose name was mentioned in the problem.

Pamela: So, that's a good first step. And then I would say, the next thing you could do after you've changed the names is then change the context. Change the contexts to things that are meaningful. But as Kyndall said, this is going to require you understanding something about your students. And some things that you can do to understand your students: You can interview your students. And one of the things we talk about in our book is empathy interviews that you can do. You can have listening conversations. Just have conversations with your students in the hall. What are they talking about in the hall? What are they talking about at lunch? What are they talking about at the bus stop? Just pay attention to those conversations, those social conversations, to figure out what's important to them. And then just do community walks. Find out what's in the community. What are popular places that kids hang out, that they go? What's meaningful to them and their families? And incorporate those contexts into problems. And then after that, if you've gotten used to changing the context, then I suggest what I call, go to a Stage Four Task. And then you try to engage their agency and help them understand that math can be a tool to use.

Mike: I would love for you to—either of you—to talk a little bit more about that last bit that you mentioned, Pam, when you talked about ways to build up kids' sense of agency. Would you be willing to indulge and just go a little bit further down into that conversation?

Pamela: Absolutely. So oftentimes, even if we have these wonderful contexts that students will solve problems and become engaged problem-solvers, there's always the question is, like, "So what now? What do I do with this? Why is this important to even get this answer?" And it has to be more than, "Well, it's going to be on the test," right? (laughs) And so, helping students understand and solve problems that help them see that they can be a part of solutions [to] things that are important to them. So, for example, I remember taking a problem. And it was something about increase in numbers. There was something about what percent did this increase? And I changed the context to the housing market because we had just actually had some storms that had come through our state and had created a lot of damage to houses and homes. And so, then the very next step was I started having them think about, "Well, how much might it cost to rebuild these homes? Were some houses damaged more than others?"

Pamela: And, "What could you possibly do to help?" Those are just some kinds of things to help kids understand that, "Oh, well, I'm not just trying to find percent increase or decrease, but there's some contexts here that matter, and it may cause me to do some more research." And even thinking about, "Well, if there are neighborhoods that were impacted, what are some things that I can do? Could there be some money that we raise? If I'm going to rebuild the house, how much might I need to spend? How much might I need to invest so that this maybe doesn't happen again?" Those are just all different types of questions to help students understand that you can use math as a part of your community. I also talk about an example of how I was teaching a unit on regression equations, and I know this is an elementary audience, but it was just an example of the fact that we give tests all the time.

Pamela: We give those state standardized tests, and I decided to use our district's data for the schools in our district, and things like that, to actually do the mathematics. And students care about that. They got to see their state scores, and they got to see the scores of their neighborhood, of friends who maybe go to a school down the street. And then not only did they get to do the math with that, then they got to have some input. I gave them that opportunity to basically talk to fellow students, talk to fellow teachers, talk to fellow administrators about, "What do you think should be different now that you've analyzed and looked at this data?"

Kyndall: And I would just add that Lisa Delpit, an education scholar, wrote this book in the early 2000s called Multiplication is for White People . And that's an extremely provocative title, but it was actually a quote from an African American student of ours. And it kind of spoke to that student's math identity. The actual quote was, "Multiplication is for white people; addition and subtraction is for Black people," right? And so that speaks to what that student's identity was about. The ability of certain people to do math based upon their racial or ethnic background. So, it is very easy to go through the U.S. educational system and come to the conclusion that mathematics is pretty much the domain of mostly white, European men, right?

Mike: Certainly.

Kyndall: When nothing could be further from the truth. There's an excellent book called The Crest of the Peacock: Non-European Roots of Mathematics that shows very clearly that mathematics is a cultural endeavor. It's a humanistic endeavor that all humans all over the planet have engaged in. And that other cultures have made significant contributions to the field of mathematics. And so, we need to do a lot better job of exposing students to that so that we can make sure that they see mathematics is as much a part of their culture as any other racial or ethnic group. And they need to see examples of people that look like them in the math textbooks, on the walls of their classrooms, as another way to help build that mathematics identity.

Mike: You know, and I think that is actually one of the things that I really appreciated about the way that you all structured the book—I know that I've heard other people who have read it say how much they appreciated being able to hear the stories from your own classrooms, the experiences that you had with students, and really being able to put those out there in a way that helps people see where there might be pitfalls and where there might be opportunities. I'm curious if either of you would be willing to share a story about culturally relevant curricula and the impact that you saw on a particular student.

Kyndall: Well, Pam has a couple of really good stories in that chapter, so I'm going to let her…

Pamela: (laughs) Yeah. So, one of the things I talk about is Jasmine. Jasmine was one of my students who, we'll just say we didn't see eye to eye on most things (laughs). Jasmine was very openly hostile towards me, and I was expending a lot of my energy just trying to get her to do anything. And she just made it very clear to me she wasn't interested in doing anything I asked her to do. And so I gave her that project that I talked about, where we decided to look at our test scores, our standardized test scores throughout the district, and applied the math content of the standard that we were using to this, to where she got to make an analysis and be able to see if there was a relationship between the percentage of Black students in our school and then our college and career readiness index, and those kinds of things.

Pamela: And I was just really amazed about the transformation that happened with her. Because previously, not only was she not willing to work with me, she didn't want to work with her classmates either (chuckles).

Mike: Mm.

Pamela: And she, as a result of working on this project, asked to be a part of a group. When she found out that she had made some mistakes on some of the data, she willingly stayed after school to fix her mistakes. And I even remember the day that the project was due. She stayed late to put her finishing touches on it. And so, I just was amazed. She was just…became pleasant. And as a result, I wanted to talk with her about the impact that this project had on her. And she said she really wanted to do it. It wasn't like it was just for a grade. She really wanted to learn the information. And the other thing that was kind of interesting is she didn't really see it as math. She didn't really think that what she was doing was really math, even though she was using Excel spreadsheets and she was using formulas. What that told me was how her perception was that school math wasn't what real math was, and that what we were doing that was connected to her community didn't feel like math. And I felt like that's something that we really need to change.

Mike: Yeah. Kendall, I saw you nodding on the other …

Kyndall: Well, I think the general public has come to believe that the only thing that counts as math is what you do in school, in a math classroom, right?

Mike: Uh-hm.

Kyndall: That all of these ways that people are engaging in mathematical thinking and reasoning all day, every day, they don't see as math. And so, they don't see themselves as math people, right? Because they were not successful at school math. Right?

Mike: Right.

Kyndall: And so how do we undo that perception and get people to recognize the myriad of ways that they're engaging in mathematical thinking and reasoning all the time?

Mike: Absolutely. Yeah. I was just going to ask you if there's anything in particular you think might be important for an elementary math educator to be thinking about when they're trying to apply the ideas, some of the suggestions that you all have when it comes to Choosing to See . Is there anything in particular that folks who are operating at the elementary level might consider or might think about that has come to y'all as you've brought the book out into the world and had people interact with it?

Pamela: Well, one thing that I've come to understand is that, while we do need to have good tasks—and the work that we ask students to do needs to be meaningful and needs to be accessible—tasks don't teach kids. And we need to think about, how do we structure how kids experience the mathematics in our classrooms? And that to me is what the framework does. It's a lens to help teachers think about, "How do I engage my students? How do I structure the instruction so that kids have a positive experience around the mathematics?" So, it should not be thought of as, "Oh, this is just once I get the math, then I'm going to go and think about this as an add-on."

Mike: Hmm.

Pamela: There are myriads of strategies out there. It's not saying that you should throw out everything that you've ever done before. It's just—look at the strategies and the things, the rituals and routines that you've been using in your classroom. And think about them in terms of this lens. If you're getting ready to do an activity, you might say, "OK, here's a routine that I normally have. How can I adapt it so I can include others as experts, so I'm not the only one that's doing all the talking? How can I engage my students so that I expect more out of them?" Right? So that they're doing more of the work? So, it's really a lens of how to think about the work that you do and the work that they do.

Mike: That totally makes sense.

Kyndall: Right. And the research shows that tracking begins very early in elementary school, right? And so elementary teachers need to be conscious of all of these different issues so that they can be on guard at the very early stages to not allow that tracking to begin.

Mike: For educators or instructional leaders who are new to the conversation, in addition to reading Choosing to See , are there other resources that you think would be helpful in supporting people in learning more about equity in the mathematics classroom?

Pamela: Well, yes, I know that I've just started reading recently—it's a new book this out called Engaging in Culturally Relevant Math Tasks: Fostering Hope in the Elementary Classroom . And it's by our good friends Lou Edward Matthews, Shelly M. Jones, and Yolanda Parker. It's at Corwin books, and I definitely recommend that—that is a great resource.

Kyndall: There's a new book that just came out. It's called Middle School Mathematics Lessons to Explore, Investigate, and Respond to Issues of Social Injustice , by Robert Berry and his colleagues. In 2020, they released a high school version of the book. And in the fall of 2022, they're planning on releasing an upper and lower elementary version of these books. And the first section of the book is really talking about the kind of pedagogy needed to implement social justice tasks. And then the second part of the book has lessons aligned to the different content strands that are social justice focused, a lot of digital resources. And so, I think that is an excellent resource for teachers.

Mike: That's fantastic. Pam and Kyndall. I want to thank you both so much for being here with us today, for sharing the book with us. It's really been a pleasure talking with both of you.

Kyndall: Thank you.

Pamela: Well, 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.

© 2023 The Math Learning Center | www.mathlearningcenter.org

Today on the podcast, we're talking with Dr. Kendra Lomax from the University of Washington about a body of research called Cognitively Guided Instruction and the promise it holds for elementary educators and students.

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

Cognitively Guided Instruction

Mike Wallus: Have you ever had an experience during your teaching career that fundamentally changed how you thought about your students and the role that you play as an educator? For me, that shift occurred during a sweltering week in July of 2007, when I attended a course on cognitively guided instruction. Cognitively guided instruction, or CGI, is a body of research that has had a massive impact on elementary mathematics over the past 20 years. Today on the podcast, we're talking with Dr. Kendra Lomax, from the University of Washington, about CGI and the promise it holds for elementary educators and students. Well, Kendra, welcome to the podcast. It's so great to have you on.

Kendra Lomax: Well, thanks for having me.

Mike: Absolutely. I'm wondering if we can start today with a little bit of background; part history lesson, part primer to help listeners understand what CGI is. So, can you just offer a brief summary of what CGI is and the questions that it's attempted to shed some light on?

Kendra: Sure, I'll give it my best try. So, CGI is short for cognitively guided instruction, and it's a body of research that began some 30 years ago with Tom Carpenter and Elizabeth Fennema. And there's lots of other scholars that since then have kind of built upon that body of research. They really tried to think about and understand how children develop mathematical ideas over time. So, they interviewed and studied and watched really carefully what young children did as they solve whole-number problems. So, you may have heard about the book Children's Mathematics , and that's where you can read a lot about cognitively guided instruction and [it] summarizes some of that research. And they really started with whole-number computation and then have kind of expanded into areas like fractions and decimals, learning about how kids develop ideas about algebraic thinking, as well as early ideas around counting and quantity.

Mike: Uh-hm.

Kendra: So, there's a couple of books that are kind of in the CGI family. Young Children's Mathematics includes those original authors, as well as Nick Johnson and Megan Franke, Angela Turrou, and Anita Wager. That fractions and decimals work was really led by Susan Empson and Linda Levi. And then, like I mentioned, Thinking Mathematically is the text by the original authors that kind of talks about algebra. So, in all of those texts that summarize this research, basically, we're trying to understand how do children develop ideas over time? And Tom and Liz really set an example for all of us to follow in how they thought about sharing this research. They had a deep respect for the wisdom of teachers and the work that they do with young children. So, you won't find any sort of prescription in the CGI research about how to teach, exactly, or a curriculum. Because their approach was to share with teachers the research that they had done when they interviewed and listened to all of these many children solving problems, and then learn from the teachers themselves. What is it that makes sense to do in response to what we now know about how children develop mathematical ideas?

Mike: I mean, it's kind of a foundational shift in some ways, right? It reframes how to even think about instruction, at least compared to the traditional paradigm, right?

Kendra: Yeah, it's less a study of how best to teach children and really a study and a curiosity about how children bring the ideas that they already have to their work in the math classroom, and how they build on those ideas over time.

Mike: Definitely. It's funny, because when I think about my first exposure I think that was the big aha, is that my job was to listen rather than to impose or tell or perfectly describe how to do something. And it's just such a sea change when you rethink the work of education.

Kendra: Definitely. And it feels really joyful, too, right? You get to be a student of your students and learn about their own thinking and be really responsive to them in the moment, which certainly provides lots of challenges for teachers. But also, I think, just a sense of genuine relationship with children and curiosity and a little bit of joy.

Mike: Definitely. So, I'm wondering if we could dig into a little bit of the whole-number work, because I think there's a bit that we were talking about with CGI, which is really the way in which you approach students, right? And the way that you listen to students for cues on what they're thinking is. But the research did reveal some ways to construct a framework for some of the things you see when children are thinking.

Kendra: So, if you read the book Children's Mathematics , you might notice or recognize some of those ideas, because CGI is one of the research bases for the Common Core state math standards. So, when you're looking through your grade-level standards and you see that they're suggesting particular problem types, number sizes, or strategies that children might use, much of that is based on the work of cognitively guided instruction, as well as other bodies of research. So, it might sound familiar when you read through the book yourself. And what CGI helps reveal is that there's a somewhat predictable sequence: That young children develop strategies for whole-number operation for working with whole number computational problems.

Mike: Yeah. Can you talk about that, Kendra?

Kendra: Yeah. So, young children are going to start out with what we call direct modeling, where they are going to directly model the context of the problem. So, if we give them a story problem, they'll act out or model or show or gesture, to show the action of the problem. So, if it describes eating something (makes eating sounds), you can imagine, right, the action that goes along with eating? And we're all very familiar with it. So, they're going to show maybe, the cookies, and then cross out the ones that get eaten…

Mike: Uh-hm.

Kendra: …right? So, they're really going to directly model the action or relationship described in the problem. And they're going to also represent all the quantities in the problem, which is different. What they learn over time is to count on or count back. So, some of the counting strategies where they learn, "Gosh, I don't want to make all the quantities in this problem." It becomes too difficult, too cumbersome. And they learn that they could count on from one of the quantities or count back. So, in that cookie example, maybe there are seven cookies on a plate, and I have two of them for dessert, right? (makes eating sounds) They go away. So, in direct modeling, they're going to show the seven cookies. They're going to remove those two cookies that get eaten, and then count how many are left. Where in counting on—so they have had lots of experiences of direct modeling—they can say, "Gosh, I don't really want to draw that seven. I'm going to imagine the seven…"

Mike: Uh-hm.

Kendra: " …And I can maybe count backwards from there."

Mike: So, like, 7, 6, 5.

Kendra: Yeah. Right. So, I don't have to make the seven. I can just imagine it. And I keep track of those two that I'm counting back.

Mike: That totally makes sense. And as a former kindergarten and first-grade teacher, it's an amazing thing to actually see that shift happen.

Kendra: Right? And it's really specialized knowledge that teachers develop to pay attention to that shift. It's easy for everybody else to kind of miss it. But for teachers, it's a really important shift to pay attention to.

Mike: I used to say to parents, when I would try to describe this, it's something that we almost aren't conscious of being able to do. But it's a gigantic step to go from imagining a quantity as a set of ones to imagining a quantity that is a number that you can count back from or count forward from. It's a gigantic leap. Even though to us, we've forgotten what big of a leap that was because it's been so long since we took it.

Kendra: Yeah. That's one thing I love about studying children's mathematics, is, like, you get to experience that wonderment all over again…

Mike: Uh-hm.

Kendra: …in the things that we kind of, as adults, take for granted in how we think about the world.

Mike: Yeah. I think you really clearly articulated the shift that kids make when they move from direct modeling, the action and the quantities, to that kind of shift in their thinking and also their efficiency of being able to count on or count back. Is there more to, kind of, the trajectory that kids are on from there?

Kendra: There is , yeah. So, after children have had lots of experiences to direct model, and then learn to become more efficient with that, and counting on or counting back, then they might start inventing. We call them invented algorithms, which is a fancy way to say that they think about the relationship between quantities and start putting them together and taking them apart in more efficient ways. So, they might use their understanding of groups of 10, right? So, in that example, with the cookies—seven cookies and eating two of them—I might know something about the relationship with fives…

Mike: Uh-hm.

Kendra: …Five and two make a seven. So, they start to develop some sense of how numbers go together, and how the operations really behave. So, in addition, I can kind of add them in any order that I want to, right? So, we see these called in the Common Core standards, Strategies Based on Place Value, Properties of Operation, and the relationship between addition, subtraction, or multiplication, division.

Mike: That's super helpful to actually connect that language in Common Core to what you might see, and how that translates into, kind of, what one might read about in some of the CGI research.

Kendra: Right. It'd be lovely if we all had the exact same (laughs) names, wouldn't it?

Mike: Definitely. One of the questions that I suspect people who might be new to this conversation are asking is, what are the conditions that I can put in place? Or what are the things that I might, as a teacher, be able to influence that would help kids move and make some of these shifts. Knowing that the answer isn't direct instruction. I could get a kid to mimic counting on, but if they're still really thinking about numbers in the sense of a direct modeler, they haven't really shifted, right? So, my wondering is, how would you describe some of the ways that teachers can help nudge children, or kind of set up situations that are there to help kids make the shift without telling, or…

Kendra: (chuckles)

Mike: …like, giving away the game?

Kendra: Totally. Yeah. That's one takeaway that I'm always on the lookout for when people hear about CGI and this trajectory that's somewhat predictable.

Mike: Uh-hm.

Kendra: Let's just teach them the next strategy then, right?

Mike: Right.

Kendra: And what's important to remember is that these are called invented algorithms for a reason. Mike : Uh-hm.

Kendra: Because children are actually inventing mathematics. It's amazing. Kindergartners are inventing mathematics. And so, our role is really to create the right opportunities for them to do that important work. And like you're saying, when they're ready for the next ideas that they're building on their existing knowledge, rather than us kind of coming in and trying to create that artificially.

Mike: Uh-hm.

Kendra: So, again, like, Liz and Tom really kind of taught us to be students of our students as well as students of teachers.

Mike: Uh-hm.

Kendra: So, what we've learned over time…some of the things that teachers have found really productive for supporting students to kind of move through this trajectory, to create increasingly efficient strategies, is really about thinking about carefully choosing the problems that we've put in front of students.

Mike: Uh-hm.

Kendra: So, paying attention to the context. Is it familiar to them? Is it reasonable for the real world? Are we helping kids see that mathematics is all around them.

Mike: Uh-hm.

Kendra: Paying attention to the quantities that we select. So, if we want them to start thinking about those relationships with five and 10, or as they get older with hundreds and thousands, that we're intentional about the quantities that we choose for those problems.

Mike: Right.

Kendra: And then, of course we know that students learn a lot from not just us, but their relationships and their discussions with their classmates. So, really orchestrating classroom discussions, thinking about choosing students to work together so that they can both learn from one another, and really just finding ways to help students connect their current thinking with the new ideas that we know are on the horizon for them.

Mike: I would love for you to say a little bit more about number choice. That is such a powerful strategy that I think is underutilized. So, I'm wondering if you could just talk about being strategic around the number choices that you offer to kids. Can you say more about that?

Kendra: Sure! It's going to depend on grade level, of course, right?

Mike: Uh-hm.

Kendra: Because they're going to be working with very different quantities early in elementary and then later on… One thing I would say, across all of the grade levels, is to not limit students whenever possible. So, sometimes we want to give problems that kids are really comfortable with, and we know they're going to be successful. But if I'm thinking of how they develop more efficient strategies, sometimes the growth comes in making it a little tricky. So, giving quantities that are just a little bit beyond where they're counting as young children, so they develop the need to learn that counting sequence. Or, as we're working with older students, if we know that particular multiplication facts are less familiar to students. Giving them that nudge by creating story context, where they can really make sense of the action of the relationship that's happening in it, but maybe choosing that times seven that we know has been tricky for kids, right?

Mike: Yep.

Kendra: So, I would just encourage people to not shy away from problems that we know pose some challenge to students. That's actually where a lot of the meat and the rigor happens. And, but then we also want to provide support inside of those, right? So, working with a partner.

Mike: Definitely.

Kendra: Or making sure they have access to those counting charts. That's one thing I would say across grade levels.

Mike: Yeah. So, you made me think of something else. It's fascinating to have this conversation, Kendra, 'cause it reminds me of all the things that I had to learn over time. And I think one of the things that I'm wondering if you could talk a little bit more about is, the types of problems and how the problem that you choose for a given group of students might influence whether they're direct modeling or they're counting on or whether they're using invented algorithms. Because I think, for me, one of the things that it took a while to make sense, is that the progression isn't necessarily linear, right? Like, if I'm counting on in a certain context, that doesn't mean I'm counting on in all contexts or direct modeling or what have you. So, I'm curious if you could talk a little bit about problem types and now how those influence what things students sometimes show us.

Kendra: Yeah. I'm glad you brought that up. When we describe, kind of, that trajectory of strategies, it sounds really nice and tidy and organized and like it is predictable in some ways. But like you're saying, it also depends on the kind of problem and the number size that we're putting in front of children. So that trajectory kind of iterates again and again throughout elementary school. So, as we pose more complex problem types…so, for example, the cookies problem where I have seven cookies, I eat two of them and the result is what's at the end of the story, right? The cookies left over.

Mike: Uh-hm.

Kendra: If I now make that problem, I have some cookies on a plate. I ate two of them, and I have five left over. All the kindergarten teachers, actually all the elementary school teachers…

Mike: (laughs) Yes.

Kendra: …can automatically recognize that's going to be a more tricky problem, right? Mike : Uh-hm.

Kendra: Where do I start!? Especially if I'm direct modeling, right? We know they start and follow the exact action of the story.

Mike: Absolutely (chuckles).

Kendra: So, as we pose more complex problem types, you're right. You're going to see that they might use less efficient strategies because they're really making sense. They're like, "Wait, what's the relationship that's happening in this story? Where do I begin? Where are the cookies at the beginning, middle, and end of this story?" So, we see that happen throughout elementary school. So, it's not that direct modeling is for kindergartners. And that invented algorithms are for fifth grade. It's that as new ideas get introduced, as we make problems more complex, maybe increasing the number size or now we're working with fractions and decimals…

Mike: Uh-hm.

Kendra: We see this happen all over again. Kids begin with direct modeling to make sense of the situation. Then they build on that and get a little bit more efficient with some counting kinds of strategies. And then over time with lots of practice with that new problem type, those new numbers, um, they develop those invented algorithms again.

Mike: So, this makes me think of something else, Kendra. How would you describe the role of representation in this process? That could mean manipulatives that students choose to use. It could mean things that they choose to draw, visual models. How does representation play in the process?

Kendra: Yeah. So, oftentimes I hear people say, "This student used cubes. That was their strategy." Or "This student used a drawing. That was their strategy." And that's really not enough information to know the mathematical work that that child is doing. Did they use cubes as a way to count on?

Mike: Uh-hm.

Kendra: Are they keeping track of only one of the quantities but using cubes to do so? Are they doing a drawing that actually represents groups of 10? And they're using ideas about place value inside of it, which is different than if they're just drawing by ones, right? So, there's lots of detail inside of those representations that's important to pay attention to.

Mike: Yeah. I'm thinking about one of my former kindergartners. I remember that I had some work that she had done in the fall, and then I had another bit of work that she had done in the spring. And the fall was this (chuckles) very detailed drawing of, like, a hundred circles. And then in the spring, she was unitizing, right? She had a bunch of circles and then within [them] had labeled that each of those were 10. And it just struck me, like, "Wow, that is a really tangible vision of how she was drawing in both cases." But her representation told a really different story about what she understood about math, about numbers, about the base 10 system.

Kendra: Right. And those might be very different starting points. As you, the teacher, you walk over and you see those two different kinds of drawings…

Mike: Uh-hm.

Kendra: …your conversation or your prompt for them…your next step for them might be pretty different.

Mike: Absolutely.

Kendra: Even though they're both drawing.

Mike: Yeah. Well, let me ask you this, 'cause I think I struggled with this a little bit when I first started really thinking about CGI. I had gone to a training and left incredibly inspired and was excited. And one of the things that I was trying to reconcile at that time is, like, I do have a curriculum resource that I'm using, and I wonder how many teachers sometimes struggle with that? I've learned these ideas about how children think, how to listen…what are some of the teacher moves I can make? And I'm also trying to integrate that with a tool that I'm using as a part of my school or my district. So, what are your thoughts about that?

Kendra: That makes a lot of sense. And I think that happens a lot of time in professional learning, where we learn a new set of ideas and then we're wrestling with how do they connect with the things I'm already doing? How do I use them in my own classroom? So, I really appreciate that challenge. I guess one way I like to think about it is that the trajectory that CGI helps us know about how children develop ideas over time is a little bit like a roadmap that I can use regardless of the curricular materials that I have in front of me. And it helps me understand what is on the horizon for that child. What's next for them and their learning? Depending on the kind of strategies that they're using and the kinds of problems that we're hoping to be giving them access to in that grade level, I can look at my curricular materials in front of me and use that roadmap to help me navigate it. So, we were talking about number selection. So, I might take that lens as I look at the curriculum in front of me and think about, "Are these the right numbers to be using? What will my students do with the problem…

Mike: Uh-hm.

Kendra: …that is suggested in my curricular materials?" To anticipate how my discussion is going to go and what kinds of strategies I might want to highlight in my discussion. So, I really like to think of it as the professional knowledge that teachers need in order to make sense of their curriculum materials and make informed decisions about how to use those really purposefully.

Mike: Yeah. The other thing that strikes me, that I'm connecting to what you said earlier, is that I could also look at the problem and think about, "Does the context actually connect with what I know about my children? Can I somehow shift the context in a way that makes it more accessible to them while still maintaining the structure, the problem, the mathematics, and such?"

Kendra: Right. Yeah. Are there small revisions I can make? Because, uh, I don't envy curriculum writers (chuckles) at all because there's no way you can write the exact right problem for every day, for every child across the country. So, as teachers, we have to make really smart decisions and make those really manageable. Because teachers are very busy people.

Mike: Sure.

Kendra: But those manageable, kind of, tweaks or revisions to make it really connected to our students' lives.

Mike: Yeah. I think the other thing that's hitting me is that, when you've started to make sense of the progression that children go through, it's a little bit like putting on a pair of glasses that allow you to see things slightly differently and understand that skill of noticing. That's universal. It doesn't necessarily come and go with a curriculum. It's something that's important. Knowing your students is always going to be something that's important for teachers, regardless of the curriculum materials they've got.

Kendra: Yep. That's right.

Mike: So, here's my, I think my last question. And it's really, it's a resource one. So, if I'm a listener who's interested in learning more about CGI, if this is really my first go at understanding the ideas, what would you recommend for someone who's just getting started thinking about this and maybe is walking away thinking, "Gosh, I'd like to learn more."

Kendra: Sure. Well, I mentioned the whole laundry list of great texts that you can dig into more. So, Children's Mathematics being the one on whole-number operation across grade levels. I find that, like, preschool through first- or second-grade teachers have found Young Children's Mathematics incredibly impactful. It helps connect ideas about counting in quantity with these ideas about problem-solving and operation. And then kind of connects them and helps us think about how to support students to develop those really important early ideas.

Mike: Uh-hm.

Kendra: Anybody who I have talked to that has read Extending Children's Mathematics: Fractions and Decimals has found it incredibly impactful.

Mike: I will add myself to that list, Kendra. It blew my mind.

Kendra: Yeah, us, too! Everybody who read it was like, "Ohhh, I see now." It points out a lot of really practical ways for us to pay attention. It offers a trajectory much like whole-number about how children develop ideas and also kind of suggests some problems that will help us support students as they're developing those ideas. So, [I] definitely recommend those. And then, Thinking Mathematically is another great text that helps us connect arithmetic and algebra, as we're thinking about how to make sure that students are set up for success as they start thinking more algebraically. And [it] digs into a little bit of—I talked about young children inventing mathematics—I think even further describes the ways that they invent important properties of operation that can be really interesting to read about.

Mike: That's fantastic. Kendra, thank you so much for joining us. It's really been a pleasure talking to you today.

Kendra: Thanks for having me.

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

Close your eyes and picture your childhood self learning math in elementary school. What memories and feelings come to mind? When you reflect on those memories, what unspoken messages did you absorb about what it meant to be good at math? And, how did those early experiences with mathematics shape your belief about yourself as a doer of math? Today on the podcast, MLC curriculum consultants Annelly Rodas and Nataki McClain talk about math identity and how educators can shape students' math identities.

Mike Wallus: Today I'd like to start our episode with a bit of a thought exercise. I'd like you to close your eyes and picture your childhood self, learning math in your elementary school. What are some of the memories and feelings that come to mind? And when you reflect on those memories, what do you think the unspoken messages you may have absorbed about what it means to be good at math were? And then, maybe most importantly, how did those early experiences with mathematics shape your belief about yourself as a doer of math? Today on the podcast, we're talking about identity; specifically, math identity. What is it? And how can we as teachers shape our students' math identities? Let's get started.

Mike: Well, hey, everyone. Welcome to Rounding Up. I'm excited to have our friends Nataki and Annelly joining us today. And I think I'll just start by welcoming the two of you. It's great to have you on the podcast.

Nataki McClain: Hi, Mike. Thank you for having us.

Annelly Rodas: Thank you, Mike.

Mike: Absolutely. So the two of you are currently curriculum consultants for the Math Learning Center. And I'm wondering before we get started with the topic of the day, can you tell us just a little bit about your teaching background and your experience in education? And, Nataki, I'm wondering if you'd be willing to go first?

Nataki: Sure. Well, I have been in education in some capacity for about 25 years. I spent 16 years in the classroom. Fourth grade was my favorite year of all time. And then I spent eight years as a math specialist. This past year, I am now a curriculum consultant for the Math Learning Center.

Mike: Annelly, how about you?

Annelly: So I started my career as a pre-K teacher at a head start program, and then I moved to the New York City public school system, where I taught second grade and fourth grade. Later, I had the opportunity to work as a math coach at my own school. And I supported pre-K to eight.

Mike: Fabulous. Thanks to both of you. So let's jump into the topic of the podcast: Cultivating a Positive Math Identity. Getting ready for this, what I found myself thinking about is that there is so much conversation in the field right now around math identity. And CTM has position statements about the importance of supporting a positive math identity. There's a ton of research that validates that need. I think I'd like to start by just asking you, from your perspective, how would you describe math identity to a listener who's new to this conversation?

Annelly: I think that it is important to understand that math identity is our own personal view on how we engage with mathematics, right? And it has to do with our disposition and our beliefs on our mathematics ability. I know for me, this topic is really close to my own personal journey in mathematics because I grew up thinking that I was not a math person and that changed with my experiences really late in life. So it has become my mission that kids get to experience math in a different way, and that they feel comfortable engaging with mathematics.

Nataki: And Nelly, um, I have to agree with you. I share a similar experience in that, I guess in my elementary school days, I didn't think of math as something that you got to either enjoy or not. It was just kind of, it's just there, and you do it, and you learn it. But then in high school, I did not have a positive experience. I was made to feel like math was not my thing. And so, Mike, to address that question about what is math identity, it really—to Nelly's point—it really is how you view yourself as a mathematician. And again, my experience in high school was such that I did not feel like I was a mathematician. So to everyone's surprise , when I go off to grad school I'm studying math, and now I'm working at the Math Learning Center, right? It's kind of a big deal. And I think it's important that everyone feel like a mathematician.

Mike: Yeah, gosh, you know what you two are saying, I suspect that it resonates with so many people who, whether they're teachers or parents or folks who are just kind of going about living their lives, think this resonates so much. I really resonate with what you said, Nataki, about this idea that math was just there.

Nataki: Uh-hm.

Mike: It was about a series of procedures that you do quickly and that you try to always find the answer as soon as possible. And get it correct the first time. And if you didn't, that meant something about who you were, and what your ultimate capacity as a mathematician was.

Nataki: Uh-hm.

Mike: And I think for a lot of folks, that really shapes their belief about what school math is and what math is in general.

Nataki: Absolutely.

Mike: Yeah. So I'm really curious when you think about the resources that helped you all build your understanding of math identity. What are some of the kind of seminal pieces of work that helped you begin to think about this idea?

Nataki: Well, Annelly and I are reading this book. It's called Choosing to See . It's written by Pamela Seda and Kyndall Brown. And I have found that this is a relevant resource, especially to our work at the Math Learning Center, because it focuses on equity specifically in the math classroom. And as you're reading it, hopefully, you'll find, like we have, that the authors do a really good job in describing those instructional strategies that help teachers to build positive math identities for students. Right away in the introduction, Kyndall Brown outlines a framework for the principles that guide equity, agency, and also identity in the classroom. And he uses an acronym. I see you care. So it's I, the letter, C-U-C-A-R-E. And that stands for Including others as experts; being Critically conscious; Understanding your students; using Culturally relevant curricula; (Assess), activate, and also to build (on) prior knowledge; Releasing control, and Expecting more. And the idea here is to be intentional about what you see, to also be compassionate and purposeful enough to respond. And when we allow this mindset to be prevalent in our classroom, it really does help to support a positive student math identity. But it also serves as a guide to help the teacher understand what, particularly, is at stake.

Annelly: And I love that resource. The two of us are reading that book and always have conversations about it. But I also think that a starting point for a teacher should be examining their own journey with mathematics, right? Like, I talked about how I didn't feel as a mathematician. And I taught, at the beginning of my career, I taught the way that I was taught: very procedural. Expecting quick answers. And the more I started putting my students at the center of my teaching, I started realizing that I was not meeting the needs of all my students. So I would say another research—and I'm going to do a plug-in here for our blog—"A Summer Dive into Teacher Math Identity." That might be something, like a starting point, right? We have to examine our own thinking and our own role before we can create those opportunities for students to develop a positive math identity.

Nataki: I like that, Annelly, that's a good one.

Mike: Hmm. Yeah. I think one thing that jumps out for me is, it would be hard for me to imagine that there's a lot of people who disagree with the aspiration of helping children build an identity about mathematics. That's positive. But I think what's hitting me is you all are kind of highlighting that there are actual practices and things that one does that actually help build that. And, Annelly, I think I'm really struck by the statement that you made, where you said, "I realized that I needed to put kids at the center of my instruction." And I'm wondering if you can just talk a little bit about, for you, in your journey as a math educator, what did it look like to do that in your classroom?

Annelly: What happened to me was that I started exploring my own math identity at the same time as I was teaching. And one of the things that I noticed is that for me, I need processing time and I needed visuals. So I started playing with that in the classroom to see what my students needed, right? I started bringing in visuals, and we started thinking about—I started thinking about—like, processing time for my kids, giving them time to think, slowing down their thinking. And that made a huge difference for my kids. And it provided a lens where I was pushed to, to think about and really pay attention to, what are the other things that they need? How can I open up space for them to share their thinking? And also, where are the opportunities for them to develop that agency as well? Where they can feel like, "I can tackle this," even though it's hard.

Mike: Hmm. Nataki, I, I was going to also offer, like, from your perspective, what did this journey look like for supporting students?

Nataki: Well, kind of similar to Annelly, you know. When I, when I am reflective of my own experiences as a math student, but also reflective in my practices as a teacher, one of the things that I noticed that was missing is the element of fun, right? And also how that fun factor makes room for accessibility. When students start having fun, then the math is accessible to them. And so one of the things that I can say that absolutely was consistent in my classroom, is that we were having fun. Now, of course, fun looks different for different people. And for me, it wasn't just, "We're being goofy and being silly." But fun meant that we are enjoying thinking about the math, doing the math, talking to our friends about the math, looking at math in different ways. In fact, I remember many days when we were at recess and students would come up to me with something that they'd noticed on the playground, right? Being that, "Oh, you know, Ms. McClain, that this merry-go-round is a circle. And it's going around and around and around and around. And it spins in the same, in the same distance from the center all the time." That's something that I didn't teach them. It was something that they noticed because they were having fun on the playground. And they were able to bring in the math concepts from the classroom into their own fun spaces.

Mike: You know, one of the things that I find myself thinking about is a really old piece of research. And gosh, I forget the actual researcher. But this idea that teaching is a cultural experience, right? That there are certain cultural narratives around mathematics education that exist just under the surface for lots of people. They're the scripts that they learned when they were in childhood. And that's the picture that shows up in people's heads when they think about math education. So part of the work really is offering kind of a counternarrative to that cultural script. Where I'm going with this is, my cultural script is: Teacher stands in front, shows me what to do, we practice it, and then I go and I sit and do 15 problems, and then two story problems at the end. And that's kind of the cultural script.

Nataki: Right.

Mike: And I suspect that it's fairly difficult to make that kind of cultural script fun. So it makes me wonder, "What did your classroom look like to make things fun?"

Nataki: Well, one of the things that was really important to me is that students could see themselves in the math that we are doing. So there wasn't a division problem that wasn't accessible to all students in the beginning, right? So we had to make it accessible. And then I would always find ways to turn everything into a game. To provide, again, that level of fun for kids. So whether it's that I've watched a game show like Jeopardy … well, "How could I use this game show to create a math lesson or a math event or an experience for students?" And so sometimes I could do that in the planning stages. OK, thinking about the content that I wanted students to learn, and then, "How can I make it fun? How can I make it engaging?" And then sometimes it just happened in the moment. You know, if you read the room and you discover that, mmm … they're not really having a lot of fun. And again, fun looks different for different people. And for me, I knew that it was fun when all students were engaged and all students had access to the learning.

Mike: So you all are really making me think about the fact that part of building identity is task structure, right? The way that you design tasks, the context that you provide that helps kids connect to it, and also really knowing your kids and knowing the fact that if I'm in second grade, you know, having the agency to actually use some of the materials and have choice around that, that's part of being fun, right? I have a question for you. When you all think about the fact that you also supported a Bridges implementation, what's your lived experience with the places where you see opportunities for building math identity within the structure of the Bridges curriculum. Um, how did that play out for you? How did that connect to the story that you're telling about your own journey?

Nataki: Kids would come barging in the room expecting Number Corner to happen. They were just so excited to discover the next pattern. Or, what are we collecting this month, right? And then, I mean, talk about fun. Work Places was just a natural place for that fun to happen. So I would say Number Corner and Work Places were the places in which I saw kids just really engage. And it was also a great time for teachers to help build that math identity in students, right? To offer support or just to be there next to students, watching them as they're playing the Work Place games. Those were two components where I saw the most where students really were engaged and having a lot of fun. And not only students. Cause I have to admit that I might have been on a couple of floors, and I might have been caught playing a couple of games, and laughing and chuckling myself (chuckles).

Mike: (Chuckles) Annelly, how about for you? Because I know that you actually, you were not only a Bridges teacher for quite a while, but you also supported the implementation in your building.

Annelly: I think that something that we saw when we implemented Bridges was the opportunity to allow kids to show their thinking. And I think that was so big, right? Like in thinking about, "There are so many subtle ways." Like when we ask kids, "Can you show me eight on your number rack," right? We're not dictating how they should think about it. They're jumping in and creating their own strategies and their own learning. And I think that that's an important way to develop that math identity. Because we are telling kids, "You can do it. You have all of the skills to do this." So I see it in that. I see it also in, when we ask kids to write their own math problems—this is something that I've been thinking about a lot—like, when we give kids the opportunity to become authors in the math classroom, we want to hear their ideas and their strategies.

Nataki: Uh-hm.

Mike: How does the role of the teacher shift in a classroom that's really supporting a positive mathematics identity? Part of what's on my mind is that idea of a cultural script, where the teacher is the knower and the place where all of the knowledge lives. And then it's really just kind of beamed out to the kids. What's the shift? If I'm trying to just reconceptualize what teaching looks like in a classroom where I am actively building a positive math identity for my students, how would you describe that?

Annelly: Like, I think that for that, I'm going to connect to my years when I was a coach. I used to love going into classrooms where I wouldn't know where the teacher was.

Nataki: Right.

Annelly: And it's even physical, right? The teacher is not in the front of the room. The teacher might be, like Nataki said, on the floor, playing with the kids. Or at a table, meeting with them. And I think that's a sign that shows you how the teacher is moving away from a teacher-center into a more of a student center. Also, when we can see kids thinking. Where we can see strategies being named after kids. Again, it seems as something so simple, but it's so powerful for them. It gives them validation that what you are thinking is important. I value your strategies. I used to say, "Even if they take you down to a rabbit hole value, they're thinking…"

Nataki: (Laughs)

Annelly: (Laughs)

Mike: That is really powerful. And, Nataki, how would you answer that question?

Nataki: Everything that Annelly said, I 100 percent agree with. I also think where there are opportunities to ask questions of students, to take those opportunities. Particularly when you have a student who doesn't always get to shine in the class, you know, when that student does something that you think the entire class should hear, find time and find moments to highlight that again. That's giving the student a different feeling about math and a different feeling about where that student finds himself or herself in that math classroom. It makes them feel like they are a mathematician. So I think asking questions and finding moments to allow all students to shine.

Mike: You know, I'm trying to put myself back into the world of a classroom teacher. I wonder if for a lot of folks, part of the hesitation is this fear of, what happens if kids say something that quote unquote is wrong or incorrect? And especially if that happens publicly in front of other children. I think there's this hesitation on the part of people. Because, again, the cultural script is, "I'll correct that and show you and tell you exactly what to do." And I wonder, when you've been faced with that spot where you have used questioning, you've been building discourse, and something just comes out of left field… When you think about a classroom again, where you're supporting identity, what does it look like in that moment for a teacher who's working to support identity, and they have some information that kids are putting out that they're concerned? Like, what do I do?

Nataki: Right.

Mike: Yeah, tell me about your thinking on that.

Nataki: Before we start to build discourse, we need to take some time at the very beginning to build a classroom community where everyone in the room feels free to share their thinking. No matter if it's quote correct or incorrect. And I always find opportunities to kind of press more when those incorrect answers come out, because we can learn a lot from those incorrect answers. We don't just learn from the things that are right. We learn from the things that are incorrect. So can you tell me more about that? Or maybe we could write the ideas on sticky notes and revisit them, right? If there are conjectures, which we talk a lot about in our classroom. Conjectures are always meant to be proven right or wrong, not just in that moment, but for as long as we are in the classroom. We're going to be thinking about the conjecture that Sally made. And the students love—and it's fun for them—when they can prove or disprove Sally's conjecture. That's fun for them. But because we've built the community, it's safe to do that.

Annelly: I love that, Nataki. I think that also creating a culture where it's OK to make a mistake and also modeling from teachers, right? Modeling that, "Oh, I made a mistake." But what I love about math is that I just think, "Cross it out and, and kind of like, think about it again." The one tip that I will give teachers that are just starting with math discourse, and they're afraid to get into gray areas: Do a turn-and-talk and listen to your kids before you ask them to share. And then you can kind of like select which kids are going to share, and you know where they're going. The other thing is that you have to do the math before you do the lesson, right? So that you know where they can go. One of the things that we used to do is, uh, we used to sit down and think about all the different ways that kids can answer a question, like a problem string. What are all the different ways kids can tackle problem strings? And then that gives you kind of like the foundation, right? Granted, you might have some kids that want to be really creative, and they might break it apart into ways that you were not even thinking about. But I think those two are, like maybe two tips, that open up the space for kids to share their ideas.

Nataki: And, Annelly, I think that's an important thing to mention because that anticipation of student responses that comes in the planning. And so it's important for teachers to remember that planning is part of your teaching. That we just don't show up and just start teaching, right? That there has to be some thought that we're giving to the anticipated responses.

Mike: Yeah. I mean, I think when you say that you, gosh, I'm so glad that we talked about this question. I mean, a few things jump out: 1) the idea of positioning student thinking as not being immediately judged right or wrong by the teacher, but as an opportunity to actually build an understanding, to actually have kids justify, to have kids turn to one another and talk about, "What is your understanding of this?" And then to build the conversation. So again, it goes back to agency, right?

Nataki: Uh-hm.

Mike: You are not the source of right or wrong. You're actually asking them to engage in thinking about that. But I think, Annelly, I'm really keying on what you said earlier about the idea that you have to anticipate where kids might go, because it actually means something. Regardless of whether they've arrived at the correct answer or whether they've arrived at something that shows partial understanding, they're telling you something, and you can use that place to help build an understanding for the whole group. Cause if one kiddo says it, it strikes me that there's probably a fairly good amount of other kiddos who might be thinking the exact same thing.

Annelly: And I think that's another way to build that math identity when we tell them, "It's OK if you just have the beginning of an idea…"

Nataki: Uh-hm.

Annelly: … Right? "Can you share with us? And we can build on that." Because what Nataki was saying before: We have the power to position kids in a positive light with the rest of the class…

Nataki: Uh-hm.

Annelly: And that it's also so important.

Mike: I just want to thank the two of you for joining us and sharing your thinking. One last question, I think before we have to close things out. You know, if I'm a listener, we've covered a lot of territory in the last bit. If I'm thinking about taking some steps in my classroom, where do you see opportunities for people to get started? Particularly if they're using the Bridges curriculum.

Nataki: I'd say one of the first places—not only a teacher, but any person in, in a school building could start—is taking a look at the blogs that are posted about math identity. One of the blogs, I think Annelly mentioned earlier is, helping teachers to be reflective of their own math journey. And I think that's an important step. So reflection, I would say, is a great place to start. And it starts perhaps by reading the blog.

Annelly: I would say don't be afraid to have conversations with your kids. And letting them lead some of those discussions.

Mike: Hey, thanks so much to both of you for joining us today. It was really a pleasure to hear your thinking and to have you on the podcast.

Annelly: Thank you, Mike, for having us.

Nataki: Yes. Thank you, Mike. This was a lot of fun. But listen, next time … can you bring cookies?

Mike: Hey, you got a deal, my friend. Thanks so much.

Nataki: Thank you. Bye now.

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.

Whether it's creating "I can statements" or developing success criteria, there's no denying that writing learning targets is a part of teacher practice. Today, Dr. Rachel Harrington from Western Oregon University talks about creating powerful and productive learning goals that impact student learning.

If you're interested in more on this topic, consider the following article for further reading: Mathematics Learning Goals Serve as a Guide

Mike Wallus: As a 17-year-veteran classroom teacher, I can't even begin to count the number of learning targets that I've written over the years. Whether it's writing 'I can' statements or developing success criteria, there's no denying that writing learning targets is an important part of teacher practice. That said, the thinking about what makes a strong learning target continues to evolve and the language that we select for those targets has implications for instructional practice. Today on the podcast, we're talking with Dr. Rachel Harrington from Western Oregon University about creating powerful and productive learning targets. Welcome to the podcast.

Rachel Harrington: Thank you for having me. I'm excited to be here.

Mike: Sure. So I'd love to just start our conversation by having you talk a little bit about how the ideas around learning targets have evolved, even just in the course of your own teaching career.

Rachel: I started out as a pre-service teacher in the late '90s and got a lot of practice in undergrad teacher education, thinking about writing those objectives. And we were always told to start with, 'The student will be able to … , ' and then we needed to have some skill and then it needed to end with a percentage of performance. So we need percent of accuracy. And so I got a lot of practice writing things that way, and we always were very strategic with our percentages. We might say 80 percent because we planned to give them five questions at the end and we wanted four out of five to be correct. And then we could check the box that the students had done what we wanted. And I felt like it was really critical. We always were kind of drilled into us that it must be measurable. You have to be able to measure that objective. And so that percentage was really important.

Rachel: In my experience though, as a teacher, that, that didn't feel as helpful. And it wasn't something that I did as a classroom teacher very often. As I transitioned into working in teacher preparation, now we have shifted the way we talk about things. Instead of saying a learning objective, we talk more about learning targets. And we talk about using active verbs that, when we phrase the learning target or the learning goal, it's using a verb that is more active and not so much 'Student will be able to … .' And so we might use verbs like compare, explain, classify, analyze, thinking more about that. And then, rather than thinking about an assessment at the end, with five questions where they get four correct, we want to think about multiple times throughout the lesson where the teacher is assessing that learning goal and the progress towards that goal. Sometimes those assessments might be more classroom-based. Other times you might be looking more at an individual student and collecting data on their progress as well. But it's more progress towards a goal rather than something that's met at the end of the lesson with a certain percentage of accuracy.

Mike: You named the thing that I think stood out for me, which is you're moving from a process where you're thinking about an outcome versus what's the action, be that cognitive or in the way that students are solving. The focus is really on what's happening and how it's happening as opposed to just an outcome.

Rachel: Uh-hm. And I feel like when I started in teacher preparation, the standards were a little more siloed by grade level. It was sort of like, this is what we do in fourth grade and it starts and ends in fourth grade. Whereas with the Common Core State Standards, we see these learning progressions that stretch across the child's whole math experience. And so I think that's shifted a little bit the way we think about targets as well and learning goals and whatever title you've given them. Now, we don't think so much as, 'What are you accomplishing at the end of today?' but sort of your progress across a learning progression and, and what progress are you making towards a longer-term goal?

Mike: I think that's a really profound shift though. There are two things that come to mind: One is really thinking about how that impacts my practice as a teacher. If I'm just thinking about what happens at the end of today, in all of these little discreet iterations, versus what's the pathway that the child is on, right? I'm really interested in, how is their thinking shifting? And that the end of the day is not the end of that shift. It's really something that happens over time. Does that make sense to you?

Rachel: Definitely. And I think it's really critical when we're teaching in a mixed-ability classroom, and we're thinking about children making progress at their own pace and not expecting every child to learn the same thing every single day, but we can have individual goals for our kids. We can have ideas about, as long as they are making progress in their math journey, then we're going to be OK with that. And we're helping them in that progress. And I think it's also more evidence as to why curriculum needs to cycle back to previously taught concepts because those concepts may or may not be mastered by all the children or understood by all the children at the end of the lesson. We're going to keep revisiting it. And children get multiple opportunities to think about this idea, and they will make progress on their own at their pace.

Mike: Well, that's in stark contrast to my own childhood math experiences. You got through your unit on fractions in fourth grade, and …

Rachel: Yep.

Mike: … if you didn't get it, well …

Rachel: So sad.

Mike: .. good, good luck in fifth grade!

Rachel: (laughs)

Mike: (laughs) Um, but it's really an entirely different way of thinking about the child's development of ideas.

Rachel: Yep. I remember teaching multiplication of fractions on a Monday followed by a division of fractions on a Tuesday. It was really just like, you know, when we moved past this idea that multiplication of fractions is a procedure that, that students will master. Then we need to start thinking about it as happening more than just on Monday.

Mike: We've already started to address the second question I had, which is: What are some of the pitfalls that schools and teachers might fall into or might encounter when they're thinking about learning targets?

Rachel: I think some folks have put pressure on teachers to take the idea of a learning target and phrase it into an 'I can' statement or a student-friendly language—which, I am not at all opposed to the idea of making things into student-friendly language. I think that's actually really critical in math class.

Mike: Uh-hm.

Rachel: But I think it can be problematic. When we start the lesson with an 'I can' statement, are we giving away the ending of the lesson right at the beginning?

Mike: Yeah.

Rachel: Are we taking away their joy of that discovery and that excitement of finding out this, understanding this new concept? I don't want to remove that magic out of math class by just saying, 'Hey, I'm going to tell you the ending right before we get started.' And I also worry a little bit that sometimes those 'I can' statements and those things that we put up on the board at the beginning of class are done under the guise of 'holding teachers accountable,' which I think is a phrase that is very (chuckles) problematic.

Rachel: I tend to err on the side of trusting teachers; that they can be trusted to know what they're doing in the classroom and that they have a goal in mind. And I assume that they are planning for teaching without telling me exactly and explicitly on the whiteboard that they are doing that. But I also recognize that the presence of that learning target or that 'I can' statement on the board at the beginning is an easy thing to check off. All of the different things that are happening in math class are really complex and really hard to understand and notice. And it can take years and tons of experience before we're able to notice all the things that are happening. And so as an administrator that maybe has limited experience teaching mathematics, I could see where it would be difficult coming into the classroom and really being able to recognize what is happening. You might look around the room and be like, 'Is this some kind of birthday party? What's going on? All these kids are cutting things out and gluing things. This doesn't look like math class.'

Rachel: But if I can see that statement written up on the board, that's something that's kind of concrete and measurable. I also just think this idea of capturing learning as a daily objective can be problematic, especially when we're thinking about building really complex ideas in mathematics. You know, that's not going to happen in one lesson, in one session of curriculum. It might build over multiple days. It might cycle back into multiple units. And so we need to make sure that students are developing alongside their peers and, but maybe not out at the same pace. And I think that's OK.

Mike: Yeah. You made me think about a couple different things, Rachel. One is the idea that the way that learning targets have been kind of introduced into classrooms really feels more like compliance as opposed to something that has value in terms of your instructional practice. And I, I've lived that world, too, as a classroom teacher. I think the other thing that really hits me from what you said is, I started thinking about whole-number multiplication, right? If I'm just thinking about the end product—meaning students being able to perform multiplication—there's so much richness that has been missed (chuckles) in that process.

Rachel: (chuckles)

Mike: I mean, we're trying to help children move from thinking additively to thinking multiplicatively. You're going to move along that kind of continuum of understanding over time. Honestly, I would say it shouldn't happen in one day.

Rachel: Yeah. What can you really learn in just one lesson? And learn, not, I wouldn't say just perform a skill.

Mike: Yeah.

Rachel: I think skills, performing a skill and memorizing an algorithm, that is something that can be taught in a really concrete chunk of time, potentially. But the real conceptual understanding of what's happening with multiplication—how it's connected to addition, how it's connected to geometric concepts and things like that—that all comes and builds. And I feel like it also builds in fits and spurts. Some kids are going to make a big leap at one point and then make some smaller steps before they make another big leap. It's not a linear progression that …

Mike: Right.

Rachel: … they're going through. And so we have to allow that to happen and give room for that to happen. And if we say everyone in the class will do this by the end of the lesson with this amount of accuracy, we don't make room for that to happen.

Mike: Yeah. I think what you're highlighting is the difference between what I would call like a learning goal and a performance goal. And I'm wondering if you could help unpack that. Because for me, when I started thinking about learning targets in that framework, it really opened my eyes to some of the places where I'd gotten it right in the classroom and some of the places where, boy, I wish I had a do over.

Rachel: Yeah. I think the language that the National Council of Teachers in Mathematics has brought to us, is this idea of contrasting performance goals with learning goals. And I find myself turning to the 'Taking Action' series of books. Specifically, K–5 when we're thinking about elementary. There's a chapter of that book I have found to be really powerful. Sadly, I think it's one that we can sometimes gloss over a little bit in our reading. Because for some folks, they look at that and they say, 'Well, I don't choose the learning goal. My curriculum chooses the learning goal or my school district tells me what the learning goal is.' But when you really look at what a learning goal is, as opposed to a performance goal, that's really not what's dictated by your curriculum or by your school district. And so in the 'Taking Action' book, I think they do a really nice job of contrasting the difference between a learning goal and a performance goal. And I would say a performance goal is sort of what I described earlier when I was talking about 'The student will be able to … '

Mike: Uh-hm. Yeah.

Rachel: … at a certain amount of accuracy. So, an example. If you do have access to the book, it talks about 'Students will solve a variety of multiplication word problems and write the related multiplication equations.' And (given) that, I could see that as the type of thing I would've written maybe with a certain amount of accuracy (laughs) at the end of it. And I would've given them maybe five word problems and then assessed if they could get at least four out of the five correct equations. And so that's a really good example of a performance goal. And, and they talk about this idea of a performance is, what is the student doing? What's something that we can look and observe and measure and count.

Mike: That's so hard though! Because what's missing in that goal is 'how'!

Rachel: Right.

Mike: You know (laughs), like …

Rachel: Or 'why'! (laughs)

Mike: (laughs) Or 'why'! Right?

Rachel: Yep, yep.

Mike: Like when you actually look at the student's work, what does that tell you about how they arrived there? And then what does that tell you about what that child needs to continue making sense of mathematics? You gave an example of a performance goal around multiplication and word problems. What might that sound like as a learning goal instead?

Rachel: So an example of that same—probably aligned to the exact same standard and the Common Core State Standards—would be that students will understand the structure of multiplication as comprising equal groups, within visual or physical representations, understand numbers and multiplication equations, and connect those representations to equations. So that learning goal really describes what you're hoping the students learn. Not just what they do, but what do they carry forward with them as they move into more and more complex mathematics? I think you'll also recognize the verbs in there are much more complex. In the previous performance goal, we talked about students solving and writing. They're solving, and they're writing. But in the learning goal, we're looking at understanding, connecting, and representing those different ways of thinking about it and bringing them together. Putting those pieces together. And again, that might be something that develops over a long period of time. They might be working on one piece of it, which is looking at an array and connecting that to an equation. But maybe later on, they're connecting the context of the task to the equation. Or they're taking a context and recognizing, 'Wouldn't an array model be a great way to solve this? And wouldn't an equation model be a great way to solve this?'

Mike: Uh-hm.

Rachel: And that's really developing over time.

Mike: Yeah. I was just going to say, you mentioned 'Taking Action.' The, the chapter on learning goals is actually my most dogeared, uh, chapter in the book. I want to read you something that I think is really powerful though. Very first chapter on learning goals, the way that they describe it is: 'Identifying what students will come to understand about mathematics rather than focusing on what students will do.' I've read that, underlined it, highlighted it. And I've got a Post-It note on that page because I think it just fundamentally changes what I think my role is as a teacher in preparing and also in a moment with children.

Rachel: Yep. It's not so much about, they're going to be able to cut this out and do this thing and perform this action. But it's really, what's the purpose? Why are we doing this? Why would they cut that out? Why would they do this action? What is that contributing to their long-term understanding? I do appreciate NCTM's guidance on this. I think they're leading the pack. And this is really cutting-edge …

Mike: Yeah.

Rachel: … thinking about how we set goals for our classroom. It's not commonly held in the field or applied in the field yet.

Mike: Uh-hm.

Rachel: But I think folks are really starting to understand its importance. That if, as we change the way we teach mathematics and the outcomes we expect for students, we have to start thinking differently about how we set up learning goals. We can't keep having these performance goals and expecting what's happening in the classroom to change. If we're really going to go towards the type of instruction we want to see in a classroom, we've got to think about learning goals instead of focus so much on just performance.

Mike: I actually had a chance to talk to DeAnn Huinker, who's one of the co-writers of 'Taking Action,' and she used the phrase, 'What are the mathematical conversations you want children to have?' And I was really struck by, like, that's a really interesting question for me to think about if I'm thinking about my learning goals. But even if I'm just thinking about planning and preparing for a lesson or a unit of study.

Rachel: Definitely. I don't think that's something that's thought a lot about. I mean, I might see for my students and their lesson plan: 'Turn and talk to your neighbor.' But if you don't really think carefully about what kind of conversation you want to happen during that turn and talk … . Or I'll see in their lesson plan that 'We will have a discussion about students' various solutions.' And what does that mean? You know, what's going to happen in that time? What's the point …

Mike: Uh-hm.

Rachel: … of that time? I can't remember who, I think it was Elham Kazemi that said something once about, 'In math class folks will present,' and it's like that old football cheer, you know, 'stand up, sit down, clap, clap, clap.' That's what we do in math class.

Mike: Yeah.

Rachel: We have kids stand up, we sit down, we all politely listen, and then we clap. And that's it. We move on. But if you really focus on those conversations that you want kids to have, what are the interesting things that you want them to be thinking about? That's a complete shift in how we've taught math.

Mike: Yeah, it really is. It makes me think about, on a practical level, if I'm a person who's listening to this podcast, what I might be starting to think about is, 'How do I take action'—no pun intended—'on this idea of thinking deeply about learning goals, integrating them into my practice?' And, for me at least, the first place I went when I read this was to think about shifting what I did in my preparation and my planning.

Rachel: Uh-hm. But I think when it comes to planning, we need to be thinking, first of all, kind of the three parts that 'Taking Action' talks about, is setting a goal that's clear. It should be clear in your mind what the children are learning. And so that can take some reading, right? It can take reading through the session, reading through the overviews, thinking about the learning progressions, always keeping your eye on that mathematical horizon, making those learning goals clear. But then also thinking about the fact that I am situating those learning goals into a learning progression. And I'm thinking about what this lesson that I'm doing on Tuesday, where does it fit in the math journey? So that makes me think about two things. First, what is this lesson building on? What foundation do these students come with that I can build on? But then also, what is it leading toward?

Rachel: Where are we going from here? And what is the important role that this idea we're looking at today plays in the whole mathematical journey? And then using that as your foundation for your instruction. So if you're finding that the activity that you had planned isn't meeting that learning goal. So it isn't helping you with this clear understanding of what you want them to know. If it isn't helping build toward something that you want them to be able to understand, then what are the changes you need to make?

Mike: Uh-hm.

Rachel: What are some things you want to adjust? Where do you want to spend more time? How do you add those conversations? Things like that.

Mike: Uh-hm. I think you led back to the thing that I wanted to unpack, which is: I worried that at different points in this conversation, people might think, 'Well, they're just suggesting that learning goals or learning targets don't really have a role.' We're not saying that. We're saying that they really stretch over time. And I think your description was really elegant in thinking about, what does this session contribute to that larger goal of understanding the meaning of multiplication? What is the intent of this session in helping that development proceed?

Rachel: Yeah. What is the big idea? What is this leading towards? Because if you don't see it, then that's when you, as a teacher, need to make some decisions. Do I need to do more reading? Do I need to do more understanding about this particular content area? Do I need to adjust the lesson itself? Is there something that I need to change or add or incorporate so that it does play a stronger role? Plus, you know your students. So if we're thinking about this session being a part of a learning progression, and it's building on something they already have, if you feel like maybe they don't have what they need to engage with today's lesson—now I'm going to think about some ways to reengage them with this content. I think especially over the next few years, that's going to be critical. But yeah, I definitely agree with you, Mike. Cause I think NCTM, the authors would say the first thing about a learning target or a learning goal is that it has to be clear, and it has to guide and be the foundation for instruction. And so, they're really important. It's just maybe the way that we've talked about them in the past hasn't been helpful.

Mike: Yeah. The other place you bring me to, Rachel, is the idea that if I'm really clear on my learning goal, what is it that children will come to understand? And where is this lesson situated in that journey? That actually has a lot of value because I can think about, 'What are some of the questions that I want to ask to try to either assess where kids are at or advance their thinking?' Or when I think about what children might do, 'Which kids do I want to strategically highlight at a closure?' So I think understanding that learning goal really does have value for folks. It's just a different way of constructing them. And then also thinking, what do you do next?

Rachel: And I also think, again, I'll take this back to the idea of assessing those learning goals. 'Cause I do think assessment and goals cannot be separated. You're going to always be thinking about that, right? Why set a goal if you don't have any way of knowing whether students are making progress towards that goal? When you establish them in that way and you think about them as less of something that's going to be accomplished by the end of this session, we allow room for students to progress at different ways and learn different things in the class. And then that's when we can have those rich conversations at the end, when we're drawing things together. If every child's going to do everything the exact same way in my classroom, then there's no opportunity for interesting conversations. The interesting conversations happen when kids are doing things differently and making progress in different ways, and heading in different directions towards the same goal.

Rachel: Then we start learning from each other. We can see what our partner is doing and try to understand what they're doing. That's when interesting math happens. And I want to encourage teachers to feel confident in thinking about these as the idea of a learning goal. And even starting to incorporate this into student-friendly language. You know, a learning goal doesn't have to be written as an 'I can' statement for kids to be able to understand it. And I also want teachers to feel confident in their abilities for advocating. Um, when they see learning goals being used in a problematic way, when we see pitfalls and things that we talked about at the beginning happening in their classroom—be confident in your abilities and your knowledge and what you know is best for students. You know your students better than anyone else does. The teacher does. And you know how to think about those individual needs and the individual growth of each child in your classroom.

Rachel: So rest assured in that confidence. But go to the resources that are available to you as well. When you're struggling with the idea of where these lessons or these concepts or these ideas you're teaching fit, go to the learning progressions, go to the 'Taking Action' book, go to the NCTM resources. Um, read your session overviews in your curriculum. Have conversations with your colleagues. Have conversations with the colleagues that teach grades above you and grades below you. That's really critical if we're think about taking away this silo idea of teaching mathematics, we need to start thinking about have these conversations across grade levels. And, and knowing, you know, if you're struggling with where this idea is going, talk to the teacher who comes next. And even just ask them, 'What reason do you think a child would need to learn this?'

Mike: Yeah.

Rachel: You know, and then they might be able to help you see where it fits in the progression.

Mike: Well, and I was going to say, look at the scope and sequence and notice, where do the ideas come back? How are they coming back? How are they being developed? And then the icing on the cake would be to do what you said. Let's take a look at how this manifests itself in the next grade or perhaps in the grade prior.

Rachel: I think that's also a role for math leaders in elementary and in the building instructional coaches, that's a vision that they can help teachers with 'cause they get the opportunity to be in multiple grades in multiple classrooms. And they also have more space to read through the progressions, and they might have more time for those sorts of things. And so I want to push math leaders to be doing that as well. Not just the classroom teachers, help your teachers to see where these ideas carry across into future grades and how they build on previous content and facilitate those conversations.

Mike: Yeah. You know, I'm so glad that you brought that up. Because it makes me think about, there are some things about the way that we've organized education that just, are givens, right? We have primarily grade-level classrooms, right? And so, I taught first grade for eight years. I intimately knew my first-grade standards. I did not clearly have a vision of necessarily how that was going to play out in second grade and third grade and fourth grade and so on. And I think that's one of the inadvertent problems that we're stuck with is, if we don't have a vertical understanding of: How are these ideas going to support children over time? It might be easy to say, 'Well, I just need them to be able to do X by the time they get out of third grade.' Not really understanding that, actually I need to have them understand X, so then they can, in fact, understand all these other concepts that are coming.

Rachel: I've just seen this year, so much, what is happening in fifth grade is dictating how you understand algebra. You know, it's like …

Mike: Yes!

Rachel: … what we see in the fifth-grade standards. If you are not really understanding those concepts, you might be OK for a little while. And then once you're into your algebra classes, you're realizing that all of that foundational knowledge came from what you learned in fifth grade and what you understand about rational numbers. And so, I totally agree. I don't think we've done a good job in education in general of those cross grade-level conversations. But I think we're getting better with this idea of having instructional leaders, instructional coaches that are really there to support the instruction …

Mike: Yeah.

Rachel: … that's happening. So I know I work with math leaders and that's one of the things I really encourage them, is not only should they know the entire curriculum or continuum, but how are they helping their classroom teachers understand that? 'Cause I think there's a lot of power in having a teacher spend eight years in first grade and really knowing those standards intimately. But there's also some value in, in once you've taught third grade going back to first grade and realizing, 'Wow, this is where it was all going.'

Mike: Absolutely. Yeah. I had a role at one point where I was a K–12 curriculum director for math.

Rachel: Oh, yeah.

Mike: And it was the most eye-opening experience because, as you said, you recognize how, if kids walk out of elementary school without a deep foundational understanding—and if it's just really a surface set of performance skills … wow—that catches up with kids when they get into sixth, seventh, and eighth grade.

Rachel: Yep. For sure. And those concepts become more abstract when we start this idea of variables and thinking about things algebraically. That if you didn't have that foundation in the concrete, the abstract is too much. It's too much to ask of kids. And so then we find ourselves reteaching and wondering, 'What happened?' And yeah, I just, I wish more conversations were happening across those grade levels.

Mike: Absolutely. Well, thank you again, Rachel.

Rachel: Yeah!

Mike: It was lovely to have you. I think a lot of folks are going to find this really helpful, and maybe validating in the experience they've had. And also a vision for what they might do in the future. And hopefully we'll have you back at some point.

Rachel: I'm always here for you. (laughs)

Mike: Thank you so much. All right, bye bye.

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.

Multilingual learners represent approximately 10% of the US K–-12 student population, and they are the fastest growing subpopulation of students in the United States. Today Dr. Erin Smith, a mathematics education professor at the University of Southern Mississippi, talks about ways to position multilingual learners as competent doers of mathematics .

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

Positioning Multilingual Learners

Mike Wallus: Multilingual learners represent approximately 10 percent of the U.S. K–12 student population. And they're the fastest growing subpopulation of students in the United States. That said, multilingual learners have been and continue to be underserved in mathematics. Today, we talk with Erin Smith, a mathematics education professor at the University of Southern Mississippi, about ways to support and position multilingual learners as competent doers of mathematics. Hey, Erin, thank you for joining us today on the podcast.

Erin Smith: Thank you so much for inviting me. I'm really happy to be here.

Mike: I was really fascinated by one of the concepts that you talked about your article . You referenced the idea of positioning, and I'm just fascinated by that because I think it has so much potential for how we support students' math identities. Can you explain positioning and how you suspect it could impact students in the classroom?

Erin: Yeah, absolutely. So positioning is a concept from positioning theory, which was developed by Rom Harré and Luk Van Langenhove. So when we talk about positioning or a position, we are really referring to a metaphorical position that you have in a conversation. So it's not necessarily like where your body is physically present, but a metaphorical position. So in the theory, they say that your position that you have impacts what is socially appropriate for you to do and say in an interaction. So in a classroom teachers and students have different positions. Teachers can do things that students can't. They can discipline students. They determine the classroom configuration. They select the tasks that students get to engage with. And in a lot of cases, teachers also get to select who gets to speak in the class, who gets floor time. So each of these decisions that teachers make can impact opportunities for students. And so when I think out positioning in particular and how useful it can be as a lens to look at how we as teachers position certain kinds of students in our classroom, and how we can use our position in the classroom to really call out the strengths of historically underserved students in mathematics, and then use that to position them as leaders in the classroom, while simultaneously also just challenging deficit narratives about who can do mathematics, who can be successful in it. And really what does it mean to do mathematics.

Mike: You know, as you were talking, what struck me as positioning in some ways related to the status that a student either has been assigned or assigned to themselves. Is that a fair comparison?

Erin: Yeah, absolutely. So in positioning we talk about both the positions that we take on ourselves and the positions that we assign others. So there is a lot of agency in that, both from a teacher perspective— like you have a lot of agencies to think about positioning—but also students can challenge the positions that you give them. And they have a lot of agency in that. So if a teacher positions a student as lacking some mathematical competency, the student can challenge that positioning by trying to demonstrate their competencies.

Mike: It's interesting because I think when we shift this to talking about multilingual learners, my suspicion is that part of the challenge that we've had is that multilingual learners have been positioned as less mathematically competent. And the strategies that you're suggesting are actually ways that we can counter that prevailing positioning or status.

Erin: Yeah. So we, in positioning theory, talk about storylines and these stories that permeate both at a larger, broader societal level, but also at a smaller individual level. So when you're talking about these stories that already exist for multilingual learners, more broadly in more social narratives, they're often really deficit-oriented. And so we can use our position as a teacher in a classroom to challenge how a particular student has been positioned in the past and also create spaces for them to carve out new stories. I think one of the things that I would like to clarify is in positioning theory, refer to them as storylines and thinking about how there are these different storylines that exist both at a societal level, but also in your classroom and at the school level that can influence the ways that you interact with students. So for instance, we have a lot of storylines about math mathematics in the U.S., what it means to be successful in mathematics. So as a teacher, you've got those storylines and your students also have those storylines that they might be drawing on. And so as a teacher, just being cognizant and aware of all these different storylines that might be percolating around and circulating, and you can help craft those stories and really call out or bring to the forefront the ones that you think are very valuable and important.

Mike: You know, it makes me think of two things. I mean, one is part of the role and part of the work is interrogating the stories that you've brought or that you've absorbed about students or different groups of students. And the other is maybe being clear about, what are the stories that you want kids to leave with, as you just said.

Erin: Yeah, I would agree that we're not walking into a classroom or an interaction with this blank slate. That's like, we have all these things that are entangled in ourselves that we're making sense of and negotiating and navigating in these interactions.

Mike: Absolutely. You know, there's a quote that really jumped out for me when I was reading your article, and I'd just like to read it aloud. 'Some people may think that multilingual learners must be proficient in English before participating in mathematical discussions. This is not the case. And ultimately puts students mathematical learning on hold.' Can you talk about why you felt it was important to address this misconception?

Erin: Oftentimes we, as teachers, conflate language competency with math competency. And I've even done this myself in my former life as a math teacher. So we might assume that because a student is at their early stages of developing a language competency, that they're also at the same time at the early stages of developing their math competencies. And we know that's just not true ( laughs ). That's not how math and language learning work. They occur at different speeds. And one does not indicate a competency in the other. And so I think it's really necessary and important to call out this assumption and also provide readers with an opportunity to reflect on like, 'Am I holding this assumption? Am I holding some of my students back? Am I doing harm for them because I'm cutting off mathematical learning opportunities because I'm conflating their language competencies with mathematics.'

Mike: Sure. So I think one of the things that also jumped out for me was the ways that teachers can set multilingual learners up for success. And one of the strategies that jumped out is the idea of rehearsal. And I'm wondering if you could talk just a bit about what you think rehearsal might look like in an elementary classroom.

Erin: So rehearsals are really a great strategy to help multilingual learners prepare to present their mathematical ideas to the whole class, especially if they are demonstrating some hesitancy or maybe they're from a culture where standing up and presenting your ideas in front of the class is not a norm. And so in an elementary classroom it might look like telling one of your multilingual learner students in advance that you want them to come to the board and share their strategy with the class. And you give them some time to rehearse and practice what they're going to say. So that could be something like you and this student are just having a conversation, and they're getting a chance to practice like that with you. Or it could be that they're practicing with a peer. It could also be something like you're asking them to write down what it is they want to say, and maybe they also have that scaffold if they need it, when they walk to the front of the classroom. You know, one of the things also that I think is really nice about this is that it doesn't need to be used in a way that is really targeting and calling out the multilingual learner, saying that they specifically need the support. And you might give your whole class, maybe a couple of minutes to like, OK, 'I want you to practice. If you were gonna come to the board and share your strategy, what would you say? I want you to practice that with your partner or your group table mates.'

Mike: Absolutely. Like great practice for everyone even if you're intent, as a teacher, is that you want to position one of those students or set them up to successfully share their thinking.

Erin: Right, right.

Mike: So one of the other things that I thought was really interesting is—and again, I think it feels like a strategy that is particularly powerful for multilingual learners, but just good practice—you really highlighted the idea of assigning student ownership to mathematical ideas when there's a conversation happening. So what does that mean and what might that sound like or look like in a classroom?

Erin: So assigning ownership means that you are publicly acknowledging the mathematical ideas that a multilingual learner possesses. I've seen teachers do this in a range of different ways. It might be something as simple as, we're having a Notice & Wonder routine and a student shares their noticing, and I'm writing their name or initials on the board. So that idea is linked to that student. That doesn't take a lot of extra work for me. It could be referring to a strategy as a student strategy, like asking the class who else used Marco's strategy and asking students to raise their hands.

Mike: Uh-hm.

So you're naming that Marco's strategy. It could be asking your students to write story problems and then putting their name next to it. So like, this is Mary Ellis' word problem that she wrote. And so you're publicly acknowledging this student has created this word problem. When teachers assign ownership of mathematical ideas to students, they're really using that as an opportunity to shift mathematical authority in the classroom off of them and on to students. And so when students have those opportunities where they become authors of mathematics, it can positively impact their mathematical identity. And it also can encourage them to continue coming up with mathematical ideas and being willing to share those mathematical ideas publicly.

Mike: Absolutely. So one of the last strategies that really struck me was something that I've seen teachers do. And I think I've done it, too, but I'd never actually had words for it. You talk about this as something called the prefacing statement. Can you explain what a prefacing statement is and why it's powerful and maybe even what it might sound like?

Erin: Yeah. So in one of my research projects, I was examining this teacher's practice, and she did this, and I noticed her doing this. And then I try to think of like how to name and capture this. So I landed on prefacing statements. And I used that word to refer to what a teacher says before a student shares their thinking or their strategy in front of the class. And so the teacher is using that as an opportunity to set the stage for the student who's presenting. And it also cues the class into what is important about what the student is going to share or (is) unique about it. And so, for example, I've seen a teacher do this, where she selected a multilingual learner to come to the board to share their strategy. And the teacher says, 'I selected Mohammed's strategy because he drew a really efficient picture.' And so, naming in advance, like, 'He drew this efficient picture. I want you to look at this and notate how great this is and how representative of an efficient picture this is.'

Mike: Yeah, I mean, in that case kind of really pointing out to them, 'There is some feature that I want you to attend to,' and then also assigning the ownership of that to the student.

Erin: Right, right. So another way it could go is, like, 'I selected Shin Hin's work to share because he represented his thinking in three different ways. So really calling out what is important mathematically about what the student is sharing. And I think that's really the important piece of, like, you really want to be specific about what it is that you're calling out in your prefacing statement, in terms of what does it mean mathematically? And what about this is a mathematical strength?

Mike: I mean, in some ways, as you say that, it really plays two roles: You're actually helping kids to attend to really specific, small, grain-size features of either the thinking or the representation that are important. And again, you're assigning the contribution clearly to the student that you're talking about.

Erin: Yeah, exactly.

Mike: Uh, you know, as I was reading this, I'm struck by the fact that these strategies have the potential for a couple things. On an individual child level, they have the ability to help a child reposition themselves or to think differently about their mathematical identity. But just on a classroom level as well, they really have the ability to push back on some of the narratives that we were talking about earlier, where marginalized kids have a particularly low status in a classroom, their ideas are kind of preset to matter less. And this is really a way to use some really practical strategies to push back on that.

Erin: Yes, absolutely.

Mike: So one of the things that jumps out is that, in addition to being powerful strategies that you can use in the moment, it seems like these are things that you might actually begin to intentionally plan when you're setting up a lesson.

Erin: Yes. So one of the things that I think is really important about understanding that there are a range of ways that you, as the teacher, hold power in the classroom. And you can leverage your position to create opportunities for students and also publicly acknowledge their competencies. So in planning, you should be considering, 'How am I going to ensure that I'm productively positioning multilingual learners in my classroom?' And then, 'What are some specific things I can embed in my lesson to ensure that happens?' So, for example, if you—going back to the earlier stuff—if you want one of your multilingual learners to present their strategy at the board. And you know from some prior classes that they're a little hesitant and reserved, so you might intentionally carve out the last five minutes of the student exploration stage for students to rehearse what they would say to a class. And so you're building in that time into your lesson and being very intentional in that work. And this might also align to your goal that you might have, that every student shares their strategy at the board. And so that's going to help you achieve that goal for each of your students. I think another thing that's important to keep in mind more broadly is that it's important to hold the same expectations for multilingual learners in your classroom as you do for your other students. And so this is also another way to think about, 'What are some things that I can do as a teacher in my classroom to ensure that, 1) I'm holding the same expectations. And 2) I'm providing appropriate scaffolding that's going to help the student reach those expectations.'

Mike: Absolutely. You started to hint at the next thing that was on my mind, which is that positioning isn't necessarily just something that happens via language. It happens via some of the other decisions like creating space and time. Are there other things in your mind that really support the idea of positioning students in a classroom?

Erin: Yeah. I think every decision that we make as a teacher can be an opportunity to position. So in thinking about just the physical space of your classroom, who is sitting where? How are seats figured? Who is sitting with who? Where do you, as a teacher, position your body in the classroom? How we structure our lessons, what kinds of pedagogical practices we decide to use … the kinds of questions that we ask. Are we asking really open-ended questions? And who are we asking those of? Are we asking those open-ended, rich questions of multilingual learners, or are we only reserving specific kinds of questions for them? Towards the end of the article, and I try to emphasize, like: We position in every interaction. We are constantly negotiating these positions, both within ourselves, the way that we position ourselves, but also how we're position and how they're responding in turn. We can use these situations to really think about the kinds of stories that we want to foster for each of our students. So what can I do in the classroom to tell a productive or a positive story for this student in mathematics?

Mike: Hmm. That's powerful. One of the questions I think that I wanted to ask before we close, 'If someone were listening to this podcast and they wanted to continue learning about support from multilingual learners, or even the idea of positioning more broadly, are there particular resources that you might point them to?'

Erin: Thank you so much for this question.

Mike: ( laughs )

Erin: ( chuckles ) Um, I would first direct them to my recently published book with my co-authors, called, 'Teaching Math to Multilingual Students, Grades K–8: Positioning English Learners for Success.' So it came out in 2021, co-published by Corwin and CTM. So that would be like a really good first place to look. And it's designed for teachers to really think about their practice and their own positioning of multilingual learners. And so I think the next step would be really engaging in some professional development with scholars who have been thinking about and doing some work with positioning in general, and then maybe directing more towards some of the original work of positioning theory as a way to kind of get a hold on, like, these different concepts of the theory.

Mike: Oh gosh, this was super fun. Erin, thank you so much for joining us today. Erin: Thank you so much for inviting me. It was a pleasure talking with you today.

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.