Rounding Up
Welcome to “Rounding Up” with the Math Learning Center. These conversations focus on topics that are important to Bridges teachers, administrators and anyone interested in Bridges in Mathematics. Hosted by Mike Wallus, VP of Educator Support at MLC.
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Season 3 | Episode 17 - Understanding the Role of Language in Math Classrooms - Guest: William Zahner
05/08/2025
Season 3 | Episode 17 - Understanding the Role of Language in Math Classrooms - Guest: William Zahner
William Zahner, Understanding the Role of Language in Math Classrooms ROUNDING UP: SEASON 3 | EPISODE 17 How can educators understand the relationship between language and the mathematical concepts and skills students engage with in their classrooms? And how might educators think about the mathematical demands and the language demands of tasks when planning their instruction? In this episode, we discuss these questions with Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University. BIOGRAPHY Bill Zahner is a professor in the mathematics department at San Diego State University and the director of the Center for Research in Mathematics and Science Education. Zahner's research is focused on improving mathematics learning for all students, especially multilingual students who are classified as English Learners and students from historically marginalized communities that are underrepresented in STEM fields. RESOURCES [BES login required] TRANSCRIPT Mike Wallus: How can educators understand the way that language interacts with the mathematical concepts and skills their students are learning? And how can educators focus on the mathematics of a task without losing sight of its language demands as their planning for instruction? We'll examine these topics with our guest, Bill Zahner, director of the Center for Research in Mathematics and Science Education at San Diego State University. Welcome to the podcast, Bill. Thank you for joining us today. Bill Zahner: Oh, thanks. I'm glad to be here. Mike: So, I'd like to start by asking you to address a few ideas that often surface in conversations around multilingual learners and mathematics. The first is the notion that math is universal, and it's detached from language. What, if anything, is wrong with this idea and what impact might an idea like that have on the ways that we try to support multilingual learners? Bill: Yeah, thanks for that. That's a great question because I think we have a common-sense and strongly held idea that math is math no matter where you are and who you are. And of course, the example that's always given is something like 2 plus 2 equals 4, no matter who you are or where you are. And that is true, I guess [in] the sense that 2 plus 2 is 4, unless you're in base 3 or something. But that is not necessarily what mathematics in its fullness is. And when we think about what mathematics broadly is, mathematics is a way of thinking and a way of reasoning and a way of using various tools to make sense of the world or to engage with those tools [in] their own right. And oftentimes, that is deeply embedded with language. Probably the most straightforward example is anytime I ask someone to justify or explain what they're thinking in mathematics. I'm immediately bringing in language into that case. And we all know the old funny examples where a kid is asked to show their thinking and they draw a diagram of themselves with a thought bubble on a math problem. And that's a really good case where I think a teacher can say, “OK, clearly that was not what I had in mind when I said, ‘Show your thinking.’” And instead, the demand or the request was for a student to show their reasoning or their thought process, typically in words or in a combination of words and pictures and equations. And so, there's where I see this idea that math is detached from language is something of a myth; that there's actually a lot of [language in] mathematics. And the interesting part of mathematics is often deeply entwined with language. So, that's my first response and thought about that. And if you look at our Common Core State Standards for Mathematics, especially those standards for mathematical practice, you see all sorts of connections to communication and to language interspersed throughout those standards. So, “create viable arguments,” that's a language practice. And even “attend to precision,” which most of us tend to think of as, “round appropriately.” But when you actually read the standard itself, it's really about mathematical communication and definitions and using those definitions with precision. So again, that's an example, bringing it right back into the school mathematics domain where language and mathematics are somewhat inseparable from my perspective here. Mike: That's really helpful. So, the second idea that I often hear is, “The best way to support multilingual learners is by focusing on facts or procedures,” and that language comes later, for lack of a better way of saying it. And it seems like this is connected to that first notion, but I wanted to ask the question again: What, if anything, is wrong with this idea that a focus on facts or procedures with language coming after the fact? What impact do you suspect that that would have on the way that we support multilingual learners? Bill: So, that's a great question, too, because there's a grain of truth, right? Both of these questions have simultaneously a grain of truth and simultaneously a fundamental problem in them. So, the grain of truth—and an experience that I've heard from many folks who learned mathematics in a second language—was that they felt more competent in mathematics than they did in say, a literature class, where the only activity was engaging with texts or engaging with words because there was a connection to the numbers and to symbols that were familiar. So, on one level, I think that this idea of focusing on facts or procedures comes out of this observation that sometimes an emergent multilingual student feels most comfortable in that context, in that setting. But then the second part of the answer goes back to this first idea that really what we're trying to teach students in school mathematics now is not simply, or only, how to apply procedures to really big numbers or to know your times tables fast. I think we have a much more ambitious goal when it comes to teaching and learning mathematics. That includes explaining, justifying, modeling, using mathematics to analyze the world and so on. And so, those practices are deeply tied with language and deeply tied with using communication. And so, if we want to develop those, well, the best way to do that is to develop them, to think about, “What are the scaffolds? What are the supports that we need to integrate into our lessons or into our designs to make that possible?” And so, that might be the takeaway there, is that if you simply look at mathematics as calculations, then this could be true. But I think our vision of mathematics is much broader than that, and that's where I see this potential. Mike: That's really clarifying. I think the way that you unpack that is if you view mathematics as simply a set of procedures or calculations, maybe? But I would agree with you. What we want for students is actually so much more than that. One of the things that I heard you say when we were preparing for this interview is that at the elementary level, learning mathematics is a deeply social endeavor. Tell us a little bit about what you mean by that, Bill. Bill: Sure. So, mathematics itself, maybe as a premise, is a social activity. It's created by humans as a way of engaging with the world and a way of reasoning. So, the learning of mathematics is also social in the sense that we're giving students an introduction to this way of engaging in the world. Using numbers and quantities and shapes in order to make sense of our environment. And when I think about learning mathematics, I think that we are not simply downloading knowledge and sticking it into our heads. And in the modern day where artificial intelligence and computers can do almost every calculation that we can imagine—although your AI may do it incorrectly, just as a fair warning [laughs]—but in the modern day, the actual answer is not what we're so focused on. It's actually the process and the reasoning and the modeling and justification of those choices. And so, when I think about learning mathematics as learning to use these language tools, learning to use these ways of communication, how do we learn to communicate? We learn to communicate by engaging with other people, by engaging with the ideas and the minds and the feelings and so on of the folks around us, whether it's the teacher and the student, the student and the student, the whole class and the teacher. That's where I really see the power. And most of us who have learned, I think can attest to the fact that even when we're engaging with a text, really fundamentally we're engaging with something that was created by somebody else. So, fundamentally, even when you're sitting by yourself doing a math word problem or doing calculations, someone has given that to you and you think that that's important enough to do, right? So, from that stance, I see all of teaching and learning mathematics is social. And maybe one of our goals in mathematics classrooms, beyond memorizing the times tables, is learning to communicate with other people, learning to be participants in this activity with other folks. Mike: One of the things that strikes me about what you were saying, Bill, is there's this kind of virtuous cycle, right? That by engaging with language and having the social aspect of it, you're actually also deepening the opportunity for students to make sense of the math. You're building the scaffolds that help kids communicate their ideas as opposed to removing or stripping out the language. That's the context in some ways that helps them filter and make sense. You could either be in a vicious cycle, which comes from removing the language, or a virtuous cycle. And it seems a little counterintuitive because I think people perceive language as the thing that is holding kids back as opposed to the thing that might actually help them move forward and make sense. Bill: Yeah. And actually that's one of the really interesting pieces that we've looked at in my research and the broader research is this question of, “What makes mathematics linguistically complex?” is a complicated question. And so sometimes we think of things like looking at the word count as a way to say, “If there are fewer words, it's less complex, and if there are more words, it's more complex.” But that's not totally true. And similarly, “If there's no context, it's easier or more accessible, and if there is a context, then it's less accessible.” And I don't see these as binary choices. I see these as happening on a somewhat complicated terrain where we want to think about, “How do these words or these contexts add to student understanding or potentially impede [it]?” And that's where I think this social aspect of learning mathematics—as you described, it could be a virtuous cycle so that we can use language in order to engage in the process of learning language. Or, the vicious cycle is, you withhold all language and then get frustrated when students can't apply their mathematics. That’s maybe the most stereotypical answer: “My kids can do this, but as soon as they get a word problem, they can't do it.” And it's like, “Well, did you give them opportunities to learn how to do this? [laughs] Or is this the first time?” Because that would explain a lot. Mike: Well, it's an interesting question, too, because I think what sits behind that in some ways is the idea that you're kind of going to reach a point, or students might reach a point, where they're “ready” for word problems. Bill: Right. Mike: And I think what we're really saying is it's actually through engaging with word problems that you build your proficiency, your skillset that actually allows you to become a stronger mathematician. Bill: Mm-hmm. Right. Exactly. And it's a daily practice, right? It's not something that you just hold off to the end of the unit, and then you have the word problems, but it's part of the process of learning. And thinking about how you integrate and support that. That's the key question that I really wrestle with. Not trivial, but I think that's the key and the most important part of this. Mike: Well, I think that's actually a really good segue because I wanted to shift and talk about some of the concrete or productive ways that educators can support multilingual learners. And in preparing for this conversation, one of the things that I've heard you stress is this notion of a consistent context. So, can you just talk a little bit more about what you mean by that and how educators can use that when they're looking at their lessons or when they're writing lessons or looking at the curriculum that they're using? Bill: Absolutely. So, in our past work, we engaged in some cycles of design research with teachers looking at their mathematics curriculum and opportunities to engage multilingual learners in communication and reasoning in the classroom. And one of the surprising things that we found—just by looking at a couple of standard textbooks—was a surprising number of contexts were introduced that are all related to the same concept. So, the concept would be something like rate of change or ratio, and then the contexts, there would be a half dozen of them in the same section of the book. Now, this was, I should say, at a secondary level, so not quite where most of the Bridges work is happening. But I think it's an interesting lesson for us that we took away from this. Actually, at the elementary level, Kathryn Chval has made the same observation. What we realized was that contexts are not good or bad by themselves. In fact, they can be highly supportive of student reasoning or they can get in the way. And it's how they are used and introduced. And so, the other way we thought about this was: When you introduce a context, you want to make sure that that context is one that you give sufficient time for the students to understand and to engage with; that is relatable, that everyone has access to it; not something that's just completely unrelated to students' experiences. And then you can really leverage that relatable, understandable context for multiple problems and iterations and opportunities to go deeper and deeper. To give a concrete example of that, when we were looking at this ratio and rate of change, we went all the way back to one of the fundamental contexts that's been studied for a long time, which is motion and speed and distance and time. And that seemed like a really important topic because we know that that starts all the way back in elementary school and continues through college-level physics and beyond. So, it was a rich context. It was also something that was accessible in the sense that we could do things like act out story problems or reenact a race that's described in a story problem. And so, the students themselves had access to the context in a deep way. And then, last, that context was one that we could come back to again and again, so we could do variations [of] that context on that story. And I think there's lots of examples of materials out there that start off with a core context and build it out. I’m thinking of some of the Bridges materials, even on the counting and the multiplication. I think there's stories of the insects and their legs and wings and counting and multiplying. And that's a really nice example of—it's accessible, you can go find insects almost anywhere you are. Kids like it. [Laughs] They enjoy thinking about insects and other icky, creepy-crawly things. And then you can take that and run with it in lots of different ways, right? Counting, multiplication, division ratio, and so on. Mike: This last bit of our conversation has me thinking about what it might look like to plan a lesson for a class or a group of multilingual learners. And I know that it's important that I think about mathematical demands as well as the language demands of a given task. Can you unpack why it's important to set math and language development learning goals for a task, or a set of tasks, and what are the opportunities that come along with that, if I'm thinking about both of those things during my planning? Bill: Yeah, that's a great question. And I want to mark the shift, right? We've gone from thinking about the demands to thinking about the goals, and where we're going to go next. And so, when I think about integrating mathematical goals—mathematical learning goals and language learning goals—I often go back to these ideas that we call the practices, or these standards that are about how you engage in mathematics. And then I think about linking those back to the content itself. And so, there's kind of a two-piece element to that. And so, when we're setting our goals and lesson planning, at least here in the great state of California, sometimes we'll have these templates that have, “What standard are you addressing?,” [Laughs] “What language standard are you addressing?,” “What ELD standard are you addressing?,” “What SEL standard are you addressing?” And I've seen sometimes teachers approach that as a checkbox, right? Tick, tick, tick, tick, tick. But I see that as a missed opportunity—if you just look at this like you're plugging things in—because as we started with talking about how learning mathematics is deeply social and integrated with language, that we can integrate the mathematical goals and the language goals in a lesson. And I think really good materials should be suggesting that to the teacher. You shouldn't be doing this yourself every day from scratch. But I think really high-quality materials will say, “Here's the mathematical goal, and here's an associated language goal,” whether it's productive or receptive functions of language. “And here's how the language goal connects the mathematical goal.” Now, just to get really concrete, if we're talking about an example of reasoning with ratios—so I was going back to that—then it might be generalized, the relationship between distance and time. And that the ratio of distance and time gives you this quantity called speed, and that different combinations of distance and time can lead to the same speed. And so, explain and justify and show using words, pictures, diagrams. So, that would be a language goal, but it's also very much a mathematical goal. And I guess I see the mathematical content, the practices, and the language really braided together in these goals. And that I think is the ideal, and at least from our work, has been most powerful and productive for students. Mike: This is off script, but I'm going to ask it, and you can pass if you want to. Bill: Mm-hmm. Mike: I wonder if you could just share a little bit about what the impact of those [kinds] of practices that you described [have been]—have you seen what that impact looks like? Either for an educator who has made the step and is doing that integration or for students who are in a classroom where an educator is purposely thinking about that level of integration? Bill: Yeah, I can talk a little bit about that. In our research, we have tried to measure the effects of some of these efforts. It is a difficult thing to measure because it's not just a simple true-false test question type of thing that you can give a multiple-choice test for. But one of the ways that we've looked for the impact [of] these types of intentional designs is by looking at patterns of student participation in classroom discussions and seeing who is accessing the floor of the discussion and how. And then looking at other results, like giving an assessment, but deeper than looking at the outcome, the binary correct versus incorrect. Also looking at the quality of the explanation that's provided. So, how [do] you justify an answer? Does the student provide a deeper or a more mathematically complete...
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Season 3 | Episode 16 - Assessment as a Shared Journey: Cultivating Partnerships with Families and Caregivers - Guest: Tisha Jones
04/17/2025
Season 3 | Episode 16 - Assessment as a Shared Journey: Cultivating Partnerships with Families and Caregivers - Guest: Tisha Jones
Tisha Jones, Assessment as a Shared Journey: Cultivating Partnerships with Families & Caregivers ROUNDING UP: SEASON 3 | EPISODE 16 Families and caregivers play an essential role in students’ success in school and in shaping their identities as learners. Therefore, establishing strong partnerships with families and caregivers is crucial for equitable teaching and learning. This episode is designed to help educators explore the importance of collaborating with families and caregivers and learn strategies for shifting to asset-based communication. BIOGRAPHY Tisha Jones is the senior manager of assessment at The Math Learning Center. Previously, Tisha taught math to elementary and middle school students as well as undergraduate and graduate math methods courses at Georgia State University. TRANSCRIPT Mike Wallus: As educators, we know that families and caregivers play an essential role in our students’ success at school. With that in mind, what are some of the ways we can establish strong partnerships with caregivers and communicate about students' progress in asset-based ways? We'll explore these questions with MLC’s [senior] assessment manager, Tisha Jones, on this episode of Rounding Up. Welcome back to the podcast, Tisha. I think you are our first guest to appear three times. We're really excited to talk to you about assessment and families and caregivers. Tisha Jones: I am always happy to talk to you, Mike, and I really love getting to share new ideas with people on your podcast. Mike: So, we've titled this episode “Assessment as a Shared Journey with Families & Caregivers,” and I feel like that title—especially the words “shared journey”—say a lot about how you hope educators approach this part of their practice. Tisha: Absolutely. Mike: So, I want to start by being explicit about how we at The Math Learning Center think about the purpose of assessment because I think a lot of the ideas and the practices and the suggestions that you're about to offer flow out of that way that we think about the purpose. Tisha: When we think about the purpose of assessment at The Math Learning Center, what sums it up best to me is that all assessment is formative, even if it's summative, which is a belief that you'll find in our Assessment Guide. And what that means is that assessment really is to drive learning. It's for the purpose of learning. So, it's not just to capture, “What did they learn?,” but it's, “What do they need?,” “How can we support kids?,” “How can we build on what they're learning?” over and over and over again. And so, there's no point where we're like, “OK, we've assessed it and now the learning of that is in the past.” We're always trying to build on what they're doing, what they've learned so far. Mike: You know, I've also heard you talk about the importance of an asset-focused approach to assessment. So, for folks who haven't heard us talk about this in the past, what does that mean, Tisha? Tisha: So that means starting with finding the things that the kids know how to do and what they understand instead of the alternative, which is looking for what they don't know, looking for the deficits in their thinking. We're looking at, “OK, here's the evidence for all the things that they can do,” and then we're looking to think about, “OK, what are their opportunities for growth?” Mike: That sounds subtle, but it is so profound a shift in thinking about what is happening when we're assessing and what we're seeing from students. How do you think that change in perspective shifts the work of assessing, but also the work of teaching? Tisha: When I think about approaching assessment from an asset-based perspective—finding the things that kids know how to do, the things that kids understand—one, I am now on a mission to find their brilliance. I am just this brilliance detective. I'm always looking for, “What is that thing that this kid can shine at?” That's one, and a different way of thinking about it just to start with. And then I think the other thing, too, is, I feel like when you find the things that they're doing, I can think about, “OK, what do I need to know? What can I do for them next to support them in that next step of growth?” Mike: I think that sounds fairly simple, but there's something very different about thinking about building from something versus, say, looking for what's broken. Tisha: For sure. And it also helps build relationships, right? If you approach any relationship from a deficit perspective, you're always focusing on the things that are wrong. And so, if we're talking about building stronger relationships with kids, coming from an asset-based perspective helps in that area too. Mike: That's a great pivot point because if we take this notion that the purpose of assessment is to inform the ways that we support student learning, it really seems like that has a major set of implications for how and what and even why we would communicate with families and caregivers. So, while I suspect there isn't a script for the type of communication, are there some essential components that you'd want to see in an asset-focused assessment conversation that an educator would have with a family or with their child's caregivers? Tisha: Well, before thinking about a singular conversation, I want to back it up and think about—over the course of the school year. And I think that when we start the communication, it has to start before that first assessment. It has to start before we've seen a piece of kids' work. We have to start building those relationships with families and caregivers. We need to invite them into this process. We need to give them an opportunity to understand what we think about assessment. How are we approaching it? When we send things home, and they haven't heard of things like “proficiency” or “meeting current expectations”—those are common words that you'll see throughout the Bridges assessment materials—if parents haven't seen that, if families and caregivers haven't heard from you on what that means for you in your classroom at your school, then they have questions. It feels unfamiliar. It feels like, “Wait, what does this mean about how my child is doing in your class?” And so, we want to start this conversation from the very beginning of the school year and continue it on continuously. And it should be this open invitation for them to participate in this process too, for them to share what they're seeing about their student at home, when they're talking about math or they're hearing how their student is talking about math. We want to know those things because that informs how we approach the instruction in class. Mike: Let's talk about that because it really strikes me that what you’re describing in terms of the meaning of proficiency or the meaning of meeting expectations—that language is likely fairly new to families and caregivers. And I think the other thing that strikes me is, families and caregivers have their own lived experience with assessment from when they were children, perhaps with other children. And that's generally a mixed bag at best. Folks have this set of ideas about what it means when the teacher contacts them and what assessment means. So, I really hear what you're saying when you're talking about, there's work that educators need to do at the start of the year to set the stage for these conversations. Let's try to get a little bit specific, though. What are some of the practices that you'd want teachers to consider when they're thinking about their communication? Tisha: So, I think that starting at the very beginning of the year, most schools do some sort of a curriculum night. I would start by making sure that assessment is a part of that conversation and making sure that you're explaining what assessment means to you. Why are you assessing? What are the different ways that you're assessing? What are some things that [families and caregivers] might see coming home? Are they going to see feedback? Are they going to see scores from assessments? But how were you communicating progress? How do they know how their student is doing? And then also that invitation, right then and there, to be a part of this process, to hear from them, to hear their concerns or their ideas around feedback or the things that they've got questions about. I would also suggest … really working hard to have that asset-based lens apply to parents and families and caregivers. I know that I have been that parent that was the last one to sign up for the parent teacher conferences, and I'm sending the apologetic email, and I'm begging for a special time slot. So, it didn't mean that I didn't care about my kids. It didn't mean that I didn't care about what they were doing. I was swamped. And so, I think we want to keep finding that asset-based lens for parents and caregivers in the same way that we do for the students. And then making sure that you're giving them good news, not just bad news. And then making sure when you're sending any communication about how a student is doing, try to be concrete about what you're seeing, right? So, trying to say, “These are the things where I see your child's strengths. These are the strengths that I'm seeing from your student. And these are the areas where we're working on to grow. And this is what we're doing here at school, and this is what you can do to support them at home.” Mike: I was really struck by a piece of what you said, Tisha, when you really made the case for not assuming that the picture that you have in your mind as an educator is clear for families when it comes to assessment. So, really being transparent about how you think about assessment, why you're assessing, and the cadence of when parents or families or caregivers could expect to hear from you and what they could expect as well. I know for a fact that if my teacher called my family when I was a kid, generally there was a look that came across their face when they answered the phone. And even if it was good news, they didn't think it was good news at the front end of that conversation. Tisha: I've been there. I had my son's fifth grade teacher call me last year, and I was like, “Oh, what is this?” [laughs] Mike: One of the things that I want to talk about before we finish this conversation is homework. I want to talk a little bit about the purpose of homework. We're having this conversation in the context of Bridges in Mathematics, which is the curriculum that The Math Learning Center publishes. So, while we can't talk about how all folks think about homework, we can talk about the stance that we take when it comes to homework: what its purpose is, how we imagine families and caregivers can engage with their students around it. Can you talk a little bit about our perspective on homework? How we think about its value, how we think about its purpose? And then we can dig a little bit into what it might look like at home, but let's start with purpose and intent. Tisha: So, we definitely recognize that there are lots of different ideas about homework, and I think that shows in how we've structured homework through our Bridges units. Most of the time, it's set up so that there's a homework [assignment] that goes with every other session, but it's still optional. So, there's no formal expectation in our curriculum that homework is given on a nightly basis or even on an every-other-night basis. We really have left that up to the schools to determine what is best practice for their population. And I think that is actually what's really the most important thing is, understanding the families and caregivers and the situations that are in your building, and making determinations about homework that makes sense for the students that you're serving. And so, I think we've set homework up in a way that makes it so that it's easy for schools to make those decisions. Mike: One of the things that I'm thinking about is that—again, I'm going to be autobiographical—when I was a kid, homework went back, it was graded, and it actually counted toward my grade at the end of the semester or the quarter or what have you. And I guess I wonder if a school or a district chose to not go about that, to not have homework necessarily be graded, I wonder if some families and caregivers might wonder, “What's the purpose?” I think we know that there can be a productive and important purpose—even if educators aren't grading homework and adding it to a percentage that is somehow determining students' grades, that it can actually still have purpose. How do you think about the purpose of homework, regardless of whether it's graded or not? Tisha: So first off, I would just like to advocate not grading homework if I can. Mike: You certainly can, yeah. Tisha: [laughs] Mike: Let's talk about that. Tisha: I think that, one, if we're talking about this idea of putting this score into an average grade or this percentage grade, I think that this is something that has so many different circumstances for kids at home. You have some students who get lots and lots of help. You get some students who do not have help available to them. Another experience that has been very common when I was teaching was that I would get messages where it was like, “We were doing homework. The kid was in tears, I was in tears. This was just really hard.” And that's just not—I don't ever want that scenario for any student, for any family, for any caregiver, for anybody trying to support a child at home. I used to tell them, “If you are getting to the point where it's that level of frustration, please just stop and send me a message, write it on the homework. Just communicate something that [says,] ‘This was too hard’ because that's information now that I can use.” And so, for me, I think about [how] homework can be an opportunity for students to practice some skills and concepts and things that they've learned at home. It's an opportunity for parents, families, caregivers to see some of the things that the kids are working on at school. Mike: What do you think is meaningful for homework? And I have kind of two bits to that. What do you think is meaningful for the child? And then, what do you think might be meaningful for the interaction between the child and their family or caregiver? What's the best case for homework? When you imagine a successful or a productive or a meaningful experience with homework at home between child and family and caregiver, what's that look like? Tisha: Well, one of the things that I've heard families say is, “I don't know how to help my child with blank.” So, then I think it is, “Well, how do we support families and caregivers in knowing what [to] do with homework when we don't know how to tell them what to do?” So, to me, it's about, how can we restructure the homework experience so that it's not this, “I have to tell you how to do it so you can get the right answer so you can get the grade.” But it's like, “How can I get at more of your thinking? How can I understand then what is happening or what you do know?” So, “We can't get to the answer. OK. So tell me about what you do know, and how can we build from there? How can we build understanding?” And that way it maybe will take some of the pressure off of families and caregivers to help their child get to the right answer. Mike: What hits me is we've really come full circle with that last statement you made because you could conceivably have a student who really clearly understands a particular problem that might be a piece of homework, [who] might have some ideas that are on the right track, but ultimately perhaps doesn't get to a fully clear answer that is perfect. And you might have a student who at a certain point in time, maybe [for them] the context or the problem itself is profoundly challenging. And in all of those cases, the question, “Tell me what you do know” or “Tell me what you're thinking” is still an opportunity to draw out the students’ ideas and to focus on the assets. Even if the work as you described it is to get them to think about, “What are the questions that are really causing me to feel stuck?” That is a productive move for a family and a caregiver and a student to engage in, to kind of wonder about, “What's going on here that's making me feel stuck?” Because then, as you said, all assessment is formative. Tisha: Mm-hmm. Mike: That homework that comes back is functioning as a formative assessment, and it allows you to think about your next moves, how you build on what the student knows, or even how you build on the questions that the student is bringing to you. Tisha: And that's such a great point, too, is there's really more value in them coming back with an incomplete assignment or there's, I don't know, maybe “more value” is not the right way to say it. But there is value in kids coming back with an incomplete assignment or an attempted assignment, but they weren't sure how to get through all the problems—as opposed to a parent who has told their student what to do to get to all of the right answers. And so, now they have all these right answers, but it doesn't really give you a clear picture of what that student actually does understand. So, I'd much rather have a student attempt the homework and stop because they got too stuck, because now I know that, than having a family [member] or a caregiver—somebody working with that student—feel like if they don't have all of the right answers, then it's a problem. Mike: I think that's really great guidance, both for teachers as they're trying to set expectations and be transparent with families. But also I think it takes that pressure off of families or caregivers who feel like their work when homework shows up, is to get to a right answer. It just feels like a much more healthy relationship with homework and a much more healthy way to think about the value that it has. Tisha: Well, in truth, it's a healthier relationship with math overall, right? That math is a process. It's not just—the value is not in just this one right answer or this paper of right answers, but it's really in, “How do we deepen our understanding?,” “How do we help students deepen their understanding and have this more positive relationship with math?” And I think that creating these homework struggles between families and caregivers and the children does not support that end goal of having a more positive relationship with math overall. Mike: Which is a really important part of what we're looking for in a child's elementary experience. Tisha: Absolutely. Mike: I think that's a great place to stop. Tisha Jones, thank you so much for joining us. We would love to have you back at some time. It has been a pleasure talking with you. Tisha: It's been great talking to you, too, Mike. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center | www.mathlearningcenter.org
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Season 3 | Episode 15 – What If I Don’t Understand Their Thinking? - Guest: Ryan Flessner
04/03/2025
Season 3 | Episode 15 – What If I Don’t Understand Their Thinking? - Guest: Ryan Flessner
Ryan Flessner, What If I Don’t Understand Their Thinking? ROUNDING UP: SEASON 3 | EPISODE 15 “What do I do if I don’t understand my student’s strategy?” This is a question teachers grapple with constantly, particularly when conferring with students during class. How educators respond in moments like these can have a profound impact on students’ learning and their mathematical identities. In this episode, we talk with Ryan Flessner from Butler University about what educators can say or do when faced with this situation. BIOGRAPHY Ryan Flessner is a professor of teacher education in the College of Education at Butler University in Indianapolis, Indiana. He holds a PhD in curriculum and instruction with an emphasis in teacher education from the University of Wisconsin–Madison; a master of arts in curriculum and teaching from Teachers College, Columbia University; and a bachelor of science in elementary education from Butler University. Prior to his time at the university level, he taught grades 3–7 in Indianapolis; New York City; and Madison, Wisconsin. RESOURCES TRANSCRIPT Mike Wallus: “What do I do if I don't understand my student’s strategy?” This is a question teachers grapple with constantly, particularly when conferring with students during class. How we respond in moments like these can have a profound impact on our students' learning and their mathematical identities. Today we'll talk with Ryan Flessner from Butler University about what educators can say or do when faced with this very common situation. Welcome to the podcast, Ryan. Really excited to talk to you today. Ryan Flessner: Thanks, Mike. I'm flattered to be here. Thank you so much for the invitation. Mike: So, this experience of working with a student and not being able to make sense of their solution feels like something that almost every teacher has had. And I'll speak for myself and say that when it happens to me, I feel a lot of anxiety. And I just want to start by asking, what would you say to educators who are feeling apprehensive or unsure about what to do when they encounter a situation like this? Ryan: Yeah, so I think that everybody has that experience. I think the problem that we have is that teachers often feel the need to have all of the answers and to know everything and to be the expert in the room. But as an educator, I learned really quickly that I didn't have all the answers. And to pretend like I did put a lot of pressure on me and made me feel a lot of stress and would leave me answering children by saying, “Let me get back to you on that.” And then I would scurry and try and find all the answers so I could come back with a knowledgeable idea. And it was just so much more work than to just simply say, “I don't know. Let's investigate that together.” Or to ask kids, “That's something interesting that I'm seeing you do. I've never seen a student do that before. Can you talk to me a little bit about that?” And just having that ability to free myself from having to have all the answers and using that Reggio-inspired practice—for those who know early childhood education—to follow the child, to listen to what he or she or they say to us and try to see. I can usually keep up with a 7- or an 8-year-old as they're explaining math to me. I just may never have seen them notate something the way they did. So, trying to ask that question about, “Show me what you know. Teach me something new.” The idea that a teacher could be a learner at the same time I think is novel to kids, and I think they respond really well to that idea. Mike: So, before we dig in a little bit more deeply about how teachers respond to student strategies if they don't understand, I just want to linger and think about the assumptions that many educators, myself included, might bring to this situation. Assumptions about their role, assumptions about what it would mean for a student if they don't know the answer right away. How do you think about some of the assumptions that are causing some of that anxiety for us? Ryan: Yeah. When the new generation of standards came out, especially in the field of math, teachers were all of a sudden asked to teach in a way that they themselves didn't learn. And so, if you have that idea that you have to have all the answers and you have to know everything, that puts you in a really vulnerable spot because how are we supposed to just magically teach things we've never learned ourselves? And so, trying to figure out ways that we can back up and try and make sense of the work that we're doing with kids, for me that was really helpful in understanding what I wanted from my students. I wanted them to make sense of the learning. So, if I hadn't made sense of it yet, how in the world could I teach them to make sense of it? And so we have to have that humility to say, “I don't know how to do this. I need to continue my learning trajectory and to keep going and trying to do a little bit better than the day that I did before.” I think that teachers are uniquely self-critical and they're always trying to do better, but I don't know if we necessarily are taught how to learn once we become teachers. Like, “We've already learned everything we have to do. Now we just have to learn how to teach it to other people.” But I don't think we have learned everything that we have to learn. There's a lot of stuff in the math world that I don't think we actually learned. We just memorized steps and kind of regurgitated them to get our A+ on a test or whatever we did. So, I think having the ability to stop and say, “I don't know how to do this, and so I'm going to keep working at it, and when I start to learn it, I'm going to be able to ask myself questions that I should be asking my students.” And just being really thoughtful about, “Why is the child saying the thing that she is?,” “Why is she doing it the way that she's doing it?,” “Why is she writing it the way that she's writing it?” And if I can't figure it out, the expert on that piece of paper is the child [herself], so why wouldn't I go and say, “Talk to me about this.”? I don't have to have all the answers right off the cuff. Mike: In some ways, what you were describing just there is a real nice segue because I've heard you say that our minds and our students' minds often work faster than we can write, or even in some cases faster than we can speak. I'm wondering if you can unpack that. Why do you think this matters, particularly in the situation that we're talking about? Ryan: Yeah, I think a lot of us, especially in math, have been conditioned to get an answer. And nobody's really asked us “Why?” in the past. And so, we've done all of the thinking, we give the answer, and then we think the job is done. But with a lot of the new standards, we have to explain why we think that way. And so, all those ideas that just flurried through our head, we have to now articulate those either in writing on paper or in speech, trying to figure out how we can communicate the mathematics behind the answer. And so, a lot of times I'll be in a classroom, and I'll ask a student for an answer, and I'll say, “How'd you get that?” And the first inclination that a lot of kids have is, “Oh, I must be wrong if a teacher is asking me why.” So, they think they're wrong. And so I say, “No, no, no. It's not that you're wrong. I'm just curious. You came to that answer, you stopped and you looked up at the ceiling for a while and then you came to me and you said the answer is 68. How did you do that?” A child will say something like, “Well, I just thought about it in my head.” And I say, “Well, what did you think about in your head?” “Well, my brain just told me the answer was 68.” And we have to actually talk to kids. And we have to teach them how to talk to us—that we're not quizzing them or saying that they're wrong or they didn't do something well enough—that we just want them to communicate with us how they're going about finding these things, what the strategies are. Because if they can communicate with us in writing, if they can communicate on paper, if they can use gestures to explain what they're thinking about, all of those tell us strengths that they bring to the table. And if I can figure out the strengths that you have, then I can leverage those strengths as I address needs that arise in my classroom. And so, I really want to create this bank of information about individual students that will help me be the best teacher that I can be for them. And if I can't ask those questions and they can't answer those questions for me, how am I going to individualize my instruction in meaningful ways for kids? Mike: We've been talking a little bit about the teacher experience in this moment, and we've been talking about some of the things that a person might say. One of the things that I'm thinking about before we dig in a little bit deeper is, just, what is my role? How do you think about the role of a teacher in the moment when they encounter thinking from a student that they don't quite understand […] yet? Part of what I'm after is, how can a teacher think about what they're trying to accomplish in that moment for themselves as a learner and also for the learner in front of them? How would you answer that question? Ryan: When I think about an interaction with a kid in a moment like that, I try to figure out, as the teacher, my goal is to try and figure out what this child knows so that I can continue their journey in a forward trajectory. Instead of thinking about, “They need to go to page 34 because we're on page 33,” just thinking about, “What does this kid need next from me as the teacher?” What I want them to get out of the situation is I want them to understand that they are powerful individuals, that they have something to offer the conversation and not just to prove it to the adult in the room. But if I can hear them talk about these ideas, sometimes the kids in the classroom can answer each other's questions. And so, if I can ask these things aloud and other kids are listening in, maybe because we're in close proximity or because we're in a small-group setting, if I can get the kids to verbalize those ideas sometimes one kid talking strikes an idea in another kid. Or another kid will say, “I didn't know how to answer Ryan when he asked me that question before, but now that I hear what it sounds like to answer that type of a question, now I get it, and I know how I would say it if it were my turn.” So, we have to actually offer kids the opportunity to learn how to engage in those moments and how to share their expertise so others can benefit from their expertise and use that in a way that's helpful in the mathematical process. Mike: One of the most practical—and, I have to say, freeing—things that I've heard you recommend when a teacher encounters student work and they're still trying to make sense of it, is to just go ahead and name it. What are some of the things you imagine that a teacher might say that just straight out name the fact that they're still trying to understand a student's thinking? Tell me a little bit about that. Ryan: Well, I think the first thing is that we just have to normalize the question “Why?” or “Tell me how you know that.” If we normalize those things—a lot of times kids get asked that question when they're wrong, and so it's an [immediate] tip of the hat that “You're wrong, now go back and fix it. There's something wrong with you. You haven't tried hard enough.” Kids get these messages even if we don't intend for them to get them. So, if we can normalize the question “Tell me why you think that” or “Explain that to me”—if we can just get them to see that every time you give me an answer whether it's right or wrong, I'm just going to ask you to talk to me about it, that takes care of half of the problem. But I think sometimes teachers get stuck because—and myself being one of them—we get stuck because we'll look at what a student is doing and they do something that we don't anticipate. Or we say, “I've shown you three different ways to get at this problem, different strategies you can use, and you're not using any of them.” And so, instead of getting frustrated that they're not listening to us, how do we use that moment to inquire into the things that we said obviously aren't useful, so what is useful to this kid? How is he attacking this on his paper? So, I often like to say to a kid, “Huh, I noticed that you're doing something that isn't up on our anchor chart. Tell me about this. I haven't seen this before. How can you help me understand what you're doing?” And sometimes it's the exact same thinking as other strategies that kids are using. So, I can pair kids together and say, “Huh, you're both talking about it in the same way, but you're writing it differently on paper.” And so, I think about how I can get kids just to talk to me and tell me what's happening so that I can help give them a notation that might be more acceptable to other mathematicians or to just honor the fact that they have something novel and interesting to share with other kids. Other questions I talk about are, I will say, “I don't understand what's happening here, and that's not your fault, that's my fault. I just need you to keep explaining it to me until you say something that strikes a chord.” Or sometimes I'll bring another kid in, and I'll have the kids listen together, and I'll say, “I think this is interesting, but I don't understand what's going on. Can you say it to her? And then maybe she'll say it in a way that will make more sense to me.” Or I'll say, “Can you show me on your paper—you just said that—can you show me on your paper where that idea is?” Because a lot of times kids will think things in their head, but they don't translate it all onto the paper. And so, on the paper, it's missing a step that isn't obvious to the viewer of the paper. And so, we'll say, “Oh, I see how you do that. Maybe you could label your table so that we know exactly what you're talking about when you do this. Or maybe you could show us how you got to 56 by writing 8 times 7 in the margin or something.” Just getting them to clarify and try to help us understand all of the amazing things that are in their head. I will often tell them too, “I love what you're saying. I don't see it on your paper, so I just want you to say it again. And I'm going to write it down on a piece of paper that makes sense to me so that I don't forget all of the cool things that you said.” And I'll just write it using more of a standard notation, whether that's a ratio table or a standard US algorithm or something. I'll write it to show the kid that thing that you're doing, there's a way that people write that down. And so, then we can compare our notations and try and figure out “What's the thing that you did?,” “How does that compare to the thing that I did?,” “Do I understand you clearly now?” to make sure that the kid has the right to say the thing she wants to say in the way that she wants to say it, and then I can still make sense of it in my own way. It's not a problem for me to write it differently as long as we're speaking the same language. Mike: I want to mark something really important, and I don't want it to get lost for folks. One of the things that jumped out is the moves that you were describing. You could potentially take up those moves if you really were unsure of how a student were thinking, if you had a general notion but you had some questions, or if you totally already understood what the student was doing. Those are questions that aren't just reserved for the point in time when you don't understand—they're actually good questions regardless of whether you fully understand it or don't understand it at all. Did I get that right? Ryan: Yes. I think that's exactly the point. One thing that I am careful of is, sometimes kids will ask me a question that I know the answer to, and there's this thing that we do as teachers where we're like, “I'm not sure. Why don't you help me figure that out?”—when the kid knows full well that you know the answer. And so, trying not to patronize kids with those questions, but to really show that I'm asking you these questions, not because I'm patronizing you. I'm asking these questions because I am truly curious about what you're thinking inside and all of the ideas that surround the things that you've written on your paper, or the things that you've said to your partner, to truly honor that the more I know about you, the better teacher I can be for you. Mike: So, in addition to naming the situation, one of the things that jumped out for me—particularly as you were talking about the students—is, what do you think the impact is on a student's thinking? But also their mathematical identity, or even the set of classroom norms, when they experience this type of questioning or these [types] of questions? Ryan: So, I think I talked a little bit about normalizing the [questions] “Why?” or “How do you know that?” And so, just letting that become a classroom norm I think is a sea-changing moment for a lot of classrooms—that the conversation is just different if the kids know they have to justify their thinking whether they're right or wrong. Half the time, if they are incorrect, they'll be able to correct themselves as they're talking it through with you. So, kids can be freed up when they're allowed to use their expertise in ways that allow them to understand that the point of math is to truly make sense of it so that when you go out into the world, you understand the situation, and you have different tools to attack it. So, what's the way that we can create an environment that allows them to truly see themselves as mathematical thinkers? And to let them know that “Your grades in other classes don't tell me much about you as a mathematician. I want to learn what really works for you, and I want to try and figure out where you struggle. And both of those things are important to me because we can use them in concert with each other. So, if I know the things you do well, I can use those to help me build a plan of instruction that will take you further in your understandings.” I think that one of the things that is really important is for kids to understand that we don't do math because we want a good grade. I think a lot of people think that the point of math is to get a good grade or to pass a test or to get into the college that you want to get into, or because sixth grade teachers want you to know this. I really want kids to understand that math is a fantastic language to use out in the world, and there are ways that we can interpret things around us if we understand some pretty basic math. And so how do we get them to stop thinking that math is about right answers and next year and to get the job I want? Well, those things may be true, but that's not the real meaning of math. Math is a way that we can live life. And so, if we don't help them understand the connections between the things that they're doing on a worksheet or in a workbook page, if we don't connect those things to the real world, what's the meaning? What's the point for them? And how do we keep them engaged in wanting to know more mathematics? So, really getting kids to think about who they are as people and how math can help them live the life that they want to live. Creating classroom environments that have routines in place that support kids in thinking in ways that will move them forward in their mathematical understanding. Trying to help them see that there's no such thing as “a math person” or “not a math person.” That everybody...
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Season 3 | Episode 14 – Supporting Neurodiverse Students in Elementary Mathematics Classrooms - Guest: Dr. Cathery Yeh
03/20/2025
Season 3 | Episode 14 – Supporting Neurodiverse Students in Elementary Mathematics Classrooms - Guest: Dr. Cathery Yeh
Dr. Cathery Yeh, Supporting Neurodiverse Students in Elementary Mathematics Classrooms ROUNDING UP: SEASON 3 | EPISODE 14 What meaning does the term neurodiverse convey and how might it impact a student’s learning experience? And how can educators think about the work of designing environments and experiences that support neurodiverse students learning mathematics? In this episode, we discuss these questions with Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin. BIOGRAPHY Dr. Cathery Yeh is an assistant professor in STEM education and a core faculty member in the Center for Asian American Studies from the University of Texas at Austin. Her research examines the intersections of race, language, and disability to provide a nuanced analysis of the constructions of ability in mathematics classrooms and education systems. TRANSCRIPT Mike Wallus: What meaning does the term neurodiverse convey and how might that language impact a student's learning experience? In this episode, we'll explore those questions. And we'll think about ways that educators can design learning environments that support all of their students. Joining us for this conversation is Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin. Welcome to the podcast, Cathery. It's really exciting to have you with us today. Cathery Yeh: Thank you, Mike. Honored to be invited. Mike: So, I wonder if we can start by offering listeners a common understanding of language that we'll use from time to time throughout the episode. How do you think about the meaning of neurodiversity? Cathery: Thank you for this thoughtful question. Language matters a lot. For me, neurodiversity refers to the natural variation in our human brains and our neurocognition, challenging this idea that there's a normal brain. I always think of… In Texas, we just had a snow day two days ago. And I think of, just as, there's no two snowflakes that are the same, there's no two brains that are exactly the same, too. I also think of its meaning from a personal perspective. I am not a special educator. I was a bilingual teacher and taught in inclusive settings. And my first exposure to the meaning of neurodiversity came from my own child, who—she openly blogs about it—as a Chinese-American girl, it was actually really hard for her to be diagnosed. Asian Americans, 1 out of 10 are diagnosed—that's the lowest of any ethnic racial group. And I'll often think about when… She's proud of her disabled identity. It is who she is. But what she noticed that when she tells people about her disabled identity, what do you think is the first thing people say when she says, “I'm neurodivergent. I have ADHD. I have autism.” What do you think folks usually say to her? The most common response? Mike: I'm going to guess that they express some level of surprise, and it might be associated with her ethnic background or racial identity. Cathery: She doesn't get that as much. The first thing people say is, they apologize to her. They say, “I'm sorry.” Mike: Wow. Cathery: And that happens quite a lot. And I say that because–and then I connected back to the term neurodiversity—because I think it's important to know its origins. It came about by Judy Singer. She's a sociologist. And about 30 years ago, she coined the term neurodiversity as an opposition to the medical model of understanding people and human difference as deficits. And her understanding is that difference is beautiful. All of us think and learn and process differently, and that's part of human diversity. So that original definition of neurodiversity was tied to the autism rights movement. But now, when we think about the term, it's expanded to include folks with ADHD, dyslexia, dyscalculia, mental health, conditions like depression, anxiety, and other neuro minorities like Tourette syndrome, and even memory loss. I wanted to name out all these things because sometimes we're looking for a really clean definition, and definitions are messy. There's a personal one. There's a societal one of how we position neurodiversity as something that's deficit, that needs to be fixed. But it's part of who one is. But it's also socially constructed. Because how do you decide when a difference becomes a difference that counts where you qualify as being neurodiverse, right? So, I think there's a lot to consider around that. Mike: You know, the answer that you shared is really a good segue because the question I was going to ask you involves something that I suspect you hear quite often is people asking you, “What are the best ways that I can support my neurodiverse students?” And it occurs to me that part of the challenge of that question is it assumes that there's this narrow range of things that you do for this narrow range of students who are different. The way that you just talked about the meaning of neurodiversity probably means that you have a different kind of answer to that question when people ask it. Cathery: I do get this question quite a lot. People email it to me, or they'll ask me. That's usually the first thing people ask. I think my response kind of matches my pink hair question. When they ask me the question, I often ask a question back. And I go, “How would you best educate Chinese children in math?” And they're like, “Why would you ask that?” The underlining assumption is that all Chinese children are the same, and they learn the same ways, they have the same needs, and also that their needs are different than the research-based equity math practices we know and have done 50–60 years of research that we've highlighted our effective teaching practices for all children. We've been part of NCTM for 20 years. We know that tasks that promote reasoning and problem solving have been effectively shown to be good for all. Using a connecting math representation—across math representations in a lesson—is good for all. Multimodal math discourse, not just verbal, written, but embodied in part who we are and, in building on student thinking, and all those things we know. And those are often the recommendations we should ask. But I think an important question is how often are our questions connecting to that instead? How often are we seeing that we assume that certain students cannot engage in these practices? And I think that's something we should prioritize more. I'm not saying that there are not specific struggles or difficulties that the neurodiversity umbrella includes, which includes ADHD, dyslexia, autism, bipolar disorder, on and on, so many things. I'm not saying that they don't experience difficulties in our school environment, but it's also understanding that if you know one neurodiverse student—you know me or my child—you only know one. That's all you know. And by assuming we're all the same, it ignores the other social identities and lived experiences that students have that impact their learning. So, I'm going to ask you a question. Mike: Fire away. Cathery: OK. What comes to your mind when you hear the term “neurodiverse student”? What does that student look like, sound like, appear like to you? Mike: I think that's a really great question. There's a version of me not long ago that would have thought of that student as someone who's been categorized as special education, receiving special education services, perhaps a student that has ADHD. I might've used language like “students who have sensory needs or processing.” And I think as I hear myself say some of those things that I would've previously said, what jumps out is two things: One is I'm painting with a really broad brush as opposed to looking at the individual student and the things that they need. And two is the extent to which painting with a broad brush or trying to find a bucket of strategies that's for a particular group of students, that that really limits my thinking around what they can do or all the brilliance that they may have inside them. Cathery: Thank you for sharing that because that's a reflection I often do. I think about when I learned about my child, I learned about myself. How I automatically went to a deficit lens of like, “Oh, no, how are we going to function in the world? How's she going to function in the world?” But I also do this prompt quite a lot with teachers and others, and I ask them to draw it. When you draw someone, what do you see? And I'll be honest, kind of like drawing a scientist, we often draw Albert Einstein. When I ask folks to draw what a neurodiverse student looks like, they're predominantly white boys, to be honest with you. And I want to name that out. It’s because students of color, especially black, brown, native students—they're disproportionately over- and under-identified as disabled in our schooling. Like we think about this idea that when most of us associate autism or ADHD mainly as part of the neurodiversity branch and as entirely within as white boys, which often happens with many of the teachers that I talk to and parents. We see them as needing services, but in contrast, when we think about, particularly our students of color and our boys—these young men—there's often a contrast of criminalization in being deprived of services for them. And this is not even what I'm saying. It's been 50 years of documented research from the Department of Ed from annual civil rights that repeatedly shows for 50 years now extreme disproportionality for disabled black and Latinx boys, in particular from suspension, expulsion, and in-school arrests. I think one of the most surprising statistics for me that I had learned recently was African-American youth are five times more likely to be misdiagnosed with conduct disorder before receiving the proper diagnosis of autism spectrum disorder. And I appreciate going back to that term of neurodiversity because I think it's really important for us to realize that neurodiversity is an asset-based perspective that makes us shift from looking at it as the student that needs to be fixed, that neurodiversity is the norm, but for us to look at the environment. And I really believe that we cannot have conversations about disability without fully having conversations about race, language, and the need to question what needs to be fixed, particularly not just our teaching, but our assessment practices. For example, we talk about neurodiversities around what we consider normal or abnormal, which is based on how we make expectations around what society thinks. One of the things that showed up in our own household—when we think about neurodiversity or assessments for autism—is this idea of maintaining eye contact. That's one of the widely considered autistic traits. In the Chinese and in the Asian household, and also in African communities, making eye contact to an adult or somebody with authority? It is considered rude. But we consider that as one of the characteristics when we engage in diagnostic tools. This is where I think there needs to be more deep reflection around how one is diagnosed, how a conversation of disability is not separate from our understanding of students and their language practices, their cultural practices. What do we consider normative? Because normative is highly situated in culture and context. Mike: I would love to stay on this theme because one of the things that stands out in that last portion of our conversation was this notion that rather than thinking about, “We need to change the child.” Part of what we really want to think about is, “What is the work that we might do to change the learning environment?” And I wonder if you could talk a bit about how educators go about that and what, maybe, some of the tools could be in their toolbox if they were trying to think in that way. Cathery: I love that question of, “What can we as teachers do? What's some actionable things?” I really appreciate Universal Design for Learning framework, particularly their revised updated version, or 3.0 version, that just came out, I think it was June or July of this year. Let me give you a little bit of background about universal design. And I'm sure you probably already know. I've been reading a lot around its origins. It came about [in the] 1980s, we know from cast.org. But I want to go further back, and it really builds from universal design and the work of architecture. So universal design was coined by a disabled architect. His name was Ronald Mace. And as I was reading his words, it really helped me better understand what UDL is. We know that UDL— Universal Design for Learning and universal design—is about access. Everybody should have access to curriculum. And that sounds great, but I've also seen classrooms where access to curriculum meant doing a different worksheet while everybody else is engaging in small group, whole group problem-based learning. Access might mean your desk is in the front of the room where you're self-isolated—where you're really close to the front of the board so you can see it really well—but you can't talk to your peers. Or that access might mean you're in a whole different classroom, doing the same set of worksheets or problems, but you're not with your grade-level peers. And when Ronald Mace talks about access, he explained that access in architecture had already been a focus in the late 1900s, around 1998, I think. But he said that universal design is really about the longing. And I think that really shifted the framing. And his argument was that we need to design a place, an environment where folks across a range of bodies and minds feel a sense of belonging there. That we don't need to adapt—the space was already designed for you. And that has been such a transformative perspective: That it shouldn't be going a different route or doing something different, because by doing that, you don't feel like you belong. But if the space is one where you can take part equally and access across the ways you may engage, then you feel a sense of belonging. Mike: The piece of what you said that I'm really contemplating right now is this notion of belonging. What occurs to me is that approaching design principles for a learning environment or a learning experience with belonging in mind is a really profound shift. Like asking the question, “What would it mean to feel a sense of belonging in this classroom or during this activity that's happening?” That really changes the kinds of things that an educator might consider going through a planning process. I'm wondering if you think you might be able to share an example or two of how you've seen educators apply universal design principles in their classrooms in ways that remove barriers in the environment and support students' mathematical learning. Cathery: Oh gosh, I feel so blessed. I spend… Tomorrow I'm going to be at a school site all day doing this. UDL is about being responsive to our students and knowing that the best teaching requires us to listen deeply to who they are, honor their mathematical brilliance, and their agency. It's about honoring who they are. I think where UDL ups it to another level, is it asks us to consider who makes the decision. If we are making all the decisions of what is best for that student, that's not fully aligned with UDL. The heart of UDL, it's around multiple ways for me to engage, to represent and express, and then students are given choice. So, one of the things that's an important part of UDL is honoring students' agency, so we do something called “access needs.” At the start of a lesson, we might go, “What do you need to be able to fully participate in math today?” And kids from kindergarten to high school or even my college students will just write out what they need. And usually, it's pretty stereotypical: “I want to talk to someone when I'm learning.” “I would like to see it and not just hear it.” And then you continually go back and you ask, “What are your access needs? What do you need to fully participate?” So students are reflecting on their own what they need to be fully present and what they believe is helpful to create a successful learning environment. So that's a very strong UDL principle—that instead of us coming up with a set of norms for our students, we co-develop that. But we're co-developing it based on students reflecting on their experience in their environment. In kindergarten, we have children draw pictures. As they get older, they can draw, they can write. But it's this idea that it's an ongoing process for me to name out what I need to be fully present. And oftentimes, they're going to say things that are pretty critical. It's almost always critical, to be honest with you, but that’s a… I would say that's a core component of UDL. We're allowing students to reflect on what they need so they can name it for themselves, and then we can then design that space together. And along the way, we have kids that name, “You know what? I need the manipulatives to be closer.” That would not come about at the start of me asking about access needs. But if we did a lesson, and it was not close by, they’ll tell me. So it's really around designing an environment where they can fully participate and be their full selves and feel a sense of belonging. So, that's one example. Another one that we've been doing is teachers and kids who have traditionally not participated the most in our classrooms or have even engaged in pullout intervention. And we'll have them walk around school, telling us about their day. “Will you walk me through your day and tell me how you feel in each of these spaces, and what are your experiences like?” And again, we're allowing the students to name out what they need. And then they're naming out… Oftentimes, with the students that we're at, where I'm working in mostly multilingual spaces, they'll say, “Oh, I love this teacher because she allows us to speak in Spanish in the room. It’s OK.” So that's going back to ideas of action, expression, engagement, where students are allowed a trans language. That's one of the language principles. But we're allowing students and providing spaces and really paying close attention to: “How do we decide how to maximize participation for our students with these set of UDL guidelines? How we are able to listen and make certain decisions on how we can strengthen their participation, their sense of belonging in our classrooms.” Mike: I think what's lovely about both of those examples—asking them to write or draw what they need or the description of, “Let's walk through the day. Let's walk through the different spaces that you learn in or the humans that you learn with”—is one, it really is listening to them and trying to make meaning of that and using that as your starting point. I think the other piece is that it makes me think that it's something that happens over time. It might shift, you might gain more clarity around the things that students need or they might gain more clarity around the things that they need over time. And those might shift a little bit, or it might come into greater focus. Like, “I thought I needed this” or “I think I needed this, but what I really meant was this.” There's this opportunity for kids to refine their needs and for educators to think about that in the designs that they create. Cathery: I really appreciate you naming that because it's all of that. It's an ongoing process where we're building a relationship with our students for us to co-design what effective teaching looks like—that it's not a one size fits all. It's disrupting this idea that what works for one works for all. It's around supporting our students to name out what they need. Now, I'm almost 50. I struggle to name out what I need sometimes, so it's not going to happen...
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Season 3 | Episode 13 – Assessment in the Early Years - Guest: Shelly Scheafer
03/06/2025
Season 3 | Episode 13 – Assessment in the Early Years - Guest: Shelly Scheafer
Assessment in the Early Years Guest: Shelly Scheafer ROUNDING UP: SEASON 3 | EPISODE 13 Mike (00:09.127) Welcome to the podcast Shelley. Thank you so much for joining us today. Shelly (00:12.956) Thank you, Mike, for having me. Mike (00:16.078) So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond? Shelly (00:30.3) There's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students, we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words, our youngest learners are just starting to connect symbolic representations to mathematical ideas, let alone, you know, put letters together to make words. So When we think of these assessments, we need to take into consideration that primary students are in the early stages of development with respect to their language, their reading, and their writing skills. And this in itself makes it challenging for them to fully articulate, write, sketch any of their mathematical thinking. So we often find that with young children in reviews, you know, individual interviews can be really helpful. But even then, there's some drawbacks. Some children find it challenging, you know, to be put on the spot, to show in the moment, you know, on demand, you know, what they know. Others, you know, just aren't fully engaged or interested because you've called them over from something that they're busy doing. Or maybe, you know, they're not yet comfortable with the setting or even the person doing the interview. So when we work with young children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving. And this is why Shelly (02:47.42) they often or some may respond differently to the same proper question, especially if the setting or the context is changed. We may find that a kindergarten student who counts to 29 on Monday may count to 69 or even 100 later in the week, kind of depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent. informative and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? You know, what do we do? You know, we're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines. And it's these authentic situations that we can look at as assessments to help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends. Mike (04:11.832) So I wanna go back to something you said and even in particular the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words do. Talk a little bit about what you were trying to convey with that, Shelley. Shelly (04:27.548) So young children are doers. When they work on a math task, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting, if they group their objects, how they line them up, do they tag them, do they move them as they count them. They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding, finger counting, and even how they cluster or match objects. So I'm going to give you an example. So let's say that I'm watching some early first graders, and they're solving the expression 6 plus 7. And the first student picks up a number rack or a rec and rec. And if you're not familiar with a number rack, it's a tool with two rows of beads. And on the first row, there are five red beads and five white beads. And on the second row, there's five red beads and five white beads. And the student solving six plus seven begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the seven. They push over five red beads and two white beads. And they haven't said a word to me. I'm just watching their actions. And I'm already able to tell, hmm, that student could subitize a group of five, because I saw him push over all five beads in one push. And that they know that six is composed of five and one, and seven is composed of five and two. And they haven't said a word. I'm just watching what they're doing. And then I might watch the student, and they see it. I see him pause, know, nothing's being said, but I start to notice this slight little head nodding. Shelly (06:26.748) And then they say 13 and they give me the answer and they're really pleased. I didn't get a lot of language from them, but boy, did I get a lot from watching how they solve that problem. And I want to contrast that observation with a student who might be solving the same expression six plus seven and they might go six and then they start popping up one finger at a time while counting seven, eight. 9, 10, 11, 12, 13. And when they get seven fingers held up, they say 13 again. They've approached that problem quite differently. But again, I get that information that they understood the equation. They were able to count on starting with six. And they kept track of their count with their fingers. And they knew to stop when seven fingers were raised. And I might even have a different student that solves the problem by thinking, hmm, and they talk to themselves or they know I'm watching and they might start talking to me. And they say, well, 6 plus 6 is 12 and 7 is 1 more than 6. So the answer is 16 or 13. And if this were being done on a paper pencil as an assessment item or they were answering on some kind of a device, all I would know about my students is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer. What skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense? Mike (08:20.84) Absolutely. I think the, the way that you described this really attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening. It makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying. and sometimes they're written communication, and there was a lot that I could have also taken in if I was attending to those things in a little bit more depth. It also strikes me that this might feel a little bit overwhelming for an educator. How do you think about what an educator, let me back that up. How can an educator know what they're looking for? Shelly (09:17.5) to start, Mike, by honoring your feelings, because I do think it can feel overwhelming at first. But as teachers begin to make informal observations, really listening to you and watching students' actions as part of just their daily practice, something that they're doing, you know, just on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect them to do, what they might say next if they see a certain actions. And after several years, let's say teaching kindergarten, if you've been a kindergarten teacher for four, five, six, 20, you know, plus years, you start to notice these patterns of behavior, things that five and six year olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know, you know, what you're looking at. But before you say anything, I know that isn't especially helpful for teachers new to the profession or new to a grade level. And fortunately, we have several researchers that have been, let's say, kid watching for 40, I don't know, 50 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. listeners might, you know, know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So. When I was sharing the thinking and actions of three students solving six plus seven, listeners familiar with cognitively guided instruction, CGI, they might have recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in my first student, they didn't say anything but gave me an answer. Shelly (11:40.068) was using direct modeling. We saw them push over five and one beads for six and then five and two beads for seven and then kind of pause at their model. And I could tell, you know, with their head nodding that they were counting quietly in their head, counting all the beads to get the answer. And, you know, that's kind of one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student. coming next time, I might work on the second strategy that I conveyed with my second student where they were able to count on. They started with that six and then they counted seven more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding. was shown with my third student when they were able to use a derived fact strategy. The student said, well, I know that 6 plus 6 is 12. I knew my double fact. And then I used that relationship of knowing that 7 is 1 more than 6. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next. And I can also think about what I might do. And so it's this knowledge of development and progressions and how children learn number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students. And they can use those, you know, to guide their instruction for that student or, you know, look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with. Mike (13:39.17) You know, as a K-1 teacher, I remember that I spent a lot of my time tracking students with things like checklists. You know, so I'd note if students quote unquote had or didn't have a skill. And I think as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning, Shelly (14:09.604) well, I think it's important to know that if, you know, it takes us belief. We have to really think about assessment and children's learning is something that is ongoing and evolving. And if we do, it just kind of becomes part of what we can do every day. We can look for opportunities to observe students skills in authentic settings. Many in the moment. types of assessment opportunities happen when we pose a question to the class and then we kind of scan looking for a response. Maybe it's something that we're having them write down on their whiteboard or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs up or a thumbs down, know, kind of to check in on their understanding. We might not be checking on every student, but we're capturing the one, you know, a few. And we can take note because we're doing this on a daily basis of who we want to check in with. What do we want to see? We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say, for example, that our lesson today includes a dot talk or a number talk, something that we're going to write down. We're going to record student thinking. And so during the lesson, the teacher is going to be busy facilitating the discussion, recording the students thinking, you know, and making all of those notes. But if we write the child's name, kind of honor their thinking and give it that caption on that public record, at the end of the lesson, you know, we can capture a picture, just, you use our phone, use an iPad, quickly take a picture of that student's thinking, and then we can record that. you know, where we're keeping track of our students. So we have, OK, another moment in time. And it's this collection of evidence that we keep kind of growing. We can also, you by capturing these public records, note whose voice and thinking were elevating in the classroom. So it kind of gives us how are they thinking and who are we listening to and making sure that we're kind of spreading that out and hearing everyone. Shelly (16:31.728) I think, Mikey, you checklists that you used. Yeah, and even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking. So one of the things we did in Bridges 3rd edition is we designed additional tools for gathering and recording information during workplaces. Mike (16:35.501) I did. Shelly (16:56.208) That's a routine where students are playing games and or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this is something you're going to do with every student. You know, or even, you know, in one day because these are spanned out over a period of four to six weeks where they can go to these games. And we might even see the students go to these activities multiple times. And so let's say that kindergarten students are playing something like the game Beat You to 10, where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that, I might just want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And so those might, you know, of objects, you know, based on the number, they spin a four, can they count out four? And those might be kinds of skills that you might have had typically on a checklist, right, Mike, for kindergarten? But I could use this activity to kind of gap. gather that note and make any comments. So just for those kids I'm looking at or maybe first graders are playing a game like sort the sum where they're drawing two different dominoes and they're supposed to find how many they have in all. And so with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from the dot? And if one set of the dots on one side and then counting on the other. Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those. So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist-like. But I can target that time to do it on students I want that information by thinking ahead of time. Shelly (19:18.684) What can I get by watching, observing these students at these games? trust, I mean, as you know, young children love it. Older children love it. When the teacher goes over and wants to watch them play, or even better, wants to engage in the game play with them, but I can still use that as an assessment. Mike (19:39.32) think that's really helpful, Shelly, for a couple reasons. First, I think it helps me rethink, like you said, one, getting really a lot clearer on like, love the, I'm gonna back that up. I think one of the things that you said was really powerful is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're gonna have students engage with, be it a game or a, Shelly (19:39.356) and Mike (20:07.96) project or some kind of activity and really thinking like, what can I get from this as a person who's trying to make sense of students thinking? And I think my checklist suddenly feels really different when I've got a clear vision of like, what can I get from this task or this game that students are playing and looking for evidence of that versus feeling like I was pulling kids over one-on-one, which I think I would still do because there's some depth that I might want to capture. But it it changes the way that I think about what I might do and also what I might get out of a task So that that really resonates for me Shelly (20:47.066) Yeah, and I think absolutely, you know, I didn't want to make individual interviews or anything sound bad because we can't do them. just, you there's the downfall of, you know, kids comfort level with that and ask them to do something on demand. But we do want more depth and it's that depth that, you know, we know who we want more depth on because of these informal types of observations that we're gathering on a daily basis in our class. You know, might, says something and we take note I want to touch bases with that thinking or I think I'm going to go observe that child during that workplace or maybe we're seeing some things happening during a game and instead of you know like stopping the game and really doing some in-depth interview with the student at that moment because you need more information I can might I might want to call them over and do that more privately at a different time so you're absolutely on there's a place there's a place for you know both Mike (21:42.466) The other thing that you made me think about is the extent to which, like one of the things that I remember thinking is like, I need to make sure if a student has got it or not got it. And I think what you're making me think can really come out of this experience of observing students in the wild, so to speak, when they're working on a task or with a partner is that I can gather a lot more evidence about the application of that idea. I can see the extent to which students are. doing something like counting on in the context of a game or a task. And maybe that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to kind of see, is this way of thinking something that students are applying in different contexts, or did it just happen at that one particular moment in time when I was with them? So that really helps me think about, I think, how those two... maybe different ways of assessing students, be it one-on-one or observing them and seeing what's happening, kind of support one another. Shelly (22:46.268) think you also made me think, you know, really hit on this idea that students, like I said, you their learning is evolving over time. And it might change with the context so that they, you know,...
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Season 3 | Episode 12 – Inside Out: Examining the Meaning and Purpose of our Questions - Guest: Dr. Victoria Jacobs
02/20/2025
Season 3 | Episode 12 – Inside Out: Examining the Meaning and Purpose of our Questions - Guest: Dr. Victoria Jacobs
Dr. Victoria Jacobs, Examining the Meaning and Purpose of our Questions ROUNDING UP: SEASON 3 | EPISODE 12 Mike (00:03): The questions educators ask their students matter. They can have a profound impact on students' thinking and the shape of their mathematical identities. Today we're examining different types of questions, their purpose and the meaning students make of them. Joining us for this conversation is Dr. Vicki Jacobs from the University of North Carolina Greensboro. Welcome to the podcast, Vicki. I'm really excited to talk with you today. Vicki (00:33): Thanks so much for having me. I'm excited to be here. Mike (00:36): So you've been examining the ways that educators use questioning to explore the details of students' thinking. And I wonder if we could start by having you share what drew you to the topic. Vicki (00:47): For me, it all starts with children's thinking because it's absolutely fascinating, but it's also mathematically rich. And so a core part of good math instruction is when teachers elicit children's ideas and then build instruction based on that. And so questioning obviously plays a big role in that, but it's hard. It's hard to do that well in the moment. So I found questioning to explore children's thinking to be a worthwhile thing to spend time thinking about and working on. Mike (01:17): Well, let's dig into the ideas that have emerged from that work. How can teachers think about the types of questions that they might ask their students? Vicki (01:24): Happy to share. But before I talk about what I've learned about questioning, I really need to acknowledge some of the many people that have helped me learn about questioning over the years. And I want to give a particular shout out to the teachers and researchers in the wonderful cognitively guided instruction or CGI community as well as my long-term research collaborators at San Diego State University. And more recently, Susan Sen. This work isn't done alone, but what have we learned about teacher questioning across a variety of projects? I'll share two big ideas and the first relates to the goals of questioning and the second addresses more directly the types of questions teachers might ask. So let's start with the goals of questioning because there are lots of reasons teachers might ask questions in math classrooms. And one common way to think about the goal of questioning is that we need to direct children to particular strategies during problem solving. (02:23): So if children are stuck or they're headed down a wrong path, we can use questions to redirect them so that they can get to correct answers with particular strategies. Sometimes that may be okay, but when we only do that, we're missing a big opportunity to tap into children's sense-making. Another way to think about the goal of questioning is that we're trying to explore children's thinking during problem solving. So think about a math task where multiple strategies are encouraged and children can approach problem solving in any way that makes sense to. So we can then ask questions that are designed to reveal how children are thinking about the problem solving, not just how well they're executing our strategies. And we can ask these questions when children are stuck, but also when they solve problems correctly. So this shift in the purpose of questioning is huge. And I want to share a quote from a teacher that I think captures the enormity of this shift. (03:26): She's a fifth grade teacher, and what she said was the biggest thing I learned from the professional development was not asking questions to get them to the answers so that I could move them up a strategy, but to understand their thinking. That literally changed my world. It changed everything. So I love this quote because it shows how transformative this shift can be because when teachers become curious about how children are thinking about problem solving, they give children more space to problem solve in multiple ways, and then they can question to understand and support children's ideas. And these types of questions are great because they increase learning opportunities for both children and teachers. So children get more opportunities to learn how to talk math in a way that's meaningful to them because they're talking about their own ideas and they also get to clarify what they did think more about important math that's embedded in their strategies and sometimes to even self-correct. And then as teachers, these types of questions give us a window into children's understandings, and that helps us determine our next steps. Questioning can have a different and powerful purpose when we shift from directing children toward particular strategies to exploring their mathematical thinking. Mike (04:54): I keep going back to the quote that you shared, and I think the details of the why and kind of the difference in the experience for students really jump out. But I'm really compelled by what that teacher said to you about how it changes everything. And I wonder if we could just linger there for a moment and you could talk about some of the things that you've seen happen for educators who have that kind of aha moment in the same way that that teacher did, how that impacts the work that they're doing with children or how they see themselves as an educator. Vicki (05:28): That's a great question. I think it's freeing in some way because it changes how educators think about what their next steps are. Every teacher has lots of pressures from standards and sometimes pacing guides and grade level teams that are working on the same page, all sorts of things that are a big part of teaching. But it puts the focus back on children and children's thinking and that my next steps should then come from there. And so in some ways, I think it gives a clearer direction for how to navigate all those various pressures that teachers have. Mike (06:14): I love that. Let's talk about part two. Vicki (06:17): Sure. So if we have the goal of questioning to explore children's thinking, how do we decide what questions to ask? So first of all, there's never a best question. There are many questioning frameworks out there that can provide lots of ideas, but what we've found is that the most productive questions always start with what children say and do. So that means I can't plan all my questions in advance, and instead I have to pay close attention to what children are saying and doing during problem solving. And to help us with that, we found a distinction between inside questions and outside questions. And that distinction has been really useful to us and also usable even during instruction. So inside questions are questions that explore details that are part of inside children's current strategies. And outside questions are questions that focus on strategies or representations that are not what children have done and may even be linked to how we as teachers are thinking about problem solving. (07:26): So I promised an example, and this is from our recent research project on teaching and learning about fractions. And we asked teachers to think about a child's written strategy for a fraction story problem. And the problem was that there are six children equally sharing four pancakes, and they need to figure out how much pancake each child can get. So we're going to talk about Joy's strategy for solving this problem. She is a fourth grader who solved the problem successfully, but in a complex and rather unconventional way. So I'm going to describe her strategy as a reminder. We have six children sharing four pancakes. So she drew the four pancakes. She split the first three pancakes into fourths and distributed the pieces to the six children, and that works out to two fourths for each child. But now she has a problem because she has one pancake left and fourths aren't going to work anymore because that's not enough pieces for her six children. (08:23): So she split the pancake first into eighths and then into 20 fourths and distributed those pieces. So each child ends up receiving two fourths, one eighth and one 24th. And when you put all those amounts together, they equal the correct amount of two thirds pancake per child. But Joy left her answer in pieces as two fourths, one eighth and one 24th, and she wrote those fractions in words rather than using symbols. Okay, so there's a lot going on in this strategy. And the specific strategy doesn't matter so much for our conversation, but the situation does. Here we have a child who has successfully solved the problem, but how she solved it and how she represented her answer are different than what we as adults typically do. So we ask teachers to think about what kind of follow-up conversation would you want to have with joy? (09:23): What types of questions would you want to ask her? And there were these two main questioning approaches, what we call inside questioning and outside questioning. So let's start with outside questioning. These teachers focused on improving Joy's strategy. So they ask follow-up questions like, is there another way you can share the four pancakes with six children? Or is your strategy the most efficient way you could share the pancakes? Or is there a way to cut bigger servings that would be more efficient? So given the complexity of Joy's strategy, we can appreciate these teachers' goals of helping joy move to a more efficient strategy. But all of these questions are pushing her to use a different strategy. So we considered them outside questions because they were outside of her current strategy. And outside questions can sometimes be productive, but they tend to get overused. And when we use them a lot, they can communicate to kids that what they're actually doing was wrong and that it needs fixing. (10:29): So let's think about the other approach of inside questioning. These teachers started by exploring what Joy had done in all of its complexity. And they ask a variety of questions. Usually it started with a general question, can you tell me what you did? But then they zoomed in on some of the many details. So for examples, they've asked how she split the pancakes. They offered questions like, why did you split the first three pancakes into four pieces? Or Tell me about the last pancake. That was the one that she split into eights and 20 fourths. Or they might ask about how she knew how to name each of the fractional amounts, especially the one 24th, because that's something that many children might've struggled with. And then there were questions about a variety of other details. Some of them are hard to explain without showing you a picture of the strategy, but the point is that the teachers took seriously what Joy had done and elevated it to the focus of the conversation. So Joy had a chance to share her reasoning and reflect on it, and the teachers could better understand Joy's approach to problem solving. So we found this distinction between inside and outside questioning to be useful to teachers and even in the midst of instruction because teachers can quickly check in with themselves. Am I asking an inside question or an outside question? Mike (11:49): Well, I have so many questions about inside and outside questions, but I want to linger on inside questions. What I found myself thinking is that for the learner, there are benefits for building number sense or conceptual understanding. The other thing that strikes me is that inside questions are also an opportunity to support students' math identity. And I wonder if that's something that you've seen in your work with teachers and with students. Vicki (12:14): Absolutely. I love your question. One of my favorite things about inside questions is that children see that their ideas are being taken seriously. And that's so empowering. It helps children believe that they can do math and that they are in charge of their mathematical thinking. I'll share a short story that was memorable for me, and this was from a while ago when I was in graduate school. So I was working on a research project and we were conducting problem solving interviews with young children. And our job was to document their strategies. So if we could see exactly what they did, we were told to write down the strategy and move on. But if we needed to clarify something, we could ask follow up questions. I was working with a first grader who had just spent a really long time solving a story problem. He had solved it successfully, and he had done that by joining many, many, many unifix cubes into a very long train. (13:10): And then he had counted them by ones multiple times. So he had been successful. I could tell exactly what he had done. So I started to move on to the next problem. So this young child looked at me a little incredulous and simply asked, don't you want to know how I did it? And he had come from a class where his math thinking was valued, and talking about children's thinking was a regular part of what they did. So he couldn't quite understand why this adult was not interested in how he had thought about the problem. Well, I was a little embarrassed and of course backtracked and listened to his full explanation. But the interaction stuck with me because it showed me how empowering it was for children to truly be listened to as math thinkers. And I think that's something we want for all children. Mike (14:00): The other thing that's hitting me in that story and in the story of joy is mea culpa. I am a person who has lived in the cult of efficiency where I looked at a student's work and my initial thought was, how do I nip the edges of this to get to more efficiency? But I really am struck by it how different the idea of asking the student to explain their thinking or the why behind it. I find myself thinking about joy, and it appears that she was intent on making sure that there were equal shares for each person. So there's ways that she could build to a different level of efficiency. But I think recognizing that there's something here that is really important to note about how and why she chose that, that would feel really meaningful as a learner. Vicki (14:44): I agree. I think what I like about inside questions is that they encourage us to, that children's thinking makes sense, even if it's different than how we think about it. It's our job to figure out how it makes sense. And then to build from there. Mike (15:03): Can you just say more about that? That feels like kind of a revelation. Vicki (15:08): Well, if we start with how kids are thinking and we take that seriously and we make that the center of the conversation, then we're acknowledging to the student and to ourselves that the child has something meaningful to bring to this conversation. And so we need to figure out how the child is thinking all the kind of kernels of mathematical strength in that thinking. And then yes, we can build from there, but we start with where they are as opposed to how we might solve the problem. Mike (15:49): If you were to offer educators a universal inside question or a few sentence frames for inside questions, is it possible to construct something like that that's generic or do you have other advice for us? Vicki (16:02): So that's a nice trick question. I wish it were that easy. I don't really think there are any universal inside questions. Perhaps the only universal one I can think of is something like, how did you solve this problem? It's a great general open-ended question. That's a good starter question in most situations. But the really powerful questions generally come from noticing mathematically important details in children's strategies. So a sentence stem that has been helpful in our work is, I noticed blank, so I wonder blank. Obviously questions don't have to be phrased exactly like this, but the idea is that we pick something that the child has done in their strategy and ask a question about the child's thinking behind that strategy detail. And that keeps us honest because the question absolutely has to begin with something in the child's strategy rather than inadvertently kind of slipping into our strategy. Mike (17:04): Vicki, what do you think about the purpose of outside questions? Are there circumstances where we would want to ask our students an outside question? Vicki (17:12): Absolutely. Sometimes we need to push children's thinking or share particular ideas, and that's okay. It's not that all outside questions are bad, it's just that we tend to overuse them and we could use them at more productive times. And by that I mean that we generally want to understand children's thinking before nudging their thinking forward with outside questions. So let's go back to the earlier example of Joy. Who was solving that problem about six children sharing four pancakes. And we had the two groups of teachers that had the different approaches to follow up questioning. There was the outside questioning that immediately zeroed in on improving Joy's strategy and the inside questioning that spent time exploring Joy's reasoning behind her strategy. So I'm thinking of two specific teachers right now. One generally took the outside questioning approach and the other inside questioning approach. And what was interesting about this pair was that they both asked the same outside question, could Joy partition the pancakes in a different way? (18:19): But they asked this question at different times and the timing really matters. So the teacher who took an outside questioning approach wanted to begin her conversation that way. She wanted to ask Joy, could she partition in a different way? But in contrast, the teacher who took an inside questioning approach wanted to ask Joy lots of questions about the details of her existing strategy, and then posed this very same question at the end to see if Joy had some new ideas for partitioning after their conversation about her existing strategy. And that feels really different to children. So the exact same question can send children different messages when outside questions are posed. First they communicate to children that what they did was wrong and needs fixing. But when outside questions are posed after a conversation about their thinking, it communicates a puzzle or a problem to be solved. (19:17): And children often are better equipped to consider this new problem having thoroughly discussed their own strategy. So I guess when I think about outside questions, I think of timing and amount. We generally want to start with inside questions, and we want most of our questions to be inside questions, but some outside questions can be productive. It's just that we overuse them. I want to mention one other thing about outside questions, and I think we often need fewer outside questions than we think we do, as long as we have space for children to learn from other children's thinking. So think about a typical lesson structure like launch, explore, discuss where children solve problems independently. And then the lesson concludes with a whole class discussion where children share their strategies and reflect on their problem solving. Will these sharing sessions serve as natural outside questions? Because children get to think about strategies that are outside of their own, but in a way that doesn't point to their own strategy as lacking in some way. So outside questions definitely have a place we just need to think about when we ask them and how many of them are really necessary. Mike (20:34): That is really helpful. I find myself thinking about my own process when I'm working on a problem, be it mathematical or organizational or what have you. When someone asks me to talk about how I've thought about it, engaging in that process in some ways primes me, right? Because I've gotten clearer on my own thinking. I suspect that the person who's asking me the question is also clearer on that, which allows them to ask a...
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Season 3 | Episode 11 – Affirming Students’ Mathematics Identities - Guest: Dr. Karisma Morton
02/06/2025
Season 3 | Episode 11 – Affirming Students’ Mathematics Identities - Guest: Dr. Karisma Morton
Dr. Karisma Morton, Understanding and Supporting Math Identity ROUNDING UP: SEASON 3 | EPISODE 11 In this episode, we will explore the connection between identity and mathematics learning. We’ll examine the factors that may have shaped our own identities and those of our students. We’ll also discuss ways to practice affirming students' identities in mathematics instruction. BIOGRAPHIES Dr. Karisma Morton is an assistant professor of mathematics education at the University of North Texas. Her research explores elementary preservice teachers’ ability to teach mathematics in equitable ways, particularly through the development of their critical racial consciousness. Findings from her research have been published in the Journal for Research in Mathematics Education and Educational Researcher. RESOURCES TRANSCRIPT Mike Wallus: If someone asked you if you were good at math, what would you say, and what justification would you provide for your answer? Regardless of whether you said yes or no, there are some big assumptions baked into this question. In this episode, we're talking with Dr. Karisma Morton about the ways the mathematics identities we formed in childhood impact our instructional practices as adults and how we can support students' mathematical identity formation in the here and now. Welcome to the podcast, Karisma. I am really excited to be talking with you about affirming our students’ mathematics identities. Karisma: Oh, I am really, really excited to be here, Mike. Thank you so much for the invitation to come speak to your audience about this. Mike: As we were preparing for this podcast, one of the things that you mentioned was the need to move away from this idea that there are math people and nonmath people. While it may seem obvious to some folks, I'm wondering if you can talk about why is this such an important thing and what type of stance educators might adopt in its place? Karisma: So, the thing is, there is no such thing as a math person, right? We are all math people. And so, if we want to move away from this idea, it means moving away from the belief that people are inherently good or bad at math. The truth is, we all engage in mathematical activity every single day, whether we realize it or not. We are all mathematicians. And so, the key is, as math teachers, we want to remove that barrier in our classrooms that says that only some students are math capable. In the math classroom, we can begin doing that by leveraging what students know mathematically, how they experience mathematics in their daily life. And then we as educators can then incorporate some of those types of activities into the everyday learning of math in our classrooms. So, the idea is to get students to realize they are capable math doers, that they are math people. And you're showing them the evidence that they are by bringing in what they're already doing. And not just that they are math doers, but that those peers that are also engaged in the classroom with them are capable math doers. And so, breaking down those barriers that say that some students are and some students aren't is really key. So, we are all math people. Mike: I love that sentiment. You know, I've seen you facilitate an activity with educators that I'm hoping that we could replicate on the podcast. You asked educators to sort themselves into one of four groups that best describe their experience when they were a learner of mathematics. And I'm wondering if you could read the categories aloud and then I'm going to ask our listeners to think about the description that best describes their own experiences. Karisma: OK, great. So, there are four groups. And so, if you believe that your experience is one where you dreaded math and you had an overall bad experience with it, then you would choose group 1. If you believe that math was difficult but you could solve problems with tutoring or help, then you would select group 2. If you found that math was easy because you were able to memorize and follow procedures but you had to practice a lot, then you'd be in group 3. And finally, if you had very few difficulties with math or you were kind of considered a math whiz, then you would select group 4. Mike: I had such a strong reaction when I participated in this activity for the first time. So, I have had my own reckoning with this experience, but I wonder what impact you've seen this have on educators. Why do it? What's the impact that you hope it has for someone who's participating? Karisma: Yeah. So, I would say that a key part of promoting that message that we started off talking about is for teachers to go back, to reflect. We have to have that experience of thinking about what it was like for us as math learners. Because oftentimes we go into the classroom and we're like, “All right, I got to do this thing.” But we don't take a minute to reflect: “What was it like for me as a math learner?” And I wanted to first also say that I did not develop this activity. This is not a Karisma original. I did see this presented at a math teacher-educator conference about five years ago by Jennifer Ward. I think she's at Kennesaw State [University] right now. But the premise is the same: We want to give teachers an opportunity to reflect over their own experiences as math learners as a good starting place for helping them to identify with each other and also with the students that they're teaching. And so, whenever I have this activity done, I have each of the participants reflect. And then they have conversations around why they chose what they chose. And this is the opportunity for them to have what we call “windows,” “mirrors,” and “sliding glass doors,” right? So, you either can see yourself in another person's experience and feel like, “Oh, I'm not alone here,” especially if it were a negative experience. Or you may get to see or take a glimpse into what someone else has experienced that was very different from your own and really get a chance to understand what it was like for them. They may have been the math whiz, and you're looking at them like they're an alien that fell from the sky because you're like, “How did that happen,” right? But you can begin to have those kinds of conversations: “Why was it like this for you?” and “It wasn't like that for me.” Or “It was the same for me, but what did it look like in your instance versus my instance?” I honestly feel like sometimes people don't realize that their experience is not necessarily unique, especially if it's coming from a math trauma perspective. Some people don't want to talk about their experience because they feel like it was just theirs. But they sometimes can begin to realize that, “Hey, you had that experience too, and let's kind of break down what that means.” Do you want to be that type of teacher? Do you want to create the type of environment where you felt like you weren't a capable math doer? So powerful, powerful exercise. I encourage your listeners to try it with a group of friends or colleagues at work and really have that conversation. Mike: Gosh, I'm just processing this. One of the things that I keep going back to is you challenging us to discard the idea that some people are inherently good at math and other people are not. And I'm making a connection that if I'm a person who identified with group 1, where I dreaded math and it was really a rough experience, what does it mean for me to discard the idea that some people are inherently good or inherently not good at math versus if I identified as a person who was treated as the math whiz and it came easy for me, again, what's required for me? It feels like there's things that we can agree with on the surface. We can agree that people are not good inherently at mathematics. But I find myself really thinking about how my own experience actually colors my beliefs and my actions, how agreeing to that on the surface and then really digging into how your own experience plays out in your practice or the ways that you interact with kids. There's some work to be done there, it seems like. Karisma: Absolutely. You hit the nail on the head there. It's important to do that work. It's really important for us to take that moment to reflect and think about how our own experience may be impacting how we're teaching mathematics to children. Mike: I think that's a great place to make a shift and talk about areas where teachers could take action to cultivate a positive mathematics identity for kids. I wonder if we can begin by talking about expectations and norms when it comes to problem solving. Karisma: Yes. So, Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin wrote this amazing book, called . And one of those equity-based practices is affirming math learners’ identities. And so, one of the ways we can do this in the math classroom is when having students engaged in problem solving. And so, one of the things that we want to be thinking about when we are having students engaged in math problem solving is we want to be promoting students' persistence and reasoning during problem solving. And you might wonder, “Well, what does that actually look like?” Well, it might be helpful to see what it doesn't look like, right? So, in the typical math classroom, we often see an emphasis on speed: who got it done quickly, who got it done first, who even got it done within the time allotted. And then also this idea of competition. So, that is really hard for kids because we all need time to process and think through our problem-solving strategies. And if we're putting value on speed, and we're putting value on competition, are we in fact putting value on a problem-solving strategy or the process of problem-solving? So, one way to affirm math learners' identities is to move away from this idea of speed and competition and foster the type of environment where we're valuing students' persistence with the problem. We're valuing students' processes in solving a problem, how they're reasoning, how they're justifying their steps or their solutions’ strategies, as opposed to who's getting done quickly. Another thing to be thinking about is reframing making mistakes. There's so many great resources about this. What comes to mind immediately is , which is really helping us to reframe the idea that we can make some mistakes, and we can revise our thinking. We can revise our reasoning, and that's perfectly OK. talks a lot about the right to make a mistake is one of the four rights of the learner in the mathematics classroom. And so, when having kids engaged in problem-solving and mathematics, mistakes should be seen more like what Olga Torres calls “celebrations,” because there are opportunities for learning to occur. We can focus on this mistake and think about and problem-solve through the mistake. “Well, how did we get here?” Use it as a moment that all students can benefit from. And so, kids then become less afraid to make mistakes because they're not ridiculed or made to feel less than because they've done so. Instead, it empowers them to know that “Hey, I made this mistake, but in actuality, this is going to help me learn. And it's also going to help my classmates.” Mike: I suspect a lot of those moments, people really appreciate when there's the “aha!” or the “oh!” What was happening before that might've been some struggle or some misconceptions or a mistake. You're making me think that we kind of have to leave space for those mistakes or those misconceptions to emerge if we really want to have those “aha!”s or those “oh!”s in our classroom. Karisma: That's exactly right. And imagine if you are the one who's like, “Oh!”—what that does for your self-confidence. And even having your peers recognize that you've come to this answer or this understanding. It almost becomes like a collective win if you have fostered a type of environment where it's less about me against you and more about all of us learning together. Mike: The other thing that came to me is that I'm thinking back to the four groups. I would've identified as a person who would fit into group 2, meaning that there were definitely points where math was difficult for me, but I could figure it out with tutoring or with help from a teacher. I start to wonder now how much of my perception was about the fact that it just took me a little bit longer to process and think about it. So, it wasn't that math was difficult. It was that I was measuring my sense of myself in mathematics around whether I was the first person, or I was fast, or I got it right away, or I got it right the first time, as opposed to really thinking about, “Do I understand this?” And to me, that really feels connected to what you're saying, which is the way that we as teachers value students' actions, their rough-draft attempts, their mistakes, and position those as part of the process—that can have a really concrete impact on how I think about myself and also how I think about what it is to do math. Well, let's shift again and talk about another area where educators could support positive identity. I’m thinking about the ways that they can engage with students' background knowledge and their life experiences. Karisma: Hmm, yeah. This is a huge one. And this really, again, comes back to recognizing that our students are whole human beings. They have experiences that we should want to leverage in the math classroom, that they don't need to keep certain parts of themselves at the door when they come in. And so, how do we take advantage of what our students are bringing to the table? And so, we want to be thinking a lot about, “Well, who is the student?” “What do they know?” “What other identities do they hold?” “What's important to them?” “What kinds of experiences do they have in their everyday life that I can bring into the math classroom?” “What are their strengths?” “What do they enjoy doing?” The truth of the matter is really great teachers do this all the time, you know? You know who your students are for the most part, right? And students come to us with a whole host of experiences that we want to leverage and come with all sorts of experiences that we could use in the math classroom. I think oftentimes we don't think about making connections between those things and how to connect them to the mathematics that's happening in the classroom. So, oftentimes we don't necessarily see a reason to connect what we know about our students to mathematics. And so, it's really just a simple extra step because really amazing teachers—which I know they're amazing teachers that are listening right now—you know who your students are. So how do we take what we know about them and bring that into the mathematics learning? Again, as with problem solving, what is it that we want to stay away from? We want to be staying away from connecting math identity only with correct answers and how fast a kid is at solving a problem. Their math identity shouldn't be dependent on how many items they got correct on an assessment. It should be more about, “Well, what is it that they know? And how are we able to use this in the math classroom?” Mike: You're making me think about how oftentimes there's this distinction that happens in people's minds between school math and math that happens everywhere in the real world. Part of what I hear you suggesting is that when you help kids connect to their real world, you're actually doing them another service and that you're helping them see, like, “Oh, these lived experiences that I might not have called mathematics, they are,” right? “I do mathematics. I'm a doer.” And part of our work in bringing that in is helping them see what's already there. Karisma: I love that. Helping them see what's already there. That's exactly right. Mike: Well, before we go, I'm wondering if you could talk about some of the resources that have informed your thinking about this and that you think might also help a person who's listening who wants to keep learning. Karisma: Yeah. There's a lot of great resources out there. The one that I rely on heavily is The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices. I really like this book because it's very accessible. It does a really great job of setting the stage for why we need to be thinking about equity-based practices. And I really enjoy how practical things are. So, the book goes through describing what a representative lesson would look like. And so, it's a really nice blueprint for teachers as they're thinking about students' identities and how to promote positive math identity amongst their students. And then I think we also mentioned Rough Draft Math by Amanda Jansen, which is a good read. And then there's also a new book that came out recently, . And this book goes even deeper by having vignettes and having specific classroom examples of what teaching in this kind of way can look like. So those are three resources off the top of my head that you could dig into and have book clubs at your schools and engage with your fellow educators and grow together. Mike: I think that's a great place to stop. Thank you so much for joining us today. This has really been a pleasure. Karisma: Oh, it's been a pleasure talking to you too. Thank you so much for this opportunity. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2025 The Math Learning Center |
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Season 3 | Episode 10 – Building Productive Partnerships - Guests: Sue Kim and Myuriel von Aspen
01/23/2025
Season 3 | Episode 10 – Building Productive Partnerships - Guests: Sue Kim and Myuriel von Aspen
Sue Kim and Myuriel Von Aspen, Building Productive Partnerships ROUNDING UP: SEASON 3 | EPISODE 10 In this episode, we examine the practice of building productive student partnerships. We’ll talk about ways educators can cultivate joyful and productive partnerships and the role the educator plays once students are engaged with their partner. BIOGRAPHIES Sue Kim is an advocate for children’s thinking and providing them a voice in learning mathematics. She received her teaching credential and master of education from Biola University in Southern California. She has been an educator for 15 years and has taught and coached across TK–5th grade classrooms including Los Angeles Unified School District and El Segundo Unified School District as well as several other Orange County, California, school districts. Myuriel von Aspen believes in fostering collaborative partnerships with teachers with the goal of advancing equitable, high-quality learning opportunities for all children. Myuriel earned a master of arts in teaching and a master of business administration from the University of California, Irvine and a bachelor of science in computer science from Florida International University. She currently serves as a math coordinator of the Teaching, Learning, and Instructional Leadership Collaborative. RESOURCES TRANSCRIPT Mike Wallus: What are the keys to establishing productive student partnerships in an elementary classroom? And how can educators leverage the learning that happens in partnerships for the benefit of the entire class? We'll explore these and other questions with Sue Kim and Myuriel von Aspen from the Orange County Office of Education on this episode of Rounding Up. Well, hi, Sue and Myuriel. Welcome to the podcast. Myuriel von Aspen: Hi, Mike. Sue Kim: Thanks for having us. Mike: Thrilled to have you both. So, I first heard you two talk about the power of student partnerships in a context that involved counting collections. And during that presentation, you all said a few things that I have been thinking about ever since. The first thing that you said was that neuroscience shows that you can't really separate emotions from the way that we learn. And I wonder what do you mean when you say that and why do you think it's important when we're thinking about student partnerships? Myuriel: Yes, absolutely. So, this idea comes directly from neuroscience research, the idea that we cannot build memories without emotions. I'm going to read to you a short quote from the NCTM [National Council of Teachers of Mathematics] publication Catalyzing Change in Early Childhood and Elementary Mathematics that says, “Emerging evidence from neuroscience strongly shows that one cannot separate the learning of mathematics content from children's views and feelings toward mathematics.” So, to me, what that says is that how children feel has a huge influence on their ability to learn math and also on how they feel about themselves as learners of math. So, depending on how they feel, they might be willing to engage in the content or not. And so, as they're engaging in counting collections and they're enjoying counting and they feel joyful and they're doing this with friends, they will learn better because they enjoy it, and they care about what they're doing and what they're learning. Mike: You know, this is a nice segue to the other thing that has been on my mind since I heard you all talk about this because I remember you said that students don't think about a task like counting collections as work, that they see it as play. And I wonder what you think the ramifications of that are for how we approach student partnership? Sue: Yeah, you know, I've been in so many classrooms across TK through fifth [grade], and when I watch kids count collections, we see joy, we see engagement in these ways. But I've also been thinking about this idea of how play is even defined, in a way, since you asked that question that they think of it as play. Kristine Mraz, teacher, author, and a consultant, has [coauthored] a book called Purposeful Play. And I remember this was the first time I hear about this reference about Vivian Paley, an American early childhood educator and researcher, stress through her career, the importance of play for children when she discovered in her work that play’s actually a very complex activity and that it is indeed hard work. It's the work of kids. It's the work of what children do. That's their life, in a sense. And so, something I've been thinking about is how kids perceive play is different than how adults perceive play. And so, they take it with seriousness. There is a complex, very intentionality behind things that they do and say. And so, when we are in our session, and we reference Megan Franke, she says that when young people are engaging with each other's ideas, what they're able to do is mathematically important. But it's also important because they're learning to learn together. They're learning to hear each other. They're developing social and emotional skills as they try and navigate and negotiate each other's ideas. And I think for kids that this could be considered play, and I think that's so fascinating because it's so meaningful to them. And even in a task like counting, they're doing all these complex things. But as adults we see them, and we’re like, “Oh, they're playing.” But they are really thinking deeply about some of these ideas while they're developing these very critical skills that we need to give opportunities for them to develop. Myuriel: I like that idea of leaning into the play that you consider maybe not as serious, but they are. Whether they're playing seriously or not, that you might take that opportunity to make it into a mathematical question or a mathematical reflection. Sue: I totally agree with you. And taking it back to that question that you asked, Mike, about, “How do we approach student partnerships then?” And I think that we need to approach it with this lens of curiosity while we let kids engage in these ways and opportunities of learning to hear each other and develop these social-emotional skills, like we said. And so, when you see kids that we think are “playing” or they're building a tower: How might we enter that space with a lens of curiosity? Because to them, I think it's serious work. We can't just think, “Oh, they're not really in the task” or “They're not doing what they were supposed to do.” But how do we lean into that space with a lens of curiosity as Megan reminded us to do, to see what mathematical things we can tap into? And I think that kids always rise to the occasion. Mike: I love that. So, let's talk about how educators can cultivate joyful and productive student partnerships. I'm going to guess that as is often the case, this starts by examining existing beliefs that I might have and some of my expectations. Sue: Yeah, I think it really begins with your outlook and your identity as a teacher. What's your outlook on what's actually possible for kids in your class? Do you believe that kids as young as 4-year-olds can take on this responsibility of engaging with each other in these intelligent ways? Unless we begin there and we really think and reflect and examine what our beliefs are about that, I think it's hard to go and move beyond that, if that makes sense. And like what we just talked about, it's being open to the curiosity of what could be the capacity of how kids learn. I've seen enough 4-year-olds in TK classrooms doing these big things. They always blow my mind, blow my expectations, when opportunities are given to them and consistently given to them. And it's a process, right? They're not going to start on day one doing some of these more complex things. But they can learn from one another, and they also learn from you as a teacher because they are really paying attention. They are attending to some of these complex ideas that we put in front of them. Mike: Well, you hit on the question that I was thinking about. Because I remember you saying that part of nurturing partnerships starts with a teacher and perhaps a pair of children at a table. Can you all paint a picture of what that might look like for educators who are listening? Sue: Yeah, so actually in one of the most recent classrooms, I went in, and this teacher allowed me to partner with her in this work. She wanted to be able to observe and do it in a structured way so that she could pick up on some details of noticing the things that kids were doing. And so, she would have a collection out, or they got to choose. She was really good about offering choice to kids, another way to really engage them. And so, they would choose. They would come together. And then she started just taking some anecdotal notes on what she heard kids saying, what she saw them doing, what they had to actually navigate through some of the things, the stuck moments that came up. From that, we were able to develop, “OK, what are some goals? We noticed Students A and B doing this and speaking in these ways. What might be the next step that we might want to put into a mini lesson or model out or have them actually share with the class what they were working on mathematically?” Whether it was organization, or how they decided they wanted to represent their count, how they counted and things like that. And so, it was just this really natural process that took place that we were able to really lean into and leverage that kids really responded to because it wasn't someone else's work or a page from a textbook. It was their work, their collection that was meaningful to them and they had a true voice and a stake in that work. Mike: I feel like there have been points in time where my understanding of building groups was almost like an engineering problem, where you needed to model what you wanted kids to do and have them rehearse it so specifically. But I think what sits at the bottom of that approach is more about compliance. And what I loved about what you described, Sue, is a process where you're building on the mathematical assets that kids are showing you during their time together—but also on the social assets that they're showing you. So, in that time when you might be observing a pair or a partnership playing together, working together with something like counting collections, you have a chance to observe the mathematics that's happening. You also have a chance to observe the social assets that you see happening. And you can use that as a way to build for that group, but also to build for the larger group of children. And that just feels really profoundly different than, I think, how I used to think about what it was to build partnerships that were “effective.” Myuriel: You know, Mike, I think it's not only compliance. It's also that control. And what it makes me think about is, when we want to model ourselves what we want students to do, instead of—exactly what you said, looking at what they're doing and bringing that knowledge, those skills, that wisdom that's in the room from the students to show to others so that they feel like their knowledge counts. The teacher is not only the only authority or the only source of knowledge in the room—we bring so much, and we can learn from each other. So, I think it's so much more productive and so effective in developing the identity of students when you are showing something that they're doing to their peers versus you as an adult telling them what to do. Mike: Yeah. Are there any particular resources that you all have found helpful for crafting mini lessons as students are learning about how to become a partnership or to be productive in a partnership? Myuriel: Yes. One book that I love, it's not specific to counting collections, but it does provide opportunities for teachers to create micro-lessons when students are listening and talking to each other. It's Hands Down, Speak Out: Listening and Talking Across Literacy and Math K–5 by Kassia [Omohundro] Wedekind and Christy [Hermann] Thompson. And the reason why I love this book is because it provides, again, these micro-lessons depending on what the teacher is noticing, whether it is that the teacher is noticing that students need support listening to each other or maybe making their ideas clear. Or maybe students need to learn how to ask questions more effectively or even reflect on setting and reflecting on the goals that they have as partners. It does provide ideas for teachers to create those micro-lessons based on what the teacher is noticing. Sue: Yeah, I guess I want to add to that, Mike, as well, the resources that Myuriel said. But also, I think this is something I really learned along the process of walking alongside this teacher, was looking at partnerships through a mathematical lens and then a social lens. And so, the mini lesson could be birthed out of watching kids in one day. It might be a social lens thinking about, “They were kind of stuck because they wanted to choose different collections. What might we do about that?” And that kind of is tied to this problem-solving type of skill and goal that we would want kids to work on. That’s definitely something that's going to come up as kids are working in partnerships. These partnerships are not perfect and pristine all the time. I think that's the nature of the job. And just as humans, they're learning how to get along, they're learning how to communicate and navigate and negotiate these things. And I think those are beautiful opportunities for kids and for teachers, then, to really lean into as goals, as mini lessons that can be out of this. And these mini lessons don't have to be long and drawn out. They can be a quick 5-, 10-minute thing. Or you can pause in the middle of counting and kind of spotlight the fact that “Mike and Brent had this problem, but we want to learn from them because they figured out how to solve it. And this is how. Let's listen to what happened.” So, these natural, not only places in a lesson that these opportunities for teaching can pop up, but that these mini lessons come straight from kids and how they are interacting and how they are taking up partnerships, whether it be mathematical or social. Mike: I think you're helping me address something that if I'm transparent about was challenging for me when I was a classroom teacher. I got a little bit nervous about what was happening and sometimes I would shut things down if I perceived partnerships to be, I don't know, overwhelming or maybe even messy. But you're making me think now that part of this work is actually noticing what are the assets that kids have in their social interactions in the way that they're playing together, collaborating together, the mathematics? And I think that's a big shift in my mind from the way that I was thinking about this work before. And I wonder, first of all, is this something that you all notice that teachers sometimes are challenged by? And two, how you talk to someone who's struggling with that question of like, “Oh my gosh, what's happening in my classroom?” Myuriel: Yes, I can totally understand how teachers might get overwhelmed. We hear this from, not only from teachers trying to do the work of counting collections, but even just using tools for students to problem-solve because it does get messy. I like the way Sue keeps emphasizing how it will be messy. When you have rich mathematical learning happening, and you're using tools and collections and you have 30 students having conversations, it definitely will get messy. But I would say that something that teachers can do to mitigate some of that messiness is to think about the logistics ahead of time and be intentional about what you are planning to do. So, some of the things that they may want to think about is: How are students going to access the counting collections? Where are you going to [put] the tools that they're going to be using? Where physically in the classrooms will students get together to have collections so that they have enough room to spread out and record and talk to each other? And just like Sue was mentioning: How do I partner students so that they do have a good experience, and they support each other? So, all of these things that might cost a bit of chaos if you don't think about them, you can actually think about each one of those ahead of time so that you do have a plan for each one of those. Another thing that teachers may want to consider thinking about is, what do they want to pay attention to when they are facilitating or walking around? There's a lot that they need to pay attention to. Just like Sue mentioned, it is important for them to pay attention to something because you want to bring what's in the room to connect it and have these mini lessons of what students actually need. And also, thinking about after the counting collections: What worked and what didn’t? And what changes do I want to make next time when I do this again? Just so that there is a process of improvement every time. Because as Sue had mentioned, it's not going to happen on day one. You are learning as a teacher, and the students are learning. So, everybody in that room is learning to make this a productive and joyful experience. Sue: Yeah, and another thing that I would definitely remind teachers about is that there's actually research out there about how important it is for kids to engage with one another's mathematical ideas. I'm so thankful that people are researching out there doing this work for us. And this goes along with what Myuriel was saying, but the expectations that we put on ourselves as teachers sometimes are too far. We're our biggest critique-ers of the work that we do. And of course we want things to go well, but to make it more low-risk for yourself. I think that when we lower those stakes, we're more prone to let kids take ownership of working together in these ways, to use language and communication that makes sense while doing math and using these cognitive abilities that are still in the process of developing. And I think they need to remember that it takes time to develop, and it's going to get there. And kids are going to learn. Kids are going to do some really big things with their understanding. But giving [yourself] space, the time to learn along with your students, I think is very critical so that you feel like it's manageable. You feel like you can do it again the next day. Mike: Tell me a little bit about how you have seen educators use things like authentic images or even video to help their students make sense of what it means to work in a partnership. What have you seen teachers do? Sue: Yeah. Not to mention how that is one sure way to get kids engaged. I don't know if you've been in a room full of first graders or kindergartners, but if you put a video image up that's them counting and showing how they are thinking about things, they are one-hundred-percent there with you. They love being acknowledged and recognized as being the doers and the sensemakers of mathematics. And it goes into this idea of how we position kids competently, and this is another way that we can do that. But capturing student thinking in photos or a short clip has really been a powerful tool to get kids to engage in each other's ideas in a deeper way. I think it allows teachers and students to pause and slow down and really focus in on the skill of noticing. I think people forget that noticing is a skill you have to teach. And you have to give opportunities for kids to actually do these things so they can see mathematically what's happening within the freeze-frame of this image, of this collection, and how we might ask questions to help facilitate and guide their thinking to think deeply about these ideas. And so, I've seen...
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Season 3 | Episode 9 – Breaking the Cycle of Math Trauma - Guest: Dr. Kasi Allen
01/09/2025
Season 3 | Episode 9 – Breaking the Cycle of Math Trauma - Guest: Dr. Kasi Allen
Dr. Kasi Allen, Breaking the Cycle of Math Trauma ROUNDING UP: SEASON 3 | EPISODE 9 If you are an educator, you’ve likely heard people say things like “I’m a math person.” While this may make you cringe, if you dig a bit deeper, many people can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you are a math person as well. Today we’re talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. BIOGRAPHY Kasi Allen serves as the vice president of learning and impact at The Ford Family Foundation. She holds a PhD degree in educational policy and a bachelor’s degree in mathematics and its history, both from Stanford University. RESOURCES TRANSCRIPT Mike Wallus: If you're an educator, I'm almost certain you've heard people say things like, “I am not a math person.” While this may make you cringe, if you dig a bit deeper, many of those folks can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you're actually a math person. Today we're talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. Well, hello, Kasi. Welcome to the podcast. Kasi Allen: Hi, Mike. Thanks for having me. Great to be here. Mike: I wonder if we could start by talking about what drew you to the topic of math trauma in the first place? Kasi: Really good question. You know, I've been curious about this topic for almost as long as I can remember, especially about how people's different relationships with math seem to affect their lives and how that starts at a very early age. I think it was around fourth grade for me probably, that I became aware of how much I liked math and how much my best friend and my sister had an absolutely opposite relationship with it—even though we were attending the same school, same teachers, and so on. And I really wanted to understand why that was happening. And honestly, I think that's what made me want to become a high school math teacher. I was convinced I could do it in a way that maybe wouldn't hurt people as much. Or it might even make them like it and feel like they could do anything that they wanted to do. But it wasn't until many years later, as a professor of education, when I was teaching teachers how to teach math, that this topic really resurfaced for me [in] a whole new way among my family, among my friends. And if you're somebody who's taught math, you're the math emergency person. And so, I had collected over the years stories of people's not-so-awesome experiences with math. But it was when I was asked to teach an algebra for elementary teachers course, that was actually the students’ idea. And the idea of this course was that we'd help preservice elementary teachers get a better window into how the math they were teaching was planting the seeds for how people might access algebra later. On the very first day, the first year I taught this class, there were three sections. I passed out the syllabus; in all three sections, the same thing happened. Somebody either started crying in a way that needed consoling by another peer, or they got up and left, or both. And I was just pretty dismayed. I hadn't spoken a word. The syllabi were just sitting on the table. And it really made me want to go after this in a new way. I mean, something—it just made me feel like something different was happening here. This was not the math anxiety that everybody talked about when I was younger. This was definitely different, and it became my passion project: trying to figure how we disrupt that cycle. Mike: Well, I think that's a good segue because I've heard you say that the term “math anxiety” centers this as a problem that's within the person. And that in fact, this isn't about the person. Instead, it's about the experience, something that's happened to people that's causing this type of reaction. Do I have that right, Kasi? Kasi: One hundred percent. And I think this is really important. When I grew up and when I became a teacher, I think that was an era when there was a lot of focus on math anxiety, the prevalence of math anxiety. Sheila Tobias wrote the famous book Overcoming Math Anxiety. This was especially a problem among women. There were dozens of books. And there were a number of problems with that work at the time, and that most of the research people were citing was taking place outside of math education. The work was all really before the field of neuroscience was actually a thing. Lots of deficit thinking that something is wrong with the person who is suffering this anxiety. And most of these books were very self-helpy. And so, not only is there something wrong with you, but you need to fix it yourself. So, it really centers all these negative emotions around math on the person that's experiencing the pain, that something's wrong with them. Whereas math trauma really shifts the focus to say, “No, no, no. This reaction, this emotional reaction, nobody's born that way.” Right? This came from a place, from an experience. And so, math trauma is saying, “No, there's been some series of events, maybe a set of circumstances, that this individual began to see as harmful or threatening, and that it's having long-lasting adverse effects. And that those long-lasting effects, this kind of triggering that starts to happen, is really beginning to affect that person's functioning, their sense of well-being when they're in the presence, in this case, of mathematics.” And I think the thing about trauma is just that. And I have to say in the early days of my doing this research, I was honestly a little bit hesitant to use that word because I didn't want to devalue some of the horrific experiences that people have experienced in times of war, witnessing the murder of a parent or something. But it's about the brain. It's how the brain is responding to the situation. And what I think we know now, even more than when I started this work, is that there is simply trauma [in] everyday life. There are things that we experience that cause our brains to be triggered. And math is unfortunately this subject in school that we require nearly every year of a young person's life. And there are things about the way it's been taught over time that can be humiliating, ridiculing; that can cause people to have just some really negative experiences that then they carry with them into the next year. And so that's really the shift. The shift is instead of labeling somebody as math anxious—“Oh, you poor thing, you better fix yourself”—it's like, “No, we have some prevalence of math trauma, and we've got to figure out how people's experiences with math are causing this kind of a reaction in their bodies and brains.” Mike: I want to take this a little bit further before we start to talk about causes and solutions. This idea that you mentioned of feeling under threat, it made me think that when we're talking about trauma, we are talking about a physiological response. Something is happening within the brain that's being manifested in the body. And I wonder if you could talk just a little bit about what happens to people experiencing trauma? What does that feel like in their body? Kasi: So, this is really important and our brains have evolved over time. We have this incredible processing capacity, and it's coupled with a very powerful filter called the amygdala. And the amygdala [has been] there from eons ago to protect us. It's the filter that says, “Hey, do not provide access to that powerful processor unless I'm safe, unless my needs are met. Otherwise, I gotta focus on being well over here.” So, we're not going to give access to that higher-order thinking unless we're safe. And this is really important because modern imaging has given us really new insights into how we learn and how our body is reacting when our brain gets fired in this way. And so, when somebody is experiencing math trauma, you know it. They sweat. Their face turns red. They cry. Their body and brain are telling them, “Get out. Get away from this thing. It will hurt you.” And I just feel like that is so important for us to remember because the amygdala also becomes increasingly sensitive to repeat negativity. So, it's one thing that you have a bad day in math, or you maybe have a teacher that makes you feel not great about yourself. But day after day, week after week, year after year, that messaging can start to make the amygdala hypersensitive to these sorts of situations. Is that what you were getting at with your question? Mike: It is. And I think you really hit on something. There's this idea of repeat negativity causing increased sensitivity, I think has real ramifications for classroom culture or the importance of the way that I show up as an educator. It's making me think a lot about culture and norms related to math in schools. I'm starting to wonder about the type of traumatizing traditions that we've had in math education that might contribute to this type of experience. What does that make you think? Kasi: Oh, for sure. Unfortunately, I think the list is a little long of the things that we may have been doing completely inadvertently. Everybody wants their students to have a great experience, and I actually think our practices have evolved. But culturally, I think there are some things about math that contribute to these “traumatizing traditions,” is what I've called them. Before we go there, I do want to say just one other thing about this trauma piece, and that is that we've learned about some things about trauma in childhood. And a lot of the trauma in childhood is about not a single life-altering event. But childhood trauma is often about these things that happened repeatedly where a child was being ridiculed, being treated cruelly. And it's about that repetition that is really seeding that trauma so deeply and that sense that they can't stop it, that they don't have control to stop the thing that is causing them pain or suffering. So, I just wanted to make sure that I tagged that because I think there is something about what we've learned about the different forms of childhood trauma that's especially salient in this situation. And so, I'll tie it to your question, which is, think about some of the things we've done in math historically. We don't do them in every place, but the ability grouping that has happened over time, it seems to go in and out of fashion. When a kid is told they're in the lower class, “Oh, this is something you're not good [at]—the slower math.” We often use speed to measure understanding, and so smarter is not faster. And there's some great quotes, Einstein among them. So that's a thing. When you gotta do it right now, it has to be one-hundred-percent right. It has to be superfast. We've often prioritized individual work over collaboration. So, you're all alone in this. In fact, if you're working with others, somehow that's cheating as opposed to collaborating. We teach kids tricks rather than teaching them how to think. And I think we deprive kids of the opportunity to have an idea. It's really hard to get excited about something where all you're doing is reproducing—reproducing something that somebody else thought of as quickly as possible and [it] needs to be one-hundred-percent [accurate]. You don't get to bring your own spin to it. And so, we focus on answers rather than people's reasoning behind the answers. That can be something that happens as well. And I think one of the things that's always gotten me is that there's only one way. Not only is there only one right answer, but there's only one way to get there, which also contributes to this idea of having to absorb somebody else's thinking rather than actualizing your own. And I absolutely know that most teachers are working to not do as much of these things in their math classrooms. And I want to be sure in having this conversation that—you know, I'm a lover of education and teachers, I taught teachers for many years. This is not about the teachers so much as the sort of culture of math and math education that we were all brought up in. And we've got to figure out how to make math something more so that kids can see themselves in it. And that it's not something that happens in a vacuum and is this performance course rather than a class where you get to solve cool problems that no one knows the exact answer to, or there's the exact right way, or that you get to get your own questions answered. Things you wonder about. That it's a chance to explore. So, I mean, ultimately, I think we just know that there's a lot of negativity that happens around math, and we accept it. And that is perhaps the most traumatizing tradition of all because that kind of repeat negativity we know affects the amygdala. It affects people's ability to access math in the long run. So, we gotta have neutral or better. Mike: So, in the field of psychology, there's this notion of generational trauma, and it's passed from generation to generation. And you're making me wonder if we're facing something similar when it comes to the field of math education. I'm wondering what you think educators might be able to do to reclaim math for themselves, especially if they're a person who potentially does have a traumatic mathematics experience and maybe some of the ways that they might create a different type of experience for their students. Kasi: Yeah, let's talk about each of those. I'm going to talk about one, the multigenerational piece, and then let's talk about how we can help ourselves and our students. One is, I think it's really very possible that that's what we're looking at in terms of math trauma. Culturally, I think we've known for a while that this is happening, with respect to math, that—you know, I've had parents come to back-to-school night and tell me that they're just not a math family. And even jokingly say, “Oh, we're all bad at math, don't be too hard on us,” and all the other things. And so, kids inherit that. And it's very common for kids to have the same attitude towards math that their parents do and also that their teachers do. And that's where I think in my mind, I really want to help every elementary teacher fall in love with math because if we look at the data, I think of any undergraduate major, it's those who major in education who report the highest rates of math anxiety and math trauma. And so, when you think about folks who feel that way about math, then being in charge of teaching it to kids in the early years, that's a lot to carry. And so, we want to give those teachers and anyone who has had this experience with math an opportunity to reclaim, regroup. And in my experience, what I've found is actually simply shifting the location of the problem is a really strong first step. When people understand that they actually aren't broken, that the feelings that they have about math don't reflect some sort of flaw in them as a human, but that it's a result of something they've experienced, a lot is unlocked. And most folks that I have worked with over my time working on this issue, they know. They know exactly the moment. They know the set of experiences that led to the reactions that they feel in their body. They can name it, and with actually fairly startling detail. So, in my teaching—and I think this is something anybody can do—is they would write a “mathography.” What is the story of your life through a math lens? What has been the story of your relationship with math over the course of your life and what windows does that give you into the places where you might need to heal? We've never had more tools to go back and sort of relearn areas of math that we thought we couldn't learn. And so often the trauma points are as math becomes more abstract. So many people have something that happened around fractions or multidigit multiplication and division. When we started—we get letters involved in math. I had somebody say, “Math was great as long as it was numbers. Then we got letters involved, and it was terrible.” And so, if people can locate, “This is where I had the problem. It's not me. I can go back and relearn some things.” I feel like that's a lot of the healing, and that, in fact, if I'm a teacher or if I'm a parent, I love my kids, whether they're my children or my students, and I'm going to work on me so that they have a better experience than I had. And I've found so many teachers embrace that idea and go to work. So, some of the things that can happen in classrooms that I think fall from this is that, first of all, the recognition that emotional safety, you can't have cognition and problem solving without it. If you have kids in your classroom who have had these negative experiences in math, you're going to need to help them unpack those and level set in order to move on. And “mathography” is also a good tool for that. Some people use breathing. Making sure that when you encounter kids that are exhibiting math anxiety, that you help them localize the problem outside of them. No one is born with math anxiety. It's the math of school that creates it. And if we ignore it, it's just going to get worse. So, some people feel like they can kind of smooth it over. I think we need to give kids the tools to unpack it and move beyond it. But it's so widespread, and I've encountered teachers who were afraid to go there. It's like the Pandora's box. My advice to them is that if you'll open the box and heal what's inside, the teaching becomes much easier. Whereas if you don't, you're fighting that uphill battle all the time. You know, students will feel more safe in classrooms where mistakes are opportunities to learn; where they're not a bad thing and where they see each other as resources, where they are not alone, and where they can collaborate and really take responsibility for each other's learning. So, some of the most powerful classrooms I've seen where there were a lot of kids who had very negative experiences with math, a teacher had succeeded in creating this learning environment, this community of learners where all the kids seem to recognize that somebody would have a good day, someone else would have a not good day, but it would be their turn for a good day a few days from now. [chuckles] So, we're all just going to take care of each other as we go. I think some things that teachers can keep a particular eye on is being sure that kids are given authentic work to do in math. It's really easy to start giving kids what we've called busywork, but work that really isn't engaging their brain. And it turns out that that boredom cycle triggers the negativity cycle, which can actually get your amygdala operating in a way that is not as far from trauma as we might all like to think. And so, while it isn't the same kind of math trauma that we're talking about here, it does affect the amygdala. And so that's something we should be aware of. And so, this is something—I think kids should learn about their brains in school. I don't know if it's the math teacher's job. But if they haven't learned about their brains yet, when you get them, I would recommend teaching kids about their brains, teaching them strategies for when they feel that kind of shutdown, that headache, like “I can't think.” Because most of the time, they actually can't. And they need to have some kind of reset. Another tip, just in terms of disrupting that trauma cycle in the classroom, is that by the time kids get to be third, fourth grade and up, they know who is good at math, or they've labeled each other. You know, “Who's good at math? Who's struggled?” Even if they are not tracked and sorted, they've...
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Season 3 | Episode 8 – Helping Our Students Build a Meaningful Understanding of Geometry - Guest: Dr. Rebecca Ambrose
12/19/2024
Season 3 | Episode 8 – Helping Our Students Build a Meaningful Understanding of Geometry - Guest: Dr. Rebecca Ambrose
Dr. Rebecca Ambrose, Helping Our Students Build a Meaningful Understanding of Geometry ROUNDING UP: SEASON 3 | EPISODE 8 As a field, mathematics education has come a long way over the past few years in describing the ways students come to understand number, quantity, place value, and even fractions. But when it comes to geometry, particularly concepts involving shape, it’s often less clear how student thinking develops. Today, we’re talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry. BIOGRAPHIES Rebecca Ambrose researches how children solve mathematics problems and works with teachers to apply what she has learned about the informal strategies children employ to differentiate and improve instruction in math. She is currently a professor at the University of California, Davis in the School of Education. RESOURCES TRANSCRIPT Mike Wallus: As a field, mathematics education has come a long way over the past few years in describing the ways that students come to understand number, place value, and even fractions. But when it comes to geometry, especially concepts involving shape, it's often less clear how student thinking develops. Today, we're talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry. Well, welcome to the podcast, Rebecca. Thank you so much for joining us today. Rebecca Ambrose: It's nice to be here. I appreciate the invitation. Mike: So, I'd like to start by asking: What led you to focus your work on the ways that students build a meaningful understanding of geometry, particularly shape? Rebecca: So, I taught middle school math for 10 years. And the first seven years were in coed classrooms. And I was always struck by especially the girls who were actually very successful in math, but they would tell me, “I like you, Ms. Ambrose, but I don't like math. I'm not going to continue to pursue it.” And I found that troubling, and I also found it troubling that they were not as involved in class discussion. And I went for three years and taught at an all-girls school so I could see what difference it made. And we did have more student voice in those classrooms, but I still had some very successful students who told me the same thing. So, I was really concerned that we were doing something wrong and that led me to graduate school with a focus on gender issues in math education. And I had the blessing of studying with Elizabeth Fennema, who was really the pioneer in studying gender issues in math education. And as I started studying with her, I learned that the one area that females tended to underperform males on aptitude tests—not achievement tests, but aptitude tests—was in the area of spatial reasoning. And you'll remember those are the tests, or items that you may have had where you have one view of a shape and then you have a choice of four other views, and you have to choose the one that is the same shape from a different view. And those particular tasks we see consistent gender differences on. I became convinced it was because we didn't give kids enough opportunity to engage in that kind of activity at school. You either had some strengths there or not, and because of the play activity of boys, that may be why some of them are more successful at that than others. And then the other thing that informed that was when I was teaching middle school, and I did do a few spatial activities, kids would emerge with talents that I was unaware of. So, I remember in particular this [student,] Stacy, who was an eighth-grader who was kind of a good worker and was able to learn along with the rest of the class, but she didn't stand out as particularly interested or gifted in mathematics. And yet, when we started doing these spatial tasks, and I pulled out my spatial puzzles, she was all over it. And she was doing things much more quickly than I could. And I said, “Stacy, wow.” She said, “Oh, I love this stuff, and I do it at home.” And she wasn't the kind of kid to ever draw attention to herself, but when I saw, “Oh, this is a side of Stacy that I didn't know about, and it is very pertinent to mathematics. And she needs to know what doorways could be open to her that would employ these skills that she has and also to help her shine in front of her classmates.” So, that made me really curious about what we could do to provide kids with more opportunities like that little piece that I gave her and her classmates back in the day. So, that's what led me to look at geometry thinking. And the more that I have had my opportunities to dabble with teachers and kids, people have a real appetite for it. There are always a couple of people who go, “Ooh.” But many more who are just so eager to do something in addition to number that we can call mathematics. Mike: You know, I'm thinking about our conversation before we set up and started to record the formal podcast today. And during that conversation you asked me a question that involved kites, and I'm wondering if you might ask that question again for our listeners. Rebecca: I'm going to invite you to do a mental challenge. And the way you think about it might be quite revealing to how you engage in both geometric and spatial reasoning. So, I invite you to picture in your mind's eye a kite and then to describe to me what you're seeing. Mike: So, I see two equilateral triangles that are joined at their bases—although as I say the word “bases,” I realize that could also lead to some follow-up questions. And then I see one wooden line that bisects those two triangles from top to bottom and another wooden line that bisects them along what I would call their bases. Rebecca: OK, I'm trying to imagine with you. So, you have two equilateral triangles that—a different way of saying it might be they share a side? Mike: They do share a side. Yes. Rebecca: OK. And then tell me again about these wooden parts. Mike: So, when I think about the kite, I imagine that there is a point at the top of the kite and a point at the bottom of the kite. And there's a wooden piece that runs from the point at the top down to the point at the bottom. And it cuts right through the middle. So, essentially, if you were thinking about the two triangles forming something that looked like a diamond, there would be a line that cut right from the top to the bottom point. Rebecca: OK. Mike: And then, likewise, there would be another wooden piece running from the point on one side to the point on the other side. So essentially, the triangles would be cut in half, but then there would also be a piece of wood that would essentially separate each triangle from the other along the two sides that they shared. Rebecca: OK. One thing that I noticed was you used a lot of mathematical ideas, and we don't always see that in children. And I hope that the listeners engaged in that activity themselves and maybe even stopped for a moment to sort of picture it before they started trying to process what you said so that they would just kind of play with this challenge of taking what you're seeing in your mind's eye and trying to articulate in words what that looks like. And that's a whole mathematical task in and of itself. And the way that you engaged in it was from a fairly high level of mathematics. And so, one of the things that I hope that task sort of illustrates is how a.) geometry involves these images that we have. And that we are often having to develop that concept image, this way of imagining it in our visual domain, in our brain. And almost everybody has it. And some people call it “the mind's eye.” Three percent of the population apparently don't have it—but the fact that 97 percent do suggests for teachers that they can depend on almost every child being able to at least close their eyes and picture that kite. I was strategic in choosing the kite rather than asking you to picture a rectangle or a hexagon or something like that because the kite is a mathematical idea that some mathematicians talk about, but it's also this real-world thing that we have some experiences with. And so, one of the things that that particular exercise does is highlight how we have these prototypes, these single images that we associate with particular words. And that's our starting point for instruction with children, for helping them to build up their mathematical ideas about these shapes. Having a mental image and then describing the mental image is where we put language to these math ideas. And the prototypes can be very helpful, but sometimes, especially for young children, when they believe that a triangle is an equilateral triangle that's sitting on, you know, the horizontal—one side is basically its base, the word that you used—they've got that mental picture. But that is not associated with any other triangles. So, if something looks more or less like that prototype, they'll say, “Yeah, that's a triangle.” But when we start showing them some things that are very different from that, but that mathematicians would call triangles, they're not always successful at recognizing those as triangles. And then if we also show them something that has curved sides or a jagged side but has that nice 60-degree angle on the top, they'll say, “Oh yeah, that's close enough to my prototype that we'll call that a triangle.” So, part of what we are doing when we are engaging kids in these conversations is helping them to attend to the precision that mathematicians always use. And that's one of our standards. And as I've done more work with talking to kids about these geometric shapes, I realize it's about helping them to be very clear about when they are referring to something, what it is they're referring to. So, I listen very carefully to, “Are they saying ‘this’ and ‘that’ and pointing to something?” That communicates their idea, but it would be more precise as like, I have to ask you to repeat what you were telling me so that I knew exactly what you were talking about. And in this domain, where we don't have access to a picture to point to, we have to be more precise. And that's part of this geometric learning that we're trying to advance. Mike: So, this is bringing a lot of questions for me. The first one that I want to unpack is, you talked about the idea that when we're accessing the mind's eye, there's potentially a prototype of a shape that we see in our mind's eye. Tell me more about what you mean when you say “a prototype.” Rebecca: The way that that word is used more generally, as often when people are designing something, they build a prototype. So, it's sort of the iconic image that goes with a particular idea. Mike: You're making me think about when I was teaching kindergarten and first grade, we had colored pattern blocks that we use quite often. And often when we talked about triangles, what the students would describe or what I believed was the prototype in their mind's eye really matched up with that. So, they saw the green equilateral triangle. And when we said trapezoid, it looked like the red trapezoid, right? And so, what you're making me think about is the extent to which having a prototype is useful, but if you only have one prototype, it might also be limiting. Rebecca: Exactly. And when we're talking to a 3- or a 4-year-old, and we're pointing to something and saying, “That's a triangle,” they don't know what aspect of it makes it a triangle. So, does it have to be green? Does it have to be that particular size? So, we’ll both understand each other when we're talking about that pattern block. But when we're looking at something that's much different, they may not know what aspect of it is making me call it a triangle” And they may experience a lot of dissonance if I'm telling them that—I'm trying to think of a non-equilateral triangle that we might all, “Oh, well, let’s”—and I'm thinking of 3-D shapes, like an ice cream cone. Well, that's got a triangular-ish shape, but it's not a triangle. But if we can imagine that sort of is isosceles triangle with two long sides and a shorter side, if I start calling that a triangle or if I show a child that kind of isosceles triangle and I say, “Oh, what's that?” And they say, “I don't know.” So, we have to help them come to terms with that dissonance that's going to come from me calling something a triangle that they're not familiar with calling a triangle. And sadly, that moment of dissonance from which Piaget tells us learning occurs, doesn't happen enough in the elementary school classroom. Kids are often given equilateral triangles or maybe a right triangle. But they're not often seeing that unusual triangle that I described. So, they're not bumping into that dissonance that'll help them to work through, “Well, what makes something a triangle? What counts and what doesn't count?” And that's where the geometry part comes in that goes beyond just spatial visualization and using your mind's eye, but actually applying these properties and figuring out when do they apply and when do they not apply. Mike: I think this is probably a good place to shift and ask you: What do we know as a field about how students' ideas about shape initially emerge and how they mature over time? Rebecca: Well, that's an interesting question because we have our theory about how they would develop under the excellent teaching conditions, and we haven't had very many opportunities to confirm that theory because geometry is so overlooked in the elementary school classroom. So, I'm going to theorize about how they develop based on my own experience and my reading of the literature on very specific examples of trying to teach kids about squares and rectangles. Or, in my case, trying to see how they describe three-dimensional shapes that they may have built from polydrons. So, their thinking tends to start at a very visual level. And like in the kite example, they might say, “It looks like a diamond”—and you actually said that at one point—but not go farther from there. So, you decomposed your kite, and you decomposed it a lot. You said it has two equilateral triangles and then it has those—mathematicians would call [them] diagonals. So, you were skipping several levels in doing that. So, I'll give you the intermediate levels using that kite example. So, one thing a child might say is that “I'm seeing two short sides and two long sides.” So, in that case, they're starting to decompose the kite into component parts. And as we help them to learn about those component parts, they might say, “Oh, it's got a couple of different angles.” And again, that's a different thing to pay attention to. That's a component part that would be the beginning of them doing what Battista called spatial structuring. Michael Battista built on the van Hiele levels to try to capture this theory about how kids’ thinking might develop. So, attention to component parts is the first place that we see them making some advances. And then the next is if they're able to talk about relationships between those component parts. So, in the case of the kite, they might say, “Oh, the two short sides are equal to each other”—so, there's a relationship there—“and they're connected to each other at the top.” And I think you said something about that. “And then the long sides are also connected to each other.” And that's looking at how the sides are related to the other sides is where the component parts start getting to become a new part. So, it's like decomposing and recomposing, which is part of all of mathematics. And then the last stage is when they're able to put the shapes themselves into the hierarchy that we have. So, for example, in the kite case, they might say, “It's got four sides, so it's a quadrilateral. But it's not a parallelogram because none of the four sides are parallel to each other.” So now I'm not just looking at component parts and their relations, but I'm using those relations to think about the definition of that shape. So, I would never expect a kid to be able to tell me, “Oh yeah, a kite is a quadrilateral that is not a parallelogram,” and then tell me about the angles and tell me about the sides without a lot of experience describing shapes. Mike: There are a few things that are popping out for me when I'm listening to you talk about this. One of them is the real importance of language and attempting to use language to build a meaningful description or to make sense of shape. The other piece that it really makes me think about is the prototypes, as you described them, are a useful starting place. They’re something to build on. But there's real importance in showing a wide variety of shapes or even “almost-shapes.” I can imagine a triangle that is a triangle in every respect except for the fact that it's not a closed shape. Maybe there's an opening or a triangle that has wavy sides that are connected at three points. Or an obtuse triangle. Being able to see multiple examples and nonexamples feels like a really important part of helping kids actually find the language but also get to the essence of, “What is a triangle?” Tell me if I'm on point or off base when I'm thinking about that, Rebecca. Rebecca: You are right on target. And in fact, Clements and Sarama wrote a piece in the NCTM Teaching Children Mathematics in about 2000 where they describe their study that found exactly what you said. And they make a recommendation that kids do have opportunities to see all kinds of examples. And one way that that can happen is if they're using dynamic geometry software. So, for example, Polypad, I was just playing with it, and you can create a three-sided figure and then drag around one of the points and see all these different triangles. And the class could have a discussion about, “Are all of these triangles? Well, that looks like a weird triangle. I've never seen that before.” And today I was just playing around with the idea of having kids create a favorite triangle in Polypad and then make copies of it and compose new shapes out of their favorite triangle. What I like about that task, and I think can be a design principle for a teacher who wants to play around with these ideas and get creative with them, is to give kids opportunities to use their creativity in making new kinds of shapes and having a sense of ownership over those creations. And then using those creations as a topic of conversation for other kids. So, they have to treat their classmates as contributors to their mathematics learning, and they're all getting an opportunity to have kind of an aesthetic experience. I think that's the beauty of geometry. It's using a different part of our brain. Thomas West talks about , and he describes people like Einstein and others who really solved problems visually. They didn't use numbers. They used pictures. And Ian Robertson talks about . So, his work is more focused on how we all could benefit from being able to visualize things. And actually, our fallback might be to engage our mind's eye instead of always wanting to talk [chuckles] about things. That brings us back to this language idea. And I think language is very important. But maybe we need to stretch it to communication. I want to engage kids in sharing with me what they notice and what they see, but it may be embodied as much as it is verbal. So, we might use our arms and our elbow to discuss angle. And well, we'll put words to it. We're also then experiencing it in our body and showing it to each other in a different way than [...] just the words and the pictures on the paper. So, people are just beginning to explore this idea of gesture. But I have seen, I worked with a teacher who was working with first graders and they...
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Season 3 | Episode 7 – How You Say It Matters: Teacher Language Choices That Support Number Sense - Guest: Dr. James Brickwedde
12/05/2024
Season 3 | Episode 7 – How You Say It Matters: Teacher Language Choices That Support Number Sense - Guest: Dr. James Brickwedde
Dr. James Brickwedde, How You Say It Matters: Teacher Language Choices That Support Number Sense ROUNDING UP: SEASON 3 | EPISODE 7 Carry the 1. Add a 0. Cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained that many educators use it without even thinking. But what’s the long-term impact of language like this on the development of our students’ number sense? Today, we’re talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. BIOGRAPHIES James Brickwedde is the director of the Project for Elementary Mathematics. He served on the faculty of Hamline University’s School of Education & Leadership from 2011–2021, supporting teacher candidates in their content and pedagogy coursework in elementary mathematics. RESOURCES TRANSCRIPT Mike Wallus: Carry the 1, add a 0, cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained, we often use it without even thinking. But what's the long-term impact of language like this on our students’ number sense? Today we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. Welcome to the podcast, James. I'm excited to be talking with you today. James Brickwedde: Glad to be here. Mike: Well, I want to start with something that you said as we were preparing for this podcast. You described how an educator’s language can play a critical role in helping students think in value rather than digits. And I'm wondering if you can start by explaining what you mean when you say that. James: Well, thinking first of primary students—so, kindergarten, second grade, that age bracket—kindergartners, in particular, come to school thinking that numbers are just piles of ones. They're trying to figure out the standard order. They're trying to figure out cardinality. There are a lot of those initial counting principles that lead to strong number sense that they are trying to integrate neurologically. And so, one of the goals of kindergarten, first grade, and above is to build the solid quantity sense—number sense—of how one number is relative to the next number in terms of its size, magnitude, et cetera. And then as you get beyond 10 and you start dealing with the place value components that are inherent behind our multidigit numbers, it's important for teachers to really think carefully of the language that they're using so that, neurologically, students are connecting the value that goes with the quantities that they're after. So, helping the brain to understand that 23 can be thought of not only as that pile of ones, but I can decompose it into a pile of 20 ones and three ones, and eventually that 20 can be organized into two groups of 10. And so, using manipulatives, tracking your language so that when somebody asks, “How do I write 23?” it's not a 2 and a 3 that you put together, which is what a lot of young children think is happening. But rather, they realize that there's the 20 and the 3. Mike: So, you're making me think about the words in the number sequence that we use to describe quantities. And I wonder about the types of tasks or the language that can help children build a meaningful understanding of whole numbers, like say, 11 or 23. James: The English language is not as kind to our learners [laughs] as other languages around the world are when it comes to multidigit numbers. We have in English 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And when we get beyond 10, we have this unique word called “eleven” and another unique word called “twelve.” And so, they really are words capturing collections of ones really then capturing any sort of tens and ones relationship. There's been a lot of wonderful documentation around the Chinese-based languages. So, that would be Chinese, Japanese, Korean, Vietnamese, Hmong follows the similar language patterns, where when they get after 10, it literally translates as “10, 1,” “10, 2.” When they get to 20, it's “2, 10”—”2, 10, 1,” “2, 10, 2.” And so, the place value language is inherent in the words that they are saying to describe the quantities. The teen numbers, when you get to 13, a lot of young children try to write 13 as “3, 1” because they're trying to follow the language patterns of other numbers where you start left to right. And so, they're bringing meaning to something, which of course is not the social convention. So, the teens are all screwed up in terms of English. Spanish does begin to do some regularizing when they get to 16 because of the name “diez y seis,” so “ten, six.” But prior to that you have, again, sort of more unique names that either don't follow the order of how you write the number or they're unique like 11 and 12 is. Somali is another interesting language in that—and I apologize to anybody who is fluent in that language because I'm hoping I'm going to articulate it correctly—I believe that there, when they get into the teens, it's “1 and 10,” “2 and 10,” is the literal translation. So, while it may not be the “10, 1” sort of order, it still is giving … the fact that there's ten-ness there as you go. So, for the classrooms that I have been in and out of—both [in] my own classroom years ago as well as the ones I still go in and out of now—I try to encourage teachers to tap the language assets that are among their students so that they can use them to think about the English numbers, the English language, that can help them wire that brain so that the various representations—the manipulatives, expanded notation cards or dice, the numbers that I write, how I break the numbers apart, say that 23 is equal to 20 plus 3—all of those models that you're using, and the language that you use to back it up with, is consistent so that, neurologically, those pathways are deeply organized. Piaget, in his learning theory, talks about young children—this is sort of the 10 years and younger—can only really think about one attribute at a time. So that if you start operating on multidigit numbers, and I'm using digitized language, I'm asking that kindergartner, first [grader], second grader to think of two things at the same time. I'm, say, moving a 1 while I also mean 10. What you find, therefore, is when I start scratching the surface of kids who were really procedural-bound, that they really are not reflecting on the values of how they've decomposed the numbers or are reconfiguring the numbers. They're just doing digit manipulation. They may be getting a correct answer, they may be very fast with it, but they've lost track of what values they're tracking. There's been a lot of research on kids’ development of multidigit operations, and it's inherent in that research about students following—the students who are more fluid with it talk in values rather than in digits. And that's the piece that has always caught my attention as a teacher and helped transform how I talked with kids with it. And now as a professional development supporter of teachers, I'm trying to encourage them to incorporate in their practice. Mike: So, I want to hang on to this theme that we're starting to talk about. I'm thinking a lot about the very digit-based language that as a child I learned for adding and subtracting multidigit numbers. So, phrases like, “Carry the 1” or “Borrow something from the 6.” Those were really commonplace. And in many ways, they were tied to this standard algorithm, where a number was stacked on top of another number. And they really obscured the meaning of addition and subtraction. I wonder if we can walk through what it might sound like or what other models might draw out some of the value-based language that we want to model for kids and also that we want kids to eventually adopt when they're operating on numbers. James: A task that I give adults, whether they are parents that I’m out doing a family math night with or my teacher candidates that I have worked with, I have them just build 54 and 38, say, with base ten blocks. And then I say, “How would you quickly add them?” And invariably everybody grabs the tens before they move to the ones. Now your upbringing, my upbringing is the same and still in many classrooms: Students are directed only to start with the ones place. And if you get a new 10, you have to borrow and you have to do all of this exchange kinds of things. But the research shows when school gets out of the way [chuckles] and students and adults are operating on more of their natural number sense, people start with the larger and then move to the smaller. And this has been found around the world. This is not just unique to US classrooms that have been working this way. If, in the standard algorithms—which really grew out of accounting procedures that needed to save space in ledger books out of the 18th, 19th centuries—they are efficient, space-saving means to be able to accurately compute. But in today's world, technology takes over a lot of that bookkeeping type of thing. An analogy I like to make is, in today's world, Bob Cratchit out of [A] Christmas Carol, Charles Dickens’s character, doesn't have a job because technology has taken over everything that he was in charge of. So, in order for Bob Cratchit to have a job, [laughs] he does need to know how to compute. But he really needs to think in values. So, what I try to encourage educators to loosen up their practice is to say, “If I'm adding 54 plus 38, so if you keep those two numbers in your mind, [chuckles] if I start with the ones and I add 4 and 8, I can get 12.” There's no reason, if I'm working in a vertical format, to not put 12 fully under the line down below, particularly when kids are first learning how to add. But then language-wise, when they go to the tens place, they're adding 50 and 30 to get 80, and the 80 goes under the 12. Now, many teachers will know that's partial sums. That's not the standard algorithm. That is the standard algorithm. The difference between the shortcut of carrying digits is only a space-saving version of partial sums. Once you go to partial sums in a formatting piece, and you're having kids watch their language—and that's a phrase I use constantly in my classrooms—is, it's not a 5 and 3 that you are working with, it's a 50 and a 30. So when you move to the language of value, you allow kids to initially, at least, get well-grounded in the partial sums formatting of their work, the algebra of the connectivity property pops out, the number sense of how I am building the quantities, how I'm adding another 10 to the 80, and then the 2, all of that begins to more fully fall into place. There are some of the longitudinal studies that have come out that students who were using more of the partial sums approach for addition, their place value knowledge fell into place sooner than the students who only did the standard algorithm and used the digitized language. So, I don't mind if a student starts in the ones place, but I want them to watch their language. So, if they're going to put down a 2, they're not carrying a 1—because I'll challenge them on that—is “What did you do to the 12 to just isolate the 2? What's left?” “Oh, you have a 10 up there and the 10 plus the 50 plus the 30 gives me 90.” So, the internal script that they are verbalizing is different than the internal digitized script that you and I and many students still learn today in classrooms around the country. So, that's where the language and the values and the number sense all begin to gel together. And when you get to subtraction, there's a whole other set of language things. So, when I taught first grade and a student would say, “Well, you can't take 8 from 4,” if I still use that 54 and 38 numbers as a reference here, my challenge to them is, “Who said?” Now, my students are in Minnesota. So, Minnesota is at a cultural advantage of knowing what happens in wintertime when temperatures drop below 0. [laughs] And so, I usually have as a representation model in my room, a number line that’s swept around the edges of the room, that started from negative 35 and went to 185. And so, there are kids who've been puzzling about those other numbers on the other side of 0. And so, somebody pops up and says, “Well, you'll get a negative number.” “What do you mean?” And then they whip around and start pointing at that number line and being able to say, “Well, if you're at 4 and you count back 8, you'll be at negative 4.” So, I am not expecting first graders to be able to master the idea of negative integers, but I want them to know the door is open. And there are some students in late first grade and certainly in second grade who start using partial differences where they begin to consciously use … the idea of negative integers. However, there [are] other students, given that same scenario, who think going into the negative numbers is too much of The Twilight Zone. [laughs] They'll say, “Well, I have 4 and I need 8. I don't have enough to take 8 from 4.” And another phrase I ask them is, “Well, what are you short?” And that actually brings us back to the accounting reference point of sort of debit-credit language of, “I'm short 4.” “Well, if you're short 4, we’ll just write ‘minus 4.’” But if they already have subtracted 30 from 50 and have 20, then the question becomes, “Where are you going to get that 4 from?” “Well, you have 20 cookies sitting on that plate there. I'm going to get that 4 out of the 20.” So again, the language around some of these strategies in subtractions shifts kids to think with alternative strategies and algorithms compared to the American standard algorithm that predominates US education. Mike: I think what's interesting about what you just said too is you're making me think about an article. I believe it was “Rules That Expire.” And what strikes me is that this whole notion that you can't take 8 away from 4 is actually a rule that expires once kids do begin to work in integers. And what you're suggesting about subtraction is, “Let's not do that. Let's use language to help them make meaning of, “Well, what if?” As a former Minnesotan, I can definitely validate that when it's 4 degrees outside and the temperature drops 8 degrees, kids can look at a thermometer and that context helps them understand. I suppose if you're a person listening to this in Southern California or Arizona, that might feel a little bit odd. But I would say that I have seen first graders do the same thing. James: And if you are more international travelers, as soon as, say, people in Southern California or southern Arizona step across into Mexico, everything is in Celsius. If those of us in the northern plains go into Canada, everything is in Celsius. And so, you see negative numbers sooner [laughs] than we do in Fahrenheit, but that's another story. Mike: This is a place where I want to talk a little bit about multiplication, particularly this idea of multiplying by 10. Because I personally learned a fairly procedural understanding of what it is to multiply by 10 or 100 or 1,000. And the language of “add a 0” was the language that was my internal script. And for a long time when I was teaching, that was the language that I passed along. You're making me wonder how we could actually help kids build a more meaningful understanding of multiplying by 10 or multiplying by powers of 10. James: I have spent a lot of time with my own research as well as working with teachers about what is practical in the classroom, in terms of their approach to this. First of all, and I've alluded to this earlier, when you start talking in values, et cetera, and allow multiple strategies to emerge with students, the underlying algebraic properties, the properties of operations begin to come to the surface. So, one of the properties is the zero property, [laughs], right? What happens when you add a number to 0 or a 0 to a number? I'm now going to shift more towards a third-grade scenario here. When a student needs to multiply four groups of 30. “I want 30 four times,” if you're using the times language. And they'd say, “Well, I know 3 times 4 is 12 and then I just add a 0.” And that's where I as a teacher reply, “Well, I thought 12 plus 0 is still 12. How could you make it 120?” And they’d say, “Well, because I put it there.” So, I begin to try to create some cognitive dissonance [laughs] over what they're trying to describe, and I do stop and say this to kids: “I see that you recognize a pattern that's happening there. But I want us to explore, and I want you to describe why does that pattern work mathematically?” So, with addition and subtraction, kids learn that they need to decompose the numbers to work on them more readily and efficiently. Same thing when it comes to multiplication. I have to decompose the numbers somehow. So if, for the moment, you come back to, if you can visualize the numbers four groups of 36. Kids would say, “Well, yeah, I have to decompose the 36 into 30 plus 6.” But by them now exploring how to multiply four groups of 30 without being additive and just adding above, which is an early stage to it. But as they become more abstract and thinking more in multiples, I want them to explore the fact that they are decomposing the 30 into factors. Now, factors isn't necessarily a third-grade standard, right? But I want students to understand that that's how they are breaking that number apart. So, I'm left with 4 times 3 times 10. And if they've explored, in this case, the associate of property of multiplication, “Oh, I did that. So, I want to do 4 times 3 because that's easy. I know that. But now I have 12 times 10.” And how can you justify what 12 times 10 is? And that's where students who are starting to move in this place quickly say, “Well, I know 10 tens are 100 and 2 tens are 20, so it's 120.” They can explain it. The explanation sometimes comes longer than the fact that they are able to calculate it in their heads, but the pathway to understanding why it should be in the hundreds is because I have a 10 times a ten there. So that when the numbers now begin to increase to a double digit times a double digit—so now let's make it 42 groups of 36. And I now am faced with, first of all, estimating how large might my number be? If I've gotten students grounded in being able to pull out the factors of 10, I know that I have a double digit times a double digit, I have a factor of 10, a factor of 10. My answer's going to be in the hundreds. How high in the hundreds? In this case with the 42 and 36, 1,200. Because if I grab the largest partial product, then I know my answer is at least above 1,200 or one thousand, two hundred. Again, this is a language issue. It's breaking things into factors of 10 so that the powers of 10 are operated on. So that when I get deeper into fourth grade, and it's a two digit times a three digit, I know that I'm going to have a ten times a hundred. So, my answer's at least going to be up in the thousands. I can grab that information and use it both from an estimation point of view, but also, strategically, to multiply the first partial product or however you are decomposing the number. Because you don't have to always break everything down into their place value components. That's another story and requires a visual [laughs] work to explain that. But going back to your question, the “add the 0,” or as I have heard, some teachers say, “Just append the 0,” they think that that's going to solve the mathematical issue. No, that doesn't. That's still masking why the pattern...
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Season 3 | Episode 5 - Building Asset-Focused Professional Learning Communities - Guests: Summer Pettigrew and Megan Williams
11/26/2024
Season 3 | Episode 5 - Building Asset-Focused Professional Learning Communities - Guests: Summer Pettigrew and Megan Williams
Summer Pettigrew and Megan Williams, Building Asset-Focused Professional Learning Communities ROUNDING UP: SEASON 3 | EPISODE 5 Professional learning communities have been around for a long time, in many different iterations. But what does it look like to schedule and structure professional learning communities that help educators understand and respond to their students' thinking in meaningful ways? Today we’re talking with Summer Pettigrew and Megan Williams from the Charleston County Public Schools about building asset-focused professional learning communities. BIOGRAPHIES Summer Pettigrew serves as an instructional coach at Springfield Elementary School in Charleston, South Carolina. Megan Williams serves as principal at Springfield Elementary School in Charleston, South Carolina. RESOURCES TRANSCRIPT Mike Wallus: Professional learning communities have been around for a long time and in many different iterations. But what does it look like to schedule and structure professional learning communities that actually help educators understand and respond to their students’ thinking in meaningful ways? Today we're talking with Summer Pettigrew and Megan Williams from the Charleston Public Schools about building asset-focused professional learning communities. Hello, Summer and Megan. Welcome to the podcast. I am excited to be talking with you all today about PLCs. Megan Williams: Hi! Summer Pettigrew: Thanks for having us. We're excited to be here. Mike: I'd like to start this conversation in a very practical place: scheduling. So, Megan, I wonder if you could talk just a bit about when and how you schedule PLCs at your building. Megan: Sure. I think it's a great place to start too, because I think without the structure of PLCs in place, you can't really have fabulous PLC meetings. And so, we used to do our PLC meetings once a week during teacher planning periods, and the teachers were having to give up their planning period during the day to come to the PLC meeting. And so, we created a master schedule that gives an hour for PLC each morning. So, we meet with one grade level a day, and then the teachers still have their regular planning period throughout the day. So, we were able to do that by building a time for clubs in the schedule. So, first thing in the morning, depending on your day, so if it's Monday and that's third grade, then the related arts teachers—and that for us is art, music, P.E., guidance, our special areas—they go to the third-grade teachers’ classrooms. The teachers are released to go to PLC, and then the students choose a club. And so, those range from basketball to gardening to fashion to [STEM]. We've had Spanish Club before. So, they participate with the related arts teacher in their chosen club, and then the teachers go to their PLC meeting. And then once that hour is up, then the teachers come back to class. The related arts teachers are released to go get ready for their day. So, everybody still has their planning period, per se, throughout the day. Mike: I think that feels really important, and I just want to linger a little bit longer on it. One of the things that stands out is that you're preserving the planning time on a regular basis. They have that, and they have PLC time in addition to it. Megan: Mm-hmm, correct. And that I think is key because planning time in the middle of the day is critical for making copies, calling parents, calling your doctor to schedule an appointment, using the restroom—those kind of things that people have to do throughout the day. And so, when you have PLC during their planning time, one or the other is not occurring. Either a teacher is not taking care of those things that need to be taken care of on the planning period or they're not engaged in the PLC because they're worried about something else that they've got to do. So, building that time in, it's just like a game changer. Mike: Summer, as a person who’s playing the role of an instructional coach, what impact do you think this way of scheduling has had on educators who are participating in the PLCs that you're facilitating? Summer: Well, it's huge. I have experienced going to a PLC on our planning [period] and just not being one-hundred-percent engaged. And so, I think having the opportunity to provide the time and the space for that during the school day allows the teachers to be more present. And I think that the rate at which we're growing as a staff is expedited because we're able to drill into what we need to drill into without worrying about all the other things that need to happen. So, I think that the scheduling piece has been one of the biggest reasons we've been so successful with our PLCs. Mike: Yeah, I can totally relate to that experience of feeling like I want to be here, present in this moment, and I have 15 things that I need to do to get ready for the next chunk of my day. So, taking away that “if-then,” and instead having an “and” when it comes to PLCs, really just feels like a game changer. Megan: And we were worried at first about the instructional time that was going to be lost from the classroom doing the PLC like this. We really were because we needed to make sure instructional time was maximized and we weren't losing any time. And so, this really was about an hour a week, right, where the teachers aren't directly instructing the kids. But it has not been anything negative at all. Our scores have gone up, our teachers have grown. They love—the kids love going to their clubs. I mean, even the attendance on the grade-level club day is so much better because they love coming in. They start the day really getting that SEL instruction. I mean, that's really a lot of what they're getting in clubs. They're hanging out with each other. They're doing something they love. Mike: Maybe this is a good place to shift and talk a little bit about the structure of the PLCs that are happening. So, I've heard you say that PLCs, as they're designed and functioning right now, they're not for planning; they're instead for teacher collaboration. So, what does that mean? Megan: Well, there's a significant amount of planning that does happen in PLC, but it's not a teacher writing his or her lesson plans for the upcoming week. So, there's planning, but not necessarily specific lesson planning, like, “On Monday I'm doing this; on Tuesday I'm doing this.” It's more looking at the standards, looking at the important skills that are being taught, discussing with each other ways that you do this. “How can I help kids that are struggling? How can I push kids that are higher?” So, teachers are collaborating and planning, but they're not really producing written lesson plans. Mike: Yeah. One of the pieces that you all talked about when we were getting ready for this interview, was this idea that you always start your PLCs with a recognition of the celebrations that are happening in classrooms. I'm wondering if you can talk about what that looks like and the impact it has on the PLCs and the educators who are a part of them. Summer: Yeah. I think our teachers are doing some great things in their classrooms, and I think having the time to share those great things with their colleagues is really important. Just starting the meeting on that positive note tends to lead us in a more productive direction. Mike: You two have also talked to me about the impact of having an opportunity for educators to engage in the math that their students will be doing or looking at common examples of student work and how it shows up in the classroom. I wonder if you could talk about what you see in classrooms and how you think that loops back into the experiences that are happening in PLCs. Summer: Yeah. One of the things that we start off with in our PLCs is looking at student work. And so, teachers are bringing common work examples to the table, and we're looking to see, “What are our students coming with? What's a good starting point for us to build skills, to develop these skills a little bit further to help them be more successful?” And I think a huge part of that is actually doing the work that our students are doing. And so, prior to giving a task to a student, we all saw that together in a couple of different ways. And that's going to give us that opportunity to think about what misconceptions might show up, what questions we might want to ask if we want to push students further, reign them back in a little bit. Just that pre-planning piece with the student math, I think has been very important for us. And so, when we go into classrooms, I'll smile because they kind of look like little miniature PLCs going on. The teacher’s facilitating, the students are looking at strategies of their classmates and having conversations about what's similar, what's different. I think the teachers are modeling with their students that productive practice of looking at the evidence and the student work and talking about how we go about thinking through these problems. Mike: I think the more that I hear you talk about that, I flash back to, Megan, what you said earlier about [how] there is planning that's happening, and there's collaboration. They're planning the questions that they might ask. They're anticipating the things that might come from students. So, while it's not, “I'm writing my lesson for Tuesday,” there is a lot of planning that's coming. It’s just perhaps not as specific as, “This is what we'll do on this particular day.” Am I getting that right? Megan: Yes. You're getting that one-hundred-percent right. Summer has teachers sometimes [take] the assessment at the beginning of a unit. We'll go ahead and take the end-of-unit assessment and the information that you gain from that, just with having the teachers take it and knowing how the kids are going to be assessed, then just in turn makes them better planners for the unit. And there's a lot of good conversation that comes from that. Mike: I mean, in some ways, your PLC design, the word that pops into my head is almost like a “rehearsal” of sorts. Does that analogy seem right? Meghan: It seems right. Summer: And just to add on to that, I think too again, providing that time within the school day for them to look at the math, to do the math, to think about what they want to ask, is like a mini rehearsal. Because typically, when teachers are planning outside of school hours, it's by themselves in a silo. But this just gives that opportunity to talk about all the possibilities together, run through the math together, ask questions if they have them. So, I think that's a decent analogy, yeah. Mike: Yeah. Well, you know what it makes me think about is competitive sports like basketball. As a person who played quite a lot, there are points in time when you start to learn the game that everything feels so fast. And then there are points in time when you've had some experience when you know how to anticipate, where things seem to slow down a little bit. And the analogy is that if you can kind of anticipate what might happen or the meaning of the math that kids are showing you, it gives you a little bit more space in the moment to really think about what you want to do versus just feeling like you have to react. Summer: And I think, too, it keeps you focused on the math at hand. You're constantly thinking about your next teacher move. And so, if you've got that math in your mind and you do get thrown off, you've had an opportunity, like you said, to have a little informal rehearsal with it, and maybe you're not thrown off as badly. [laughs] Mike: Well, one of the things that you’ve both mentioned when we've talked about PLCs is the impact of a program called OGAP. I'm wondering if you can talk about what OGAP is, what it brought to your educators, and how it impacted what’s been happening in PLCs. Megan: I'll start. In terms—OGAP stands for “Ongoing Assessment Project”. Summer can talk about the specifics, but we rolled it out as a whole school. And I think there was power in that: everybody in your school taking the same professional development at the same time, speaking the same language, hearing the same things. And for us, it was just a game changer. Summer: Yeah, I taught elementary math for 12 years before I knew anything about OGAP, and I had no idea what I was doing until OGAP came into my life. All of the light bulbs that went off with this very complex elementary math that I had no idea was a thing, it was just incredible. And so, I think the way that OGAP plays a role in PLCs is that we're constantly using the evidence in our student work to make decisions about what we do next. We're not just plowing through a curriculum, we're looking at the visual models and strategies that Bridges expects of us in that unit. We're coupling it with the content knowledge that we get from OGAP and how students should and could move along this progression. And we're planning really carefully around that, thinking about, “If we give this task and some of our students are still at a less sophisticated strategy and some of our students are at a more sophisticated strategy, how can we use those two examples to bridge that gap for more kids?” And we're really learning from each other's work. It's not the teacher up there saying, “This is how you'd solve this problem.” But it's a really deep dive into the content. And I think the level of confidence that OGAP has brought our teachers as they've learned to teach Bridges has been like a powerhouse for us. Mike: Talk a little bit about the confidence that you see from your teachers who have had an OGAP experience and who are now using a curriculum and implementing it. Can you say more about that? Summer: Yeah. I mean, I think about our PLCs, the collaborative part of it, we're having truly professional conversations. It's centered around the math, truly, and how students think about the math. And so again, not to diminish the need to strategically lesson plan and come up with activities and things, but we're talking really complex stuff in PLCs. And so, when we look at student work and we sort that work on the OGAP progression, depending on what skill we're teaching that week. We're able to really look at, “Gosh, the kid is, he's doing this, but I'm not sure why.” And then we can talk a little bit about, “Well, maybe he's thinking about this strategy, and he got confused with that part of it.” So, it really, again, is just centered around the student thinking. The evidence is in front of us, and we use that to plan accordingly. And I think it just one-ups a typical PLC because our teachers know what they're talking about. There's no question in, “Why am I teaching how to add on an open number line?” We know the reasoning behind it. We know what comes before that. We know what comes after that, and we know the importance of why we're doing it right now. Mike: Megan, I wanted to ask you one more question. You are the instructional leader for the building, the position you hold is principal. I know that Summer is a person who does facilitation of the PLCs. What role do you play or what role do you try to play in PLCs as well? Megan: I try to be present at every single PLC meeting and an active participant. I do all the assessments. I get excited when Summer says, “We're taking a test.” I mean, I do everything that the teachers do. I offer suggestions if I think that I have something valuable to bring to the table. I look at student work. I just do everything with everybody because I like being part of that team. Mike: What impact do you think that that has on the educators who are in the PLC? Megan: I mean, I think it makes teachers feel that their time is valuable. We're valuing their time. It's helpful for me too. When I go into classrooms, I know what I'm looking for. I know which kids I want to work with. Sometimes I'm like, “Ooh, I want to come in and see you do that. That's exciting.” It helps me plan my day, and it helps me know what's going on in the school. And I think it also is just a nonjudgmental, nonconfrontational time for people to ask me questions. I mean, it's part of me trying to be accessible as well. Mike: Summer, as the person who’s the facilitator, how do you think about preparing for the kind of PLCs that you've described? What are some of the things that are important to know as a facilitator or to do in preparation? Summer: So, I typically sort of rehearse myself, if you will, before the PLC kicks off. I will take assessments, I will take screeners. I'll look at screener implementation guides and think about the pieces of that that would be useful for our teachers if they needed to pull some small groups and reengage those kids prior to a unit. What I really think is important though, is that vertical alignment. So, looking at the standards that are coming up in a module, thinking about what came before it: “What does that standard look like in second grade?” if I'm doing a third grade PLC. “What does that standard look like in fourth grade?” Because teachers don't have time to do that on their own, and I think it's really important for that collective efficacy, like, “We're all doing this together. What you did last year matters. What you're doing next year matters, and this is how they tie together.” I kind of started that actually this year, wanting to know more myself about how these standards align to each other and how we can think about Bridges as a ladder among grade levels. Because we were going into classrooms, and teachers were seeing older grade levels doing something that they developed, and that was super exciting for them. And so, having an understanding of how our state standards align in that way just helps them to understand the importance of what they're doing and bring about that efficacy that we all really just need our teachers to own. It's so huge. And just making sure that our students are going to the next grade prepared. Mike: One of the things that I was thinking about as I was listening to you two describe the different facets of this system that you've put together is how to get started. Everything from scheduling to structure to professional learning. There's a lot that goes into making what you all have built successful. I think my question to you all would be, “If someone were listening to this, and they were thinking to themselves, ‘Wow, that's fascinating!’ What are some of the things that you might encourage them to do if they wanted to start to take up some of the ideas that you shared?” Megan: It's very easy to crash and burn by trying to take on too much. And so, I think if you have a long-range plan and an end goal, you need to try to break it into chunks. Just making small changes and doing those small changes consistently. And once they become routine practices, then taking on something new. Mike: Summer, how about you? Summer: Yeah, I think as an instructional coach, one of the things that I learned through OGAP is that our student work is personal. And if we're looking at student work without the mindset of, “We're learning together,” sometimes we can feel a little bit attacked. And so, one of the first things that we did when we were rolling this out and learning how to analyze student work is we looked at student work that wasn't necessarily from our class. We asked teachers to save student work samples. I have folders in my office of different student work samples that we can practice sorting and have conversations about. And that's sort of where we started with it. Looking at work that wasn't necessarily our students’ gave us an opportunity to be a little bit more open about what we wanted to say about it, how we wanted to talk about it. And it really does take some practice to dig into student thinking and figure out,...
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Season 3 | Episode 6 – Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades - Guests: Drs. Jody Guarino and Chepina Rumsey
11/21/2024
Season 3 | Episode 6 – Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades - Guests: Drs. Jody Guarino and Chepina Rumsey
Drs. Jody Guarino and Chepina Rumsey, Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades ROUNDING UP: SEASON 3 | EPISODE 6 Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergartners, first-, or second graders. What would it look like to encourage these practices with our youngest learners? Today, we’ll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2. BIOGRAPHIES Chepina Rumsey, PhD, is an associate professor of mathematics education at the University of Northern Iowa (UNI). Jody Guarino is currently a mathematics coordinator at the Orange County Department of Education and a lecturer at the University of California, Irvine. RESOURCES . Teaching Children Mathematics, 25(4), 208–217. TRANSCRIPT Mike Wallus: Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergarten, first-, and second graders. What would it look like to encourage these practices with our youngest learners? Today, we'll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2. Welcome to the podcast, Chepina and Jody. Thank you so much for joining us today. Jody Guarino: Thank you for having us. Chepina Rumsey: Yeah, thank you. Mike: So, I'm wondering if we can start by talking about the genesis of your work, particularly for students in grades K–2. Jody: Sure. Chepina had written a paper about argumentation, and her paper was situated in a fourth grade class. At the time, I read the article and was so inspired, and I wanted to use it in an upcoming professional learning that I was going to be doing. And I got some pushback with people saying, “Well, how is this relevant to K–2 teachers?” And it really hit me that there was this belief that K–2 students couldn't engage in argumentation. Like, “OK, this paper's great for older kids, but we're not really sure about the young students.” And at the time, there wasn't a lot written on argumentation in primary grades. So, we thought, “Well, let's try some things and really think about, ‘What does it look like in primary grades?’ And let's find some people to learn with.” So, I approached some of my recent graduates from my teacher ed program who were working in primary classrooms and a principal that employed quite a few of them with this idea of, “Could we learn some things together? Could we come and work with your teachers and work with you and just kind of get a sense of what could students do in kindergarten to second grade?” So, we worked with three amazing teachers—Bethany, Rachael, and Christina—in their first years of teaching, and we worked with them monthly for two years. We wanted to learn, “What does it look like in K–2 classrooms?” And each time we met with them, we would learn more and get more and more excited. Little kids are brilliant, but also their teachers were brilliant, taking risks and trying things. I met with one of the teachers last week, and the original students that were part of the book that we've written now are actually in high school. So, it was just such a great learning opportunity for us. Mike: Well, I'll say this, there are many things that I appreciated about the book, about Nurturing Math Curiosity with Learners in Grades K–2, and I think one of the first things was the word “with” that was found in the title. So why “with” learners? What were y'all trying to communicate? Chepina: I'm so glad you asked that, Mike, because that was something really important to us when we were coming up with the title and the theme of the book, the message. So, we think it's really important to nurture curiosity with our students, meaning we can't expect to grow it in them if we're not also growing it in ourselves. So, we see that children are naturally curious and bring these ideas to the classroom. So, the word “with” was important because we want everyone in the classroom to grow more curious together. So, teachers nurturing their own math curiosity along with their students is important to us. One unique opportunity we tried to include in the book is for teachers who are reading it to have opportunities to think about the math and have spaces in the book where they can write their own responses and think deeply along with the vignettes to show them that this is something they can carry to their classroom. Mike: I love that. I wonder if we could talk a little bit about the meaning and the importance of argumentation? In the book, you describe four layers: noticing and wondering, conjecture, justification, and extending ideas. Could you share a brief explanation of those layers? Jody: Absolutely. So, as we started working with teachers, we'd noticed these themes or trends across, or within, all of the classrooms. So, we think about noticing and wondering as a space for students to make observations and ask curious questions. So, as teachers would do whatever activity or do games, they would always ask kids, “What are you noticing?” So, it really gave kids opportunities to just pause and observe things, which then led to questions as well. And when we think about students conjecturing, we think about when they make general statements about observations. So, an example of this could be a child who notices that 3 plus 7 is 10 and 7 plus 3 is 10. So, the child might think, “Oh wait, the order of the addends doesn't matter when adding. And maybe that would even work with other numbers.” So, forming a conjecture like, “This is what I believe to be true.” The next phase is justification, where a student can explain either verbally or with writing or with tools to prove the conjecture. So, in the case of the example that I brought up, 3 plus 7 and 7 plus 3, maybe a student even uses their fingers, where they're saying, “Oh, I have these 3 fingers and these 7 fingers, and whichever fingers I look at first, or whichever number I start with, it doesn't matter. The sum is going to be the same.” So, they would justify in ways like that. I've seen students use counters, just explaining it. Oftentimes, they use language and hand motions and all kinds of things to try to prove what they're saying works. Or sometimes they'll find, just, really look for, “Can I find an example where that doesn't work?” So, just testing their conjecture would be justifying. And then the final stage, extending ideas, could be extending that idea to all numbers. So, in the idea of addition in the commutative property, and they come to discover that they might realize, “Wait a minute, it also works for 1 plus 9 and 9 plus 1.” They could also think, “Does it work for other operations? So, not just with addition, but maybe I can subtract like that too. Does that make a difference if I'm subtracting 5 take away 2 versus 2 take away 5.” So, just this idea of, “Now [that] I've made sense of something, what else does it work with or how can I extend that thinking?” Mike: So, the question that I was wondering about as you were talking is, “How do you think about the relationship between a conjecture and students’ justification?” Jody: I've seen a lot of kids—so, sometimes they make conjectures that they don't even realize are conjectures, and they're like, “Oh, wait a minute, this pattern's happening, and I think I see something.” And so often they're like, “OK, I think that every time you add two numbers together, the sum is greater than the two numbers.” And so, then this whole idea of justifying, we often ask them, “How could you convince someone that that's true?” Or, “Is that always true?” And now they actually have to take and study it and think about, “Is it true? Does it always work?” Which, Mike, in your question, often leads back to another conjecture or refining their conjecture. It's kind of this cyclical process. Mike: That totally makes sense. I was going to use the words “virtuous cycle,” but that absolutely helps me understand that. I wonder if we can go back to the language of conjecture, because that feels really important to get clear on and to both understand and start to build a picture of. So, I wonder if you could offer a definition of conjecture for someone who’s unfamiliar with the term or talk about how students understand conjecture. Chepina: Yeah. So, a conjecture is based on our exploration with the patterns and observations. So, through that exploration, we might have an idea that we believe to be true. We are starting to notice things and some language that students start to use—things like, “Oh, that's always going to work” or “Sometimes we can do that.” So, there starts to be this shift toward an idea that they believe is going to be true. It's often a work in progress, so it needs to be explored more in order to have evidence to justify why that's going to be true. And through that process, we can modify our conjecture, or we might have an idea, like this working idea of a conjecture, that then when we go to justify it, we realize, “Oh, it's not always true the way we thought, so we have to make a change.” So, the conjecture is something that we believe to be true, and then we try to convince other people. So, once we introduce that with young mathematicians, they tend to latch on to that idea that it's this really neat thing to come up with a conjecture. And so, then they often start to come up with them even when we're not asking and get excited about, “Wait, I have a conjecture about the numbers and story problems,” where that wasn't actually where the lesson was going, but then they get excited about it. And that idea that we can take our patterns and observations, create a conjecture, and have this cyclical thing that happens. We had a second grade student make what she called a “conjecture cycle.” So, she drew a circle with arrows and showed, “We can have an idea, we can test it, we can revise it, and we can keep going to create new information.” So, those are some examples of where we've seen conjectures and kids using them and getting excited and what they mean. And yeah, it's been really exciting. Mike: What is hitting me is that this idea of introducing conjectures and making them, it really has the potential to change the way that children understand mathematics. It has the potential to change from, “I'm seeking a particular answer” or “I'm memorizing a procedure” or “I'm doing a thing at a discrete point in time to get a discrete answer.” It feels culturally very different. It changes what we're talking about or what we're thinking about. Does that make sense to the two of you? Chepina: Yeah, it does. And I think it changes how they view themselves. They're mathematicians who are creating knowledge and seeking knowledge rather than memorizing facts. Part of it is we do want them to know their facts—but understand them in this deep way with the structure behind it. And so, they're creating knowledge, not just taking it in from someone else. Mike: I love that. Jody: Yeah, I think that they feel really empowered. Mike: That's a great pivot point. I wonder if the two of you would be willing to share a story from a K–2 classroom that could bring some of the ideas we've been talking about to life for people who are listening. Jody: Sure, I would love to. I got to spend a lot of time in these teachers' classrooms, and one of the days I spent in a first grade [classroom], the teacher was Rachael Gildea, and she had led a choral count with her first graders. And they were counting by [10s] but starting with 8. So, like, “8, 18, 28, 38, 48… .” And as the kids were counting, Rachael was charting. And she was charting it vertically. So, below 8 was written 18, and then 28. And she wrote it as they counted. And one of her students paused and said, “Oh, they're all going to end with 8.” And Rachael took that student's conjecture. So, a lot of other conjectures or a lot of other ideas were shared. Students were sharing things they noticed. “Oh, looking at the tens place, it's counting 1, 2, 3,” and all sorts of things. But this one particular student, who said, “They're all going to end in 8,” Racheal took that student’s—the actual wording—the language that the student had used, and she turned it into the task that the whole class then engaged in. Like, “Oh, this student thought or thinks it's always going to end in 8. That's her conjecture; how can we prove it?” And I happened to be in her classroom the day that they tested it. And it was just a wild scene. So, students were everywhere: at tables, laying down on the carpet, standing in front of the chart, they were examining it or something kind of standing with clipboards. And there was all kinds of buzz in the classroom. And Rachael was down on the carpet with the students listening to them. And there was this group of girls, I think three of them, that sort of screamed out, “We got it!” And Rachael walked over to the girls, and I followed her, and they were using base ten blocks. And they showed her, they had 8 ones, little units, and then they had the 10 sticks. And so, one girl would say, they'd say, “8, 18, 28,” and one of the girls was adding the 10 sticks and almost had this excitement, like she discovered, I don't know, a new universe. It was so exciting. And she was like, “Well, look, you don't ever change them. You don't change the ones, you just keep adding tens.” And it was so magical because Rachael went over there and then right after that she paused the class and she's like, “Come here, everyone. Let's listen to these girls share what they discovered.” And all of the kids were sort of huddled around, and it was just magical. And they had used manipulatives, the base ten blocks, to make sense of the conjecture that came from the choral count. And I thought it was beautiful. And so, I did choral counts in my classroom and never really thought about, “OK, what's that next step beyond, like, ‘Oh, this is exciting. Great things happen with numbers.’” Mike: What's hitting me is that there's probably a lot of value in being able to use students' conjectures as reference points for potential future lessons. I wonder if you have some ideas or if you've seen educators create something like a public space for conjectures in their classroom. Chepina: We've seen amazing work around conjectures with young mathematicians. In that story that Jody was telling us about Rachael, she used that lesson—she used that conjecture in the next lesson to bring it together. It fit so perfectly with the storyline for that unit, and the lesson, and where it was going to go next. But sometimes ideas can be really great, but they don't quite fit where the storyline is going. So, we've encouraged teachers and [have] seen this happen in the classrooms we've worked in, where they have a conjecture wall in their classroom, where ideas can be added with Post-it Notes, have a station where [there are] Post-it Notes and pencil right there. And students can go and write their idea, put their name on it, stick it to the wall. And so, conjectures that are used in the lesson can be put up there, but ones that aren't used yet could be put up there. And so, if there was a lesson where a great idea emerges in the middle, and it doesn't quite fit in, the teacher could say, “That's a great idea. I want to make sure we come back to it. Could you add it to the conjecture wall?” And it gives that validation that their idea is important, and we're going to come back to it instead of just shutting it down and not acknowledging it at all. So, we have them put their names on to share. It's their expertise. They have value in our classroom. They add something to our community. Everyone has something important to share. So, that public space, I think, is really important to nurture that community where everyone has something to share. And we're all learning together. We're all exploring, conjecturing. Jody: And I've been too in those classrooms, that Chepina is referring to with conjecture walls, and kids actually will come in, they'll be doing math, and they'll go to recess or lunch and come back in and ask for a Post-it to add a conjecture like this—I don't know, one of my colleagues uses the [words] “mathematical residue.” They continue thinking about this, and their thoughts are acknowledged. And there's a space for them. Mike: So, as a former kindergarten/first grade teacher, I'm seeing a picture in my head. And I'm wondering if you could talk about setting the stage for this type of experience, particularly the types of questions that can draw out conjectures and encourage justification? Jody: Yeah. So, as we worked with teachers, we found so many rich opportunities. And now looking back, those opportunities are probably in all classrooms all the time. But I hadn't realized in my experience that I'm one step away from this. (chuckles) So, as teachers engaged in instructional routines, like the example of choral counting I shared from Rachael's classroom, they often ask questions like: “What do you notice?,” “Why do you think that's happening?,” “Will that always happen?,” “How do you know?,” ”How can you prove it will always work?,” “How can you convince a friend?” And those questions nudge children naturally to go to that next step when we're pushing, asking an advancing question in response to something that a student said. Mike: You know, one of the things that occurs to me is that those questions are a little bit different even than the kinds of questions we would ask if we were trying to elicit a student's strategy or their conceptual understanding, right? In that case, it seems like we want to understand the ideas that were kind of animating a student's strategy or the ideas that they were using or even how they saw a mental model unfolding in their head. But the questions that you just described, they really do go back to this idea of generalizing, right? Is there a pattern that we can recognize that is consistently the same or that doesn't change? And it's pressing them to think about that in a way that's different even than conceptual-based questions. Does that make sense? Jody: It does, and it makes me think about—I believe it's Vicki Jacobs and Joan Case who do a lot of work with questioning. They ask this question too: “As a teacher, what did that child say that gave you permission to ask that question?” Where often, I want to take my question somewhere else, but really all of these questions are nudging kids in their own thinking. So, when they're sharing something, it's like, “Well, do you think that will always work?” It's still grounded in what their ideas were but sort of taking them to that next place. Mike: So, one of the things that I'm also wondering about is a scaffold called “language frames.” How do students or a teacher use language frames to support argumentation? Chepina: Yeah, I think that communication is such a big part of argumentation. And we found language frames can help support students to share their ideas by having this common language that might be different than the way they talk about other things with their friends or in other subjects. So, using the language frames as a scaffold that supports students in communicating by offering them a model for that discussion. When...
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Season 3 | Episode 4 - Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics - Guest: Beth Hulbert
10/24/2024
Season 3 | Episode 4 - Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics - Guest: Beth Hulbert
Beth Hulbert, Making Sense of Unitizing: The Theme That Runs Through Elementary Mathematics ROUNDING UP: SEASON 3 | EPISODE 4 During their elementary years, students grapple with many topics that involve relationships between different units. In fact, unitizing serves as a foundation for much of the mathematics students encounter during their elementary years. Today, we’re talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms. BIOGRAPHY Beth Hulbert is an independent consultant focused on mathematics curriculum, instruction, and assessment at the K–8 level. She has been involved in all aspects of the Ongoing Assessment Project (OGAP) since its inception. Beth is coauthor of A Focus on Multiplication and Division: Bringing Research to the Classroom. The book was written to communicate how students develop their understanding of the concepts of multiplication and division. RESOURCES TRANSCRIPT Mike Wallus: During their elementary years, students grapple with many topics that involve relationships between different units. This concept, called “unitizing,” serves as a foundation for much of the mathematics that students encounter during their elementary years. Today, we're talking with Beth Hulbert from the Ongoing Assessment Project (OGAP) about the ways educators can encourage unitizing in their classrooms. Welcome to the podcast, Beth. We are really excited to talk with you today. Beth Hulbert: Thanks. I'm really excited to be here. Mike: I'm wondering if we can start with a fairly basic question: Can you explain OGAP and the mission of the organization? Beth: Sure. So, OGAP stands for the Ongoing Assessment Project, and it started with a grant from the National Science Foundation to develop tools and resources for teachers to use in their classroom during math that were formative in nature. And we began with fractions. And the primary goal was to read, distill, and make the research accessible to classroom teachers, and at the same time develop tools and strategies that we could share with teachers that they could use to enhance whatever math program materials they were using. Essentially, we started by developing materials, but it turned into professional development because we realized teachers didn't have a lot of opportunity to think deeply about the content at the level they teach. The more we dug into that content, the more it became clear to us that content was complicated. It was complicated to understand, it was complicated to teach, and it was complicated to learn. So, we started with fractions, and we expanded to do work in multiplicative reasoning and then additive reasoning and proportional reasoning. And those cover the vast majority of the critical content in K–8. And our professional development is really focused on helping teachers understand how to use formative assessment effectively in their classroom. But also, our other goals are to give teachers a deep understanding of the content and an understanding of the math ed research, and then some support and strategies for using whatever program materials they want to use. And we say all the time that we're a program blind — we don't have any skin in the game about what program people are using. We are more interested in making people really effective users of their math program. Mike: I want to ask a quick follow-up to that. When you think about the lived experience that educators have when they go through OGAP’s training, what are the features that you think have an impact on teachers when they go back into their classrooms? Beth: Well, we have learning progressions in each of those four content strands. And learning progressions are maps of how students acquire the concepts related to, say, multiplicative reasoning or additive reasoning. And we use those to sort, analyze, and decide how we're going to respond to evidence in student work. They're really maps for equity and access, and they help teachers understand that there are multiple right ways to do some mathematics, but they're not all equal in efficiency and sophistication. Another piece they take away of significant value is we have an item bank full of hundreds of short tasks that are meant to add value to, say, a lesson you taught in your math program. So, you teach a lesson, and you decide, “What is the primary goal of this lesson?” And we all know, no matter what the program is you're using, that every lesson has multiple goals, and they're all in varying degrees of importance. So partly, picking an item in our item bank is about helping yourself think about what was the most critical piece of that lesson that I want to know about that's critical for my students to understand for success tomorrow. Mike: So, one big idea that runs through your work with teachers is this concept called “unitizing.” And it struck me that whether we're talking about addition, subtraction, multiplication, fractions, that this idea just keeps coming back and keeps coming up. I'm wondering if you could offer a brief definition of unitizing for folks who may not have heard that term before. Beth: Sure. It became really clear as we read the research and thought about where the struggles kids have, that unitizing is at the core of a lot of struggles that students have. So, unitizing is the ability to call something “1,” say, but know it's worth maybe 1 or 100 or a 1,000, or even 1/10. So, think about your numbers in a place value system. In our base ten system, when a 1 is in the tenths place, it's not worth 1 anymore; it's worth 1 of 10. And so that idea that the 1 isn't the value of its face value, but it's the value of its place in that system. So, base ten is one of the first big ways that kids have to understand unitizing. Another kind of unitizing would be money. Money's a really nice example of unitizing. So, I can see one thing, it's called a nickel, but it's worth 5. And I can see one thing that's smaller, and it's called a dime, and it's worth 10. And so, the idea that 1 would be worth 5 and 1 would be worth 10, that's unitizing. And it's an abstract idea, but it provides the foundation for pretty much everything kids are going to learn from first grade on. And when you hear that kids are struggling, say, in third and fourth grade, I promise you that one of their fundamental struggles is a unitizing struggle. Mike: Well, let's start where you all started when you began this work in OGAP. Let's start with multiplication. Can you talk a little bit about how this notion of unitizing plays out in the context of multiplication? Beth: Sure. In multiplication, one of the first ways you think about unitizing is, say, in the example of 3 times 4. One of those numbers is a unit or a composite unit, and the other number is how many times you copy or iterate that unit. So, your composite unit in that case could be 3, and you're going to repeat or iterate it 4 times. Or your composite unit could be 4, and you're going to repeat or iterate it 3 times. When I was in school, the teacher wrote “3 times 4” up on the board and she said, “3 tells you how many groups you have, and 4 tells you how many you put in each group.” But if you think about the process you go through when you draw that in that definition, you draw 1, 2, 3 circles, then you go 1, 2, 3, 4; 1, 2, 3, 4; 1, 2, 3, 4; 1, 2, 3, 4. And in creating that model, you never once thought about a unit, you thought about single items in a group. So, you counted 1, 2, 3, 4 three times, and there was never really any thought about the unit. In a composite unit way of thinking about it, you would say, “I have a composite unit of 3, and I'm going to replicate it 4 times.” And in that case, every time, say, you stamped that — you had this stamp that was 3 — every time you stamped it, that one action would mean 3, right? 1 to 3, 1 to 3, 1 to 3, 1 to 3. So, in really early number work, kids think 1 to 1. When little kids are counting a small quantity, they'll count 1, 2, 3, 4. But what we want them to think about in multiplication is a many-to-1 action. When each of those quantities happens, it's not one thing. Even though you make one action, it's 4 things or 3 things, depending upon what your unit is. If you needed 3 times 8, you could take your 3-times-4 and add four more 3-times-4s to that. So, you have your four 3s and now you need four more 3s. And that allows you to use a fact to get a fact you don't know because you've got that unit and that understanding that it's not by 1, but by a unit. When it gets to larger multiplication, we don't really want to be working by drawing by 1s, and we don't even want to be stamping 27 nineteen times, right? But it's a first step into multiplication, this idea that you have a composite unit, and in the case of 3 times 4 and 3 times 7, seeing that 3 is common. So, there's your common composite unit. You [need] four of them for 3 times 4, and you need seven of them for 3 times 7. So, it allows you to see those relationships, which if you look at the standards, the relationships are the glue. So, it's not enough to memorize your multiplication facts. If you don't have a strong relationship understanding there, it does fall short of a depth of understanding. Mike: I think it was interesting to hear you talk about that, Beth, because one of the things that struck me is some of the language that you used, and I was comparing it in my head to some of the language that I've used in the past. So, I know I've talked about 3 times 4, but I thought it was really interesting how you used “iterations of” or “duplicated” … Beth: “Copies.” Mike: … or “copies,” right? What you make me think is that those language choices are a little bit clearer. I can visualize them in a way that “3 times 4” is a little bit more abstract or obscure. I may be thinking of that wrong, but I'm curious how you think the language that you use when you're trying to get kids to think about composite numbers matters. Beth: Well, I'll say this, that when you draw your 3 circles and count 4 dots in each circle, the result is the same model than if you thought of it as a unit of 3 stamped 4 times. In the end, the model looks the same, but the physical and mental process you went through is significantly different. So, you thought when you drew every dot, you were thinking about, “1, 1, 1, 1, 1, 1, 1.” When you thought about your composite unit copied or iterated, you thought about this unit being repeated over and over. And that changes the way you're even thinking about what those numbers mean. And one of those big, significant things that makes addition different than multiplication when you look at equations is, in addition, those numbers mean the same thing. You have three things, and you have four things, and you're going to put them together. If you had 3 plus 4, and you changed that 4 to a 5, you're going to change one of your quantities by 1, impacting your answer by 1. In multiplication, if you have 3 times 4, and you change that 4 to a 5, your factor increases by 1, but your product increases by the value of your composite unit. So, it's a change of the other factor. And that is [a] significant change in how you think about multiplication, and it allows you to pave the way, essentially, to proportional reasoning, which is that replicating [of] your unit. Mike: One of the things I'd appreciated about what you said was it's a change in how you're thinking. Because when I think back to Mike Wallus, classroom teacher, I don't know that I understood that as my work. What I thought of my work at that point in time was, “I need to teach kids how to use an algorithm or how to get an answer.” But I think where you're really leading is we really need to be attending to, “What's the thinking that underlies whatever is happening?” Beth: Yes. And that's what our work is all about, is, “How do you give teachers a sort of lens into, or a look into, how kids are thinking and how that impacts whether they can employ more efficient and sophisticated relationships and strategies in their thinking?” And it's not enough to know your multiplication facts. And the research is pretty clear on the fact that memorizing is difficult. If you're memorizing 100 single facts just by memory, the likelihood you're not going to remember some is high. But if you understand the relationship between those numbers, then you can use your 3 times 4 to get your 3 times 5 or your 3 times 8. So, the language that you use is important, and the way you leave kids thinking about something is important. And this idea of the composite unit, it's thematic, right? It goes through fractions and additive and proportional, but it's not the only definition of multiplication. So, you've got to also think of multiplication as scaling — that comes later, but you also have to think of multiplication as area and as dimensions. But that first experience with multiplication has to be that composite-unit experience. Mike: You've got me thinking already about how these ideas around unitizing that students can start to make sense of when they're multiplying whole numbers, that that would have a significant impact when they [start] to think about fractions or rational numbers. Can you talk a little bit about unitizing in the context of fractions, Beth? Beth: Sure. The fraction standards have been most difficult for teachers to get their heads around because the way that the standards promote thinking about fractions is significantly different than the way most of us were taught fractions. So, in the standards and in the research, you come across the term “unit fraction,” and you can probably recognize the unitizing piece in the unit fraction. So, a unit fraction is a fraction where 1 is in the numerator, it's one unit of a fraction. So, in the case of 3/4, you have three of the one-fourths. Now, this is a bit of a shift in how we were taught. Most of us were taught, “Oh, we have 3/4. It means you have 4 things, but you only keep 3 of them,” right? We learned about the name “numerator” and the name “denominator.” And, of course, we know in fractions, in particular, kids really struggle. Adults really struggle. Fractions are difficult because they seem to be a set of numbers that don't have anything in common with any other numbers. But once you start to think about unitizing and that composite unit, there's a standard in third grade that talks about “decompose any fraction into the sum of unit fractions.” So, in the case of 5/6, you would identify the unit fraction as 1/6, and you would have five of those 1/6. So, your unit fraction is 1/6, and you're going to iterate it or copy it or repeat it 5 times. Mike: I can hear the parallels between the way you described this work with whole numbers, right? I have 1/4, and I've duplicated or copied that 5 times, and that's what 5/4 is. It feels really helpful to see the through line between how we think about helping kids think about composite numbers and multiplying with whole numbers to what you just described with unit fractions. Beth: Yeah, and even the language, that language in fractions, is similar too. So, you talk about that 5 one-fourths. You decompose the five-fourths into 5 of the one-fourths, or you recompose those 5 one-fourths. This is a fourth-grade standard. You recompose those 5 one-fourths into 3 one-fourths (or three-fourths) and 2 one-fourths (or two-fourths). So, even reading a fraction like seven-eighths as “7 one-eighths,” helps to really understand what that seven-eighths means, and it keeps you from reading it as “7 out of 8.” Because when you read a fraction as “7 out of 8,” it sounds like you're talking about a whole number over another whole number. And so again, that connection to the composite unit in multiplication extends to that composite unit or that unit fraction or unitizing in multiplication. And really, even when we talk about multiplying fractions, we talk about multiplying, say, a whole number times a fraction, “5 times one-fourth,” that would be the same as saying, “I'm going to repeat one-fourth 5 times,” as opposed to, we were told, “Put a 1 under the 5 and multiply across the numerator and multiply across the denominator.” But that didn't help kids really understand what was happening. Mike: So, this progression of ideas that we've talked about from multiplication to fractions, what you've got me thinking about is, “What does it mean to think about unitizing with younger kids?” Particularly perhaps, kids in kindergarten, first, or second grade. I'm wondering how, or what you think educators could do to draw out the big ideas about units and unitizing with students in those grade levels? Beth: Well, really we don't expect kindergartners to strictly unitize because it's a relatively abstract idea. The big focus in kindergarten is for a student to understand “four” means “4” — “4 ones” — and “seven” means “7 ones.” But where we do unitize is in the use of our models in early grades. In kindergarten, the use of a 5-frame or a [10-frame]. So, let's use the [10-frame] to count by 10s: 10, 20, 30. And then, how many [10-frames] did it take us to count to 30? It took three. There's the beginning of your unitizing idea. The idea that we would say, “It took three of the [10-frames] to make 30” is really starting to plant that idea of unitizing: 3 can mean 30. And in first grade, when we start to expose kids to coin values, time, telling time, you know. One of the examples we use is, “[When] was 1 minus 1 [equal to] 59?” And that was, “When you read for 1 hour and your friend read for 1 minute less than you, how long did they read?” So, all time is really a unitizing idea. So, all measures — measure conversion, time, money, and the big one in first grade is base ten. And first grade and second grade have the opportunity to solidify strong base ten so that when kids enter third grade, they've already developed a concept of unitizing within the base ten system. In first grade, the idea that in a number like 78, the 7 is actually worth more than the 8, even though at face value, the 7 seems less than the 8. The idea that 7 is greater than the 8 in a number like 78 is unitizing. In second grade, when we have a number like 378, we can unitize that into 3 hundreds, 7 tens, and 8 ones — or 37 tens and 8 ones — and there's your re-unitizing. And that's actually a standard in second grade. Or 378 ones. So, in first and second grade, really what teachers have to commit to is developing really strong, flexible base ten understanding. Because that's the first place kids have to struggle with, this idea of the face value of a number isn't the same as the place value of a number. Mike: Yeah, yeah. So, my question is, would you describe that as the seeds of unitizing? Like, conserving? That's the thing that popped into my head, is, “Maybe that's what I'm actually starting to do when I'm trying to get kids to go from counting each individual 1 and naming the total when they say the last 1.” Beth: So, there are some early number concepts that need to be solidified for kids to be able to unitize, right? So, conservation is certainly one of them. And we work on conservation all throughout elementary school. As numbers get larger, as they have different features to them, they're more complex. Conservation doesn't get fixed in kindergarten; it's just pre-K and K are the places where we start to build that really early understanding with small quantities. There's cardinality, hierarchical inclusion, those are all concepts that we focus on and develop in the earliest grades that feed into a child's ability to unitize. So, the thing...
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Season 3 | Episode 3 - Choice as a Foundation for Student Engagement - Guest: Drs. Zandra De Arajuo and Amber Candela
10/10/2024
Season 3 | Episode 3 - Choice as a Foundation for Student Engagement - Guest: Drs. Zandra De Arajuo and Amber Candela
Drs. Zandra De Arajuo and Amber Candela, Choice as a Foundation for Student Engagement ROUNDING UP: SEASON 3 | EPISODE 3 As educators, we know offering students choice has a big impact on their engagement, identity, and sense of autonomy. That said, it's not always clear how to design choice into activities, especially when using curriculum materials. Today, we’re talking with Drs. Zandra De Araujo and Amber Candela about some of the ways educators can design choice into their students’ learning experiences. BIOGRAPHY Zandra de Araujo serves as the mathematics principal at the Lastinger Center for Learning. Her research examines teachers’ instruction in algebra with students who are primarily English learners. Amber Candela is an assistant professor of mathematics education at the University of Missouri–St. Louis (UMSL). She teaches mathematics methods classes for prospective elementary, middle, and high school teachers in the teacher education program at UMSL. TRANSCRIPT Mike Wallus: As an educator, I know that offering my students choice has a big impact on their engagement, their identity, and their sense of autonomy. That said, I've not always been sure how to design choice into the activities in my classroom, especially when I'm using curriculum. Today, we're talking with Drs. Zandra de Araujo and Amber Candela about some of the ways educators can design choice into their students' learning experiences. Welcome back to the podcast, Zandra and Amber. It is really exciting to have you all with us today. Zandra de Araujo: Glad to be back. Amber Candela: Very excited to be here. Mike: So, I've heard you both talk at length about the importance of choice in students' learning experiences, and I wonder if we can start there. Before we talk about the ways you think teachers can design choice in a learning experience, can we just talk about the “why”? How would you describe the impact that choice has on students' learning experiences? Zandra: So, if you think about your own life, how fun would it be to never have a choice in what you get to do during a day? So, you don't get to choose what chores to do, where to go, what order to do things, who to work with, who to talk to. Schools are a very low-choice environment, and those tend to be punitive when you have a low-choice environment. And so, we don't want schools to be that way. We want them to be very free and open and empowering places. Amber: And a lot of times, especially in mathematics, students don't always enjoy being in that space. So, you can get more enjoyment, engagement, and if you have choice with how to engage with the content, you'll have more opportunity to be more curious and joyful and have hopefully better experiences in math. Zandra: And if you think about being able to choose things in your day makes you better able to make choices. And so, I think we want students to be smart consumers and users and creators of mathematics. And if you're never given choice or opportunity to kind of own it, I think that you're at a deficit. Amber: Also, if we want problem-solving people engaged in mathematics, it needs to be something that you view as something you were able to do. And so often we teach math like it's this prepackaged thing, and it's just your role to memorize this thing that I give you. You don't feel like it's yours to play with. Choice offers more of those opportunities for kids. Zandra: Yeah, it feels like you're a consumer of something that's already made rather than somebody who's empowered to create and use and drive the mathematics that you're using, which would make it a lot more fun. Mike: Yeah. You all are hitting on something that really clicked for me as I was listening to you talk. This idea that school, as it's designed oftentimes, is low choice. But math, in particular, where historically it has really been, “Let me show you what to do. Let me have you practice the way I showed you how to do it,” rinse and repeat. It's particularly important in math, it feels like, to break out and build a sense of choice for kids. Zandra: Absolutely. Mike: Well, one of the things that I appreciate about the work that both of you do is the way that you advocate for practices that are both really, really impactful and also eminently practical. And I'm wondering if we can dive right in and have you all share some of the ways that you think about designing choice into learning experiences. Amber: I feel like I want “eminently practical” on a sticker for my laptop. (Zandra laughs) Because I find that is a very satisfying and positive way to describe the work that I do because I do want it to be practical and doable within the constraints of schooling as it currently is, not as we wish it to be. Which, we do want it to be better and more empowering for students and teachers. But also, there are a lot of constraints that we have to work within. So, I appreciate that. Zandra: I think that choice is meant to be a way of empowering students, but the goal for the instruction should come first. So, I'm going to talk about what I would want from my students in my classroom and then how we can build choice in. Because choice is kind of like the secondary component. So, first you have your learning goals, your aims as a teacher. And then, “How do we empower students with choice in service of that goal?” So, I'll start with number sense because that's a hot topic. I'm sure you all hear a lot about it at the MLC. Mike: We absolutely do. Zandra: So, one of the things I think about when teachers say, “Hey, can you help me think about number sense?” It's like, “Yes, I absolutely can.” So, our goal is number sense. So, let's think about what that means for students and how do we build some choice and autonomy into that. So, one of my favorite things is something like, “Give me an estimate, and we can Goldilocks it,” for example. So, it could be a word problem or just a symbolic problem, and say, “OK, give me something that you know is either wildly high, wildly low, kind of close, kind of almost close but not right. So, give me an estimate, and it could be a wrong estimate or a close estimate, but you have to explain why.” So, it takes a lot of number sense to be able to do that. You have infinitely many options for an answer, but you have to avoid the one correct answer. So, you have to actually think about the one correct answer to give an estimate. Or if you're trying to give a close estimate, you're kind of using a lot of number sense to estimate the relationships between the numbers ahead of time. The choice comes in because you get to choose what kind of estimate you want. It's totally up to you. You just have to rationalize your idea. Mike: That's awesome. Amber, your turn. Amber: Yep. So related to that is a lot of math goes forwards. We give kids the problem, and we want them to come up with the answer, right? And so, a lot of the work that we've been doing is, “OK, if I give you the answer, can you undo the problem?” I'll go multiplication. So, we do a lot with, “What's 7 times 8?” And there's one answer, and then kids are done. And you look for that answer as the teacher, and once that answer has been given, you're kind of like, “OK, here. I'm done with what I'm doing.” But instead, you could say, “Find me numbers whose product is 24.” Now you've opened up what it comes to. There's more access for students. They can come up with more than one solution, but it also gets kids to realize that math doesn't just go one way. It's not, “Here's the problem, find the answer.” It’s “Here’s the answer, find the problem.” And that also goes to the number sense. Because if students are able to go both ways, they have a better sense in their head around what they're doing and undoing. And you can do it with a lot of different problems. Zandra: And I'll just add in that that's not specific to us. Barb Dougherty had a really nice article in, I think, Teaching Children Mathematics, about reversals at some point. And other people have shown this idea as well. So, we're really taking ideas that are really high uptake, we think, and sharing them again with teachers to make sure that they've seen ways that they can do it in their own classroom. Mike: What strikes me about both of these is, the structure is really interesting. But I also think about what the output looks like when you offer these kinds of choices. You're going to have a lot of kids doing things like justifying or using language to help make sense of the “why.” “Why is this one totally wrong?” and “Why is this one kind of right?” and “Why is this close, but maybe not exact?” And to go to the piece where you're like, “Give me some numbers that I can multiply together to get to 24.” There's more of a conversation that comes out of that. There's a back and forth that starts to develop, and you can imagine that back and forth bouncing around with different kids rather than just kind of kid says, teacher validates, and then you're done. Zandra: Yeah, I think one of the cool things about choice is giving kids choice means that there's more variety and diversity of ideas coming in. And that's way more interesting to talk about and rationalize and justify and make sense of than a single correct answer or everybody's doing the same thing. So, I think, not only does it give kids more ownership, it has more access. But also, it just gives you way more interesting math to think and talk about. Mike: Let's keep going. Zandra: Awesome. So, I think another one, a lot of my work is with multilingual students. I really want them to talk. I want everybody to talk about math. So, this goes right to what you were just saying. So, one of the ways that we can easily say, “OK, we want more talk, so how do we build that in through choice?” is to say, “Let's open up what you choose to share with the class.” So, there have been lots of studies done on the types of questions that teachers ask, [which] tend to be closed, answer-focused, like single-calculation kind of questions. So, “What is the answer?” “Who got this?” You know, that kind of thing? Instead, you can give students choices, and I think a lot of teachers have done something akin to this with sentence starters or things. But you can also just say, instead of a sentence starter to say what your answer is — like, “I agree with X because of Y” — you can also say, “You can share an incorrect answer that you know is wrong because you did it, and it did not work out.” “You can also share where you got stuck because that's valuable information for the class to have.” “You could also say, ‘I don't want to really share my thinking, my solution because it's not done, but I'll show you my diagram. And so, let me show you a visual.’” And just plop it up on the screen. So, there are a lot of different things; you could share a question that you have because you’re not sure, and it's just a related question. Instead of always sharing answers, let kids open up what they may choose to share, and you'll get more kids sharing. Because answers are kind of scary because you're expecting a correct answer often. And so, when you share and open up, then it's not as scary. And everybody has something to offer because they have a choice that speaks to them. Amber: And kids don't want to be wrong. People don't want to be wrong. I don't want to give you a wrong answer. And we went to the University of Georgia together, but Les Steffe always would say, “No child is ever wrong. They're giving you an answer with a purpose behind what that answer is. They don't actually believe that's a wrong answer that they're giving you.” And so, if you open up the space — and teachers, we want spaces to be safe, we want kids to want to come in and share. But are we actually structuring spaces in that way? And so, some of the ideas that we're trying to come up with, we're saying that “We actually do value what you're saying when you choose to give us this. It's your choice of offering it up and you can say whatever it is you want to say around that.” But it's not as evaluative or as high stakes as trying to get the right answer and just like, “Am I right? Did I get it right?” And then what the teacher might say after that. Zandra: I would add on that kids do like to give wrong answers if that's what you're asking for. They don't like to give wrong answers if you're asking for a right one and they're accidentally wrong. So, I think back to my first suggestion: If you ask for a wrong answer and they know it's wrong, they're likely to chime right in because the right answer is the wrong answer, and there are multiple, infinite numbers of them. Mike: You know, it makes me think there's this set of ideas that we need to normalize mistakes as being productive things. And I absolutely agree with that. I also think that when you're asking for the right answer, it's really hard to kind of be like, “Oh, my mistake was so productive.” On the other hand, if you ask for an error or a place where someone's stuck, that just feels different. It feels like an invitation to say, “I've actually been thinking about this. I'm not there. I may be partly there. I'm still engaged. This is where I'm struggling.” That just feels different than providing an answer where you're just like, “Ugh.” I'm really struck by that. Zandra: Yeah, and I think it's a culture thing. So, a lot of teachers say to me that it's hard to have kids work in groups because they kind of just tell each other the answers. But they're modeling what they experience as learners in the classroom. “I often get told the answers,” that's the discourse that we have in the classroom. So, if you open up the discourse to include these things like, “Oh, I'm stuck here. I'm not sure where to go here.” They get practice saying, “Oh, I don't know what this is. I don't know how to go from here.” Instead of just going straight to the answer. And I think it'll spread to the group work as well. Mike: It feels like there's value for every other student in articulating, “I'm certain that this one is wrong, and here's why I know that.” There's information in there that is important for other kids. And even the idea of “I'm stuck here,” right? That's really a great formative assessment opportunity for the teacher. And it also might validate some of the other places where kids are like, “Yeah. Me too.” Zandra: Mm-hmm. Amber: Right, absolutely. Mike: What's next, my friend? Amber: I remember very clearly listening to Zandra present about choice, her idea of choice of feedback. And this was very powerful to me because I had never thought about asking my students how they wanted to receive the feedback I'd be giving them on the problems that they solved. And this idea of students being able to turn something in and then say, “This is how I'd like to receive feedback” or “This is the feedback I'd like to receive” becomes very powerful because now they're the ones in charge of their own learning. And so much of what we do, kids should get to say, “This is how I think that I will grow better, is if you provide this to me.” And so, having that opportunity for students to say, “This is how I'll be a better learner, if you give it to me in this way. And I think if you helped me with this part that would help the whole rest of it.” Or “I don't actually want you to tell me the answer. I am stuck here. I just need a little something to get me through. But please don't tell me what the answer is because I still want to figure it out for myself.” And so, allowing kids to advocate for themselves and teaching them how to advocate for themselves to be better learners, how to advocate for themselves to learn and think about what I need to learn this material and be a student or be a learner in society will just ultimately help students. Zandra: Yeah, I think as a student, I don't like to be told the answers. I like to figure things out, and I will puzzle through something for a long time. But sometimes I just want a model or a hint that'll get me on the right path, and that's all I need. But I don't want you to do the problem for me or take over my thinking. If somebody asked me, “What do you want?” I might say, “Oh, a model problem,” or something like that. But I don't think we ask kids a lot. We just do whatever we think as an adult. Which is different because we're not learning it for the first time. We already know what it is. Mike: You're making me think about the range of possibilities in a situation like that. One is I could notice a student who is working through something and just jump in and take over and do the problem for them essentially and say, “Here, this is how you do it.” Or I guess just let them go, let them continue to work through it. But potentially there could be some struggle, and there might be some frustration. I am really kind of struck by the fact that I wonder how many of us as teachers have really thought about the kinds of options that exist between those two far ends of the continuum. What are the things that we could offer to students rather than just “Let me take over” or productive struggle, but perhaps it's starting to feel unproductive? Does that make sense? Zandra: Yeah, I think it does. I mean, there are so many different ways. I would ask teachers to recenter themselves as the learner that's getting feedback. So, if you have a principal or a coach coming into your room, they've watched a lesson, sometimes you're like, “Oh, that didn't go well. I don't need feedback on that. I know it didn't go well, and I could do better.” But I wonder if you have other things that you notice just being able to take away a part that you know didn't go well. And you're like, “Yep, I know that didn't go well. I have ideas for improving it. I don't really want to focus on that. I want to focus on this other thing.” Or “I've been working really hard on discourse. I really want feedback on the student discourse when you come in.” That's really valuable to be able to steer it — not taking away the other things that you might notice, but really focusing in on something that you've been working on is pretty valuable. And I think kids often have these things that maybe they haven't really thought about a lot, but when you ask them, they might think about it. And they might grow this repertoire of things that they're kind of working on personally. Amber: Yeah, and I just think it's getting at, again, we want students to come out of situations where they can say, “This is how I learn” or “This is how I can grow,” or “This is how I can appreciate math better.” And by allowing them to say, “It'd be really helpful if you just gave me some feedback right here,” or “I'm trying to make this argument, and I'm not sure it's coming across clear enough,” or “I'm trying to make this generalization; does it generalize?” We're also maybe talking about some upper-level kids, but I still think we can teach elementary students to advocate for themselves also. Like, “Hey, I try this method all the time. I really want to try this other method. How am I doing with this? I tried it. It didn't really seem to work, but where did I make a mistake? Could you help me out with that? Because I think I want to try this method instead.” And so, I think there are different ways that students can allow for that. And they can say: “I know this answer is wrong. I'm not sure how this answer is wrong. Could you please help me understand my thinking?” or “How could I go back and think about my thinking?” Zandra: Yeah. And I think when you said upper level, you meant upper grades. Amber:...
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Season 3 | Episode 2 - Principles for Responsive Curriculum Use - Guest: Dr. Corey Drake
09/19/2024
Season 3 | Episode 2 - Principles for Responsive Curriculum Use - Guest: Dr. Corey Drake
Dr. Corey Drake, Principles for Responsive Curriculum Use ROUNDING UP: SEASON 3 | EPISODE 2 When it comes to curriculum, educators are often told to implement with fidelity. But what does fidelity mean, and where does that leave educators who want to be responsive to the students in their classrooms? Today we’re talking with Dr. Corey Drake about principles for responsive curriculum use that invite educators to respond to the students in their classrooms while still implementing curriculum with integrity. BIOGRAPHY Corey Drake is the senior director for professional learning at The Math Learning Center. She began her career in education as a middle school special education teacher in the Chicago Public Schools. Corey is committed to supporting teachers in using curriculum materials to teach diverse groups of students in equitable ways. RESOURCES TRANSCRIPT Mike Wallus: When it comes to curriculum, educators are often told to implement with fidelity. But what does fidelity mean? And where does that leave educators who want to be responsive to students in their classrooms? Today we're talking with Dr. Corey Drake about principles for responsive curriculum use that invite educators to respond to the students in their classrooms while still implementing curriculum with integrity. One of the age-old questions that educators grapple with is how to implement a curriculum in ways that are responsive to the students in their classroom. It's a question I thought a lot about during my years as a classroom teacher, and it's one that I continue to discuss with my colleague at MLC, Dr. Corey Drake. As a former classroom teacher and a former teacher educator who only recently began working for an organization that publishes curriculum, Corey and I have been trying to carve out a set of recommendations that we hope will help teachers navigate this question. Today on the podcast, we'll talk about this question of responsive curriculum use and offer some recommendations to support teachers in the field. Welcome back to the podcast, Corey. I'm excited to have you with us again. Corey Drake: It's great to be with you again. Mike: So, I've been excited about this conversation for a while because this question of, “What does it mean to be responsive to students and use a curriculum?” is something that teachers have been grappling with for so long. And you and I often hear phrases like “implementation with fidelity” used when folks are trying to describe their expectations when a curriculum's adopted. Corey: Yeah, I mean, I think this is a question teachers grapple with. It's a question I've been grappling with for my whole career, from different points of view from when I was a classroom teacher and a teacher educator and now working at The Math Learning Center. But I think this is the fundamental tension: “How do you use a set of published curriculum materials while also being responsive to your students?” And I think ideas like implementation with fidelity didn't really account for the responsive-to-your-students piece. Fidelity has often been taken up as meaning “following curriculum materials page by page, word for word, task for task.” We know that's not actually possible. You have to make decisions, you have to make adaptations as you move from a written page to an enacted curriculum. But still the idea of fidelity was to be as close as possible to the written page, whereas ideas like implementation with integrity or responsive curriculum use are starting with what's written on the page, staying consistent with the key ideas of what's on the page, but doing it in a way that's responsive to the students who are sitting in front of you. And that's really kind of the art and science of curriculum use. Mike: Yeah, I think one of the things that I used to think was that it was really a binary choice between something like fidelity, where you were following things in what I would've described as a lockstep fashion. Or the alternative, which would be, “I'm going to make everything up.” And you've helped me think, first of all, about what might be some baseline expectations from a large-scale curriculum. What are we actually expecting from curriculum around design, around the audience that it's written for? I wonder if you could share with the audience some of the things that we've talked about when it comes to the assets and also the limitations of a large-scale curriculum. Corey: Yeah, absolutely. And I will say, when you and I were first teachers probably, and definitely when we were students, the conversation was very different. We had different curriculum materials available. There was a very common idea that good teachers were teachers who made up their own curriculum materials, who developed all of their own materials. But there weren't the kinds of materials out there that we have now. And now we have materials that do provide a lot of assets, can be rich tools for teachers, particularly if we release this expectation of fidelity and instead think about integrity. So, some of the assets that a high-quality curriculum can bring are the progression of ideas, the sequence of ideas and tasks that underlies almost any set of curriculum materials; that really looks at, “How does student thinking develop across the course of a school year?” And what kinds of tasks, in what order, can support that development of that thinking. That's a really important thing that individual teachers or even teams of teachers working on their own, that would be very hard for them to put together in that kind of coherent, sequential way. So, that's really important. A lot of curriculum materials also bring in many ideas that we've learned over the last decades about how children learn mathematics: the kinds of strategies children use, the different ways of thinking that children bring. And so, there's a lot that both teachers and students can learn from using curriculum materials. At the same time, any published set of large-scale curriculum materials are, by definition, designed for a generic group of students, a generic teacher in a generic classroom, in a generic community. That's what it means to be large scale. That's what it means to be published ahead of time. So, those materials are not written for any specific student or teacher or classroom or community. And so, that's the real limitation. It doesn't mean that the materials are bad. The materials are very good. But they can't be written for those specific children in that specific classroom and community. That's where this idea that responsive curriculum use and equitable instruction always have to happen in the interactions between materials, teachers, and students. Materials by themselves cannot be responsive. Teachers by themselves cannot responsively develop the kinds of ideas in the ways that curriculum can, the ways they can when using curriculum as a tool. And, of course, students are a key part of that interaction. And so, it's really thinking about those interactions among teachers, students, and materials and thinking about, “What are the strengths the materials bring? What are the strengths the teacher brings?” The teacher brings their knowledge of the students. The teacher brings their knowledge of the context. And the students bring, of course, their engagement and their interaction with those materials. And so, it's thinking about the strengths they each bring to that interaction, and it's in those interactions that equitable and responsive curriculum use happens. Mike: One of the things that jumps out from what you said is this notion that we're not actually attempting to fix “bad curriculum.” We're taking the position that curriculum has a set of assets, but it also has a set of limitations, and that's true regardless of the curriculum materials that you're using. Corey: Absolutely. This is not at all about curriculum being bad or not doing what it's supposed to do. This assumes that you're using a high-quality curriculum that does the things we just talked about that has that progression of learning, those sequences of tasks that [bring] ideas about how children learn and how we learn and teach mathematics. And then, to use that well and responsively, the teacher then needs to work in ways, make decisions to enact that responsively. It's not about fixing the curriculum. It's about using the curriculum in the most productive and responsive ways possible. Mike: I think that's good context, and I also think it's a good segue to talk about the three recommendations that we want educators to consider when they're thinking about, “What does it mean to be responsive when you're using curriculum?” So, just to begin with, why don't we just lay them out? Could you unpack them, Corey? Corey: Yeah, absolutely. But I will say that this is work you and I have developed together and looking at the work of others in the field. And we've really come up with, I think, three key criteria for thinking about responsive curriculum use. One is that it maintains the goals of the curriculum. So again, recognizing that one of the strengths of curriculum is that it's built on this progression of ideas and that it moves in a sequential way from the beginning of the year to the end of the year. We want teachers to be aware of, to understand what the goals are of any particular session or unit or year, and to stay true to those goals, to stay aligned with those goals. But at the same time, doing that in ways that open up opportunities for voice and choice and sensemaking for the specific students who are in front of them in that classroom. And then the last is, we're really concerned with and interested in supporting equitable practice. And so, we think about responsive curriculum use as curriculum use that reflects the equity-based practices that were developed by Julia Aguirre and her colleagues. Mike: I think for me, one of the things that hit home was thinking about this idea that there's a mathematical goal and that goal is actually part of a larger trajectory that the curriculum's designed around. And when I've thought about differentiation in the past, what I was really thinking about was replacement that fundamentally altered the instructional goal. And I think the challenge in this work is to say, “Am I clear on the instructional goal? And do the things that I'm considering actually maintain that for kids, or are they really replacing them or changing them in a way that will alter or impact the trajectory?” Corey: I think that's such a critical point. And it's not easy work. It's not always clear even in materials that have a stated learning goal or learning target for a session. There's still work to do for the teacher to say, “What is the mathematical goal? Not the activity, not the task, but what is the goal? What is the understanding I'm trying to support for my students as they engage in this activity?” And so, you're right. I think the first thing is, teachers have to be super clear about that because all the rest of the decisions flow from understanding, “What is the goal of this activity? What are the understandings that I am trying to develop and support with this session?” And then I can make decisions that are enhancing and providing access to that goal, but not replacing it. I'm not changing the goal for any of my students. I'm not changing the goal for my whole group of students. Instead, I'm recognizing that students will need different ways into that mathematics. Students will need different kinds of supports along the way. But all of them are reaching toward or moving toward that mathematical goal. Mike: Yeah. When I think about some of the options, like potentially, number choice; if I'm going to try to provide different options in terms of number choice, is that actually maintaining a connection to the mathematical goal, or have I done something that altered it? Another thing that occurs to me is the resources that we share with kids for representation, be it manipulatives or paper-pencil, even having them talk about it — any of those kinds of choices. To what extent do they support the mathematical goal, or do they veer away from it? Corey: Yeah, absolutely. And there are times when different numbers or different tools or different models will alter the mathematical goal because part of the mathematical goal is to learn about a particular tool or a particular representation. And there are other times when having a different set of numbers or a different set of tools or models will only enhance students' access to that mathematical goal because maybe the goal is understanding something like two-digit addition and developing strategies for two-digit addition. Well then, students could reach that goal in a lot of different ways. And some students will be working just with decade numbers, and some students will be working with decades and ones, and some students will need number pieces, and others will do it mentally. But if the goal is developing strategies, developing your understanding of two-digit addition, then all of those choices make sense, all of those choices stay aligned with the goal. But if the goal is to understand how base ten pieces work, then providing a different model or telling students they don't need to use that model would, of course, fundamentally alter the goal. So, this is why it's so critically important that we support teachers in understanding making sense of the goal, figuring out, how do they figure that out? How do you open a set of curriculum materials, look at a particular lesson, and understand what the mathematical goal of that lesson is? And it's not as simple as just looking for the statement of the learning goal and the learning target. But it's really about, “What are the understandings that I think will develop or are intended to develop through this session?” Mike: I feel like we should talk a little bit about context because context is such a powerful tool, right? If you alter the context, it might help kids surface some prior knowledge that they have. What I'm thinking about is this task that exists in Bridges where we're having kids look at a pet store where there are arrays of different sorts and kinds of dog foods or dog toys or cat toys. And I remember an educator saying to me, “I wonder if I could shift the context.” And the question that I asked her is, “If you look at this image that we're using to launch the task, what are the particular parts of that image that are critical to maintain if you're going to replace it with something that's more connected to your students?” Corey: Connecting to your students, using context to help students access the mathematics, is so important and such an empowering thing for teachers and students. But you're asking exactly the right question. And of course, that all relates to “What's the mathematical goal?” again. Because if I know that, then I can look for the features of the context that's in the textbook and see the ways in which that context was designed to support students in reaching that mathematical goal. But I can also look at a different context that might be more relevant to my students, that might provide them better access to the mathematics. And I can look at that context through the lens of that mathematical goal and see, “Does this context also present the kinds of features that will help my students understand and make sense of the mathematical goal?” And if the answer is yes, and if that context is also then more relevant to my students or more connected to their lives, then great. That's a wonderful adaptation. That's a great example of responsive curriculum use. If now I'm in a context that's distracting or leading me away from the mathematical goal, that's where we run into adaptations that are less responsive and less productive. Mike: Well, and to finish the example, the conversation that this led to with this educator was she was talking about looking for bodegas in her neighborhood that her children were familiar with, and we end up talking quite a bit about the extent to which she could find images from the local bodega that had different kinds of arrays. She was really excited. She actually did end up finding an image. And she came back, and she shared that this really had an impact on her kids. They felt connected to it, and the mathematical goal was still preserved. Corey: I love that. I think that's a great example. And I think the other thing that comes up sometimes when we present these ideas, is maybe you want to find a different context that is more relevant to your students that they know more about. Sometimes you might look at a context that's presented in the textbook and say, “I really love the mathematical features here. I really see how knowing something about this context could help my students reach the mathematical goal. But I'm going to have to do some work ahead of time to help my students understand the context, to provide them some access to that, to provide them some entry points.” So, in your example, maybe we're going to go visit a pet store. Maybe we're going to look at images from different kinds of stores and notice how things are arranged on shelves, and in arrays, and in different combinations. So, I think there are always a couple of choices. One is to change the context. One is to do some work upfront to help your students access the context so that they can then use that context to access the mathematics. But I think in both cases, it's about understanding the goal of the lesson and then understanding how the features of the context relate to that goal. Mike: Let's shift and just talk about the second notion, this idea of opening up space for students' voice or for sensemaking when you're using curriculum. For me at least, I often try to project ideas for practice into a mental movie of myself in a classroom. And I wonder if we could work to help people imagine what this idea of opening space for voice or sensemaking might look like. Corey: I think a lot of times those opportunities for opening up voice and choice and sensemaking are not in the direct action steps or the direct instructions to teachers within the lesson, but they're kind of in the in-between. So, “I know I need to introduce this idea to my students, but how am I going to do that? What is that going to look like? What is that going to sound like? What are students going to be experiencing?” And so, asking yourself that question as the lesson plays out is, I think, where you find those opportunities to open up that space for student voice and choice. It's often about looking at that and saying, Am I going to tell students this idea? Or am I going to ask them? Are students going to develop their strategy and share it with me or turn it in on a piece of paper? Or are they going to turn and talk to a partner? Are they going to share those ideas with a small group, with a whole group? What are they going to listen for in each other's strategies? How am I going to ask them to make connections across those strategies? What kinds of tools am I going to make available to them? What kinds of choices are they going to have throughout that process? And so, I think it's having that mental movie play through as you read through the lesson and thinking about those questions all the way through. “Where are my students going to have voice? How are they going to have choice? How are students going to be sensemaking?” And often thinking about, “Where can I step back, as the teacher, to open up that space for student voice or student choice?” Mike: You're making me think about a couple things. The first one that really jumped out was this idea that part of voice is not necessarily always having the conversation flow from teacher to student, but having a...
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Season 3 | Episode 1 - Grouping Practices That Promote Efficacy & Knowledge Transfer - Guest: Dr. Peter Liljedahl
09/05/2024
Season 3 | Episode 1 - Grouping Practices That Promote Efficacy & Knowledge Transfer - Guest: Dr. Peter Liljedahl
Dr. Peter Liljedahl, Grouping Practices That Promote Efficacy & Knowledge Transfer ROUNDING UP: SEASON 3 | EPISODE 1 We know from research that student collaboration can have a powerful impact on learning. That said, how we group students for collaboration matters — a lot. Today we’re talking with Dr. Peter Liljedahl, author of Building Thinking Classrooms in Mathematics , about how educators can form productive, collaborative groups in their classrooms. BIOGRAPHY Peter Liljedahl is a professor of mathematics education in the Faculty of Education at Simon Fraser University and author of the best-selling book Building Thinking Classrooms in Mathematics, Grades K–12: 14 Teaching Practices for Enhancing Learning . RESOURCES TRANSCRIPT Mike Wallus: We know from research that student collaboration can have a powerful impact on learning. That said, how we group students for collaboration matters — a lot. Today we're talking with Dr. Peter Liljedahl, author of Building Thinking Classrooms in Mathematics, about how educators can form productive, collaborative groups in their classrooms. Hello, Peter. Welcome to the podcast. Peter Liljedahl: Thanks for having me. Mike: So, to offer our listeners some background, you've written a book, called Building Thinking Classrooms in Mathematics , and I think it's fair to say that it's had a pretty profound impact on many educators. In the book, you address 14 different practices. And I'm wondering if you could weigh in on how you weigh the importance of the different practices that you addressed? Peter: Well, OK, so, first of all, 14 is a big number that publishers don't necessarily like. When we first started talking with Corwin about this, they were very open. But I know if you think about books, if there's going to be a number in the title, the number is usually 3, 5, or 7. It's sometimes 8 — but 14 is a ridiculous number. They can't all be that valuable. What's important about the fact that it's 14, is that 14 is the number of core practices that every teacher does. That's not to say that there aren't more or less for some teachers, but these are core routines that we all do. We all use tasks. We all create groups for collaboration. We all have the students work somewhere. We all answer questions. We do homework, we assign notes, we do formative, summative assessment. We do all of these things. We consolidate lessons. We launch lessons. These are sort of the building blocks of what makes our teaching. And through a lot of time in classrooms, I deduced this list of 14. Robert Kaplinsky, in one of his blog posts, actually said that he thinks that that list of 14 probably accounts for 95 percent of what happens in classrooms. And my research was specifically about, how do we enact each of those 14 so that we can maximize student thinking? So, what kind of tasks get students to think? How can we create groups so that more thinking happens? How can we consolidate a lesson so we get more thinking? How can we do formative and summative assessments so the students are thinking more? So, the book is about responding to those 14 core routines and the research around how to enact each of those to maximize thinking. Your question around which one is, “How do we put weight on each of these?” They're all important. But, of course, they're not all equally impactful. Building thinking classrooms is most often recognized visually as the thing where students are standing at whiteboards working. And, of course, that had a huge impact on student engagement and thinking in the classroom: getting them from sitting and working at desks to getting them working at whiteboards. But in my opinion, it's not the most impactful. It is hugely impactful, but the one that actually makes all of thinking classroom function is how we form collaborative groups, which is Chapter 2. And it seems like that is such an inconsequential thing. We've been doing groups for forever, and we got this figured out. We know how to do this. But do we really? Do we really have it figured out? Because my research really showed that if we want to get students thinking, then the ways we've been doing it aren't working. Mike: I think that's a great segue. And I want to take a step back, Peter. Before we talk about grouping, I want to ask what might be an obvious question. But I wonder if we can talk about the “why” behind collaboration. How would you describe the value or the potential impact of collaboration on students' learning experiences? Peter: That's a great question. We've been doing collaborative work for decades. And by and large, we see that it is effective. We have data that shows that it's effective. And when I say “we,” I don't mean me or the people I work with. I mean “we, in education,” know that collaboration is important. But why? What is it about collaboration that makes it effective? There are a lot of different things. It could be as simple as it breaks the monotony of having to sit and listen. But let's get into some really powerful things that collaboration does. Number one, about 25 years ago, we all were talking about metacognition. We know that metacognition is so powerful and so effective, and if we get students thinking about their thinking, then their thinking actually improves. And metacognition has been shown time and time again to be impactful in learning. Some of the listeners might be old enough to remember the days where we were actually trying to teach students to be metacognitive, and the frustration that that created because it is virtually impossible. Being reflective about your thinking while you're thinking is incredibly hard to do because it requires you to be both present and reflective at the same time. We're pretty good at being present, and we're pretty good about reflecting on our experiences. But to do both simultaneously is incredibly hard to do. And to teach someone to do it is difficult. But I think we've also all had that experience where a student puts up their hand, and you start walking over to them, and just as you get there, they go, “Never mind.” Or they pick up their book, and they walk over to you, and just as they get to you, they just turn around and walk back. I used to tell my students that they're smarter when they're closer to me. But what's really going on there is, as they’ve got their hand up, or as they're walking across the room toward you as a teacher, they're starting to formulate their thoughts to ask a question. They're preparing to externalize their thinking. And that is an incredibly metacognitive process. One of the easiest forms of metacognition, and one of the easiest ways to access metacognition, is just to have students collaborate. Collaborating requires students to talk. It requires them to organize their thoughts. It requires them to prepare their thinking and to think about their thinking for the purposes of externalization. It is an incredibly accessible way of creating metacognition in your classroom, which we already know is effective. So, that's one reason I think collaboration is really, really vital. Another one comes from the work on register. So, “register” is the level of sophistication with which we speak about something. So, if I'm in a classroom, and I'm talking to kindergarten students, I set a register that is accessible to them. When I talk to my undergraduates, I use a different register. My master's students, my PhD students, my colleagues, I'm using different registers. I can be talking about the same thing, but the level of sophistication with which I'm going to talk about those things varies depending on the audience. And as much as possible, we try to vary our register to suit the audience we have. But I think we've also all had that instructor who's completely incapable of varying their register, the one who just talks at you as if you're a third-year undergraduate when you're really a grade 8 student. And the ability to vary our register to a huge degree is going to define what makes us successful as a teacher. Can we meet our learners where they're at? Can we talk to them from the perspective that they're at? Now we can work at it, and very adept teachers are good at it. But even the best teachers are not as good at getting their register to be the same as students. So, this is another reason collaboration is so effective. It allows students to talk and be talked to at their register, which is the most accessible form of communication for them. And I think the third reason that collaboration is so important is the difference between, what I talk in my book about, the difference between absolute and tentative knowledge. So, I'm going to make two statements. You tell me which one is more inviting to add a comment to. So, statement number one is, “This is how to do it, or this is what I did.” That's statement number one. Statement number two is, “I think that one of the ways that we may want to try, I'm wondering if this might work.” Which one is more inviting for you to contribute to? Mike: Yes, statement number two, for many, many reasons, as I'm sitting here thinking about the impact of those two different language structures. Peter: So, as teachers, we tend to talk in absolutes. The absolute communication doesn't give us anything to hold onto. It's not engaging. It's not inviting. It doesn't bring us into the conversation. It's got no rough patches — it's just smooth. But when that other statement is full of hedging, it's tentative. It's got so many rough patches, so many things to contribute to, things I want to add to, maybe push back at or push further onto. And that's how students talk to each other. When you put them in collaborative groups, they talk in tentative discourse, whereas teachers, we tend to talk in absolutes. So, students are always talking to each other like that. When we put them in collaborative groups, they're like, “Well, maybe we should try this.” “I'm wondering if this'll work.” “Hey, have we thought about this?” “I wonder if…” Right? And it's so inviting to contribute to. Mike: That's fascinating. I'm going to move a little bit and start to focus on grouping. So, in the book, you looked really closely at the way that we group students for collaborative problem solving and how that impacts the way students engage in a collaborative effort. And I'm wondering if you could talk a little bit about the type of things that you were examining. Peter: OK. So, you don't have to spend a lot of time in classrooms before you see the two dominant paradigms for grouping. So, the first one we tend to see a lot at elementary school. So, that one is called “strategic grouping.” Strategic grouping is where the teacher has a goal and then they're going to group their students to satisfy that goal. So, maybe my goal is to differentiate, so I'm going to make ability groups. Or maybe my goal is to increase productivity, so I'm going to make mixed-ability groups. Or maybe my goal is to just have peace and quiet, so I'm going to keep those certain students apart. Whatever my goal is, I'm going to create the groups to try to achieve that goal, recognizing that how students behave in the classroom has a lot to do with who they're partnered with. So that's strategic grouping. It is the dominant grouping paradigm we see in elementary school. By the time we get to high school, we tend to see more of teachers going, “Work with who you want.” This is called “self-selected groupings.” And this is when students are given the option to group themselves any way they want. And alert: They don't group themselves for academic reasons, they group themselves for social reasons. And I think every listener can relate to both of those forms of grouping. It turns out that both of those are highly ineffective at getting students to think. And ironically, for the exact same reason. We surveyed hundreds of students who were in these types of grouping settings: strategic grouping or self-selected groupings. We asked one question: “If you knew you were going to work in groups today, what is the likelihood you would offer an idea?” That was it. And 80 percent of students said that they were unlikely or highly unlikely to offer an idea, and that was the exact same, whether they were in strategic groupings or self-selected groupings. The data cut the same. Mike: That's amazing, Peter. Peter: Yeah, and it's for the same reason, it turns out; that whether students were being grouped strategically or self-selected, they already knew what their role was that day. They knew what was expected of them. And for 80 percent of the students, their role is not to think. It's not to lead. Their role is to follow, right? And that's true whether they're grouping themselves socially, where they already know the social hierarchy of this group, or they're being grouped strategically. We interviewed hundreds of students. And after grade 3, every single student could tell us why they were in the group this teacher placed them in. They know . They know what you think of them. You're communicating very clearly what you think their abilities are through the way you group them, and then they live down to that expectation. So, that's what we were seeing in classrooms was that strategic grouping may be great at keeping the peace. And self-selected grouping may be fabulous for getting students to stop whining about collaboration. But neither of them was effective for getting students to think. In fact, they were quite the opposite. They were highly ineffective for getting students to think. Mike: So, I want to keep going with this. And I think one of the things that stood out for me as I was reading is this notion that regardless of the rationale that a teacher might have for grouping, there's almost always a mismatch between what the teacher's goals are and what the students’ goals are. I wonder if you could just unpack this and maybe explain this a bit more. Peter: So, when you do strategic grouping, do you really think the students are with the students that they want to be with? One of the things that we saw happening in elementary school was that strategic grouping is difficult. It takes a lot of effort to try to get the balance right. So, what we saw was teachers largely doing strategic grouping once a month. They would put students into a strategic group, and they would keep them in that group for the entire month. And the kids care a lot about who they're with, when you're going to be in a group for a month. And do you think they were happy with everybody that was in that group? If I'm going to be with a group of students for a month, I'd rather pick those students myself. So, they're not happy. You've created strategic groupings. And, by definition, a huge part of strategic grouping is keeping kids who want to be together away from each other. They're not happy with that. Self-selected groupings, the students are not grouping themselves for academic reasons. They're just grouping themselves for social reasons so that they can socialize, so they talk, so they can be off topic, and all of these things. And yes, they're not complaining about group work, but they're also not being productive. So, the students are happy. But do you think the teacher's happy? Do you think the teacher looks out across that room and goes, “Yeah, there were some good choices made there.” No, nobody's happy, right? If I'm grouping them strategically, that's not matching their goals. That's not matching their social goals. When they're grouping themselves in self-selected ways, that's matching their social goals but not matching my academic goals for them. So, there's always going to be this mismatch. The teacher, more often than not, has academic goals. The students, more often than not, have social goals. There are some overlaps, right? There are students who are like, “I'm not happy with this group. I know I'm not going to do well in this group. I'm not going to be productive.” And there are some teachers who are going, “I really need this student to come out of their shell, so I need to get them to socialize more.” But other than that, by and large, our goals as teachers are academic in nature. The goals as students are social in nature. Mike: I think one of the biggest takeaways from your work on grouping, for me at least, was the importance of using random groups. And I have to admit, when I read that there was a part of me, thinking back to my days as a first grade teacher, that felt a little hesitant. As I read, I came to think about that differently. But I'm wondering if you can talk about why random groups matter, the kind of impact that they have on the collaborative experience and the learning experience for kids. Peter: All right, so going back to the previous question. So, we have this mismatch. And we have also that 80 percent of students are not thinking; 80 percent of students are entering into that group not prepared to offer an idea. So those are the two problems that we're trying to address here. So, random groups. Random wasn't good enough. It had to be visibly random. The students had to see the randomness because when we first tried it, we said, “Here's your random groups.” They didn't believe we were being random. They just thought we were being strategic. So, it has to be visibly random, and it turns out it has to be frequent as well. About once every 45 to 75 minutes. See, when students are put into random groups, they don't know what their role is. So, we're solving this problem. They don't know what their role is. When we started doing visibly random groups frequently, within three weeks we were running that same survey: “If you know you're going to work in groups today, what is the likelihood you would offer an idea?” Remember the baseline data was that 80 percent of students said that they were unlikely or highly unlikely, and, all of a sudden, we have 100 percent of students saying that they're likely or highly likely. That was one thing that it solved. It shifted this idea that students were now entering groups willing to offer an idea, and that's despite 50 percent of them saying, “It probably won't lead to a solution, but I'm going to offer an idea.” Now why is that? Because they don't know what their role is. So, right on the surface, what random groups does, is it shatters this idea of preconceived roles and then preconceived behaviors. So, now they enter the groups willing to offer an idea, willing to be a contributor, not thinking that their role is just to follow. But there's a time limit to this because within 45 to 75 minutes, they're going to start to fall into roles. In that first 45 minutes, the roles are constantly negotiated. They're dynamic. So, one student is being the leader, and the others are being the follower. And now, someone else is a leader, the others are following. Now everyone is following. They need some help from some external source. Now everyone is leading. We’ve got to resolve that. But there is all of this [dynamism] and negotiation going on around the roles. But after 45 to 75 minutes, this sort of stabilizes, and now you have sort of a leader and followers, and that's when we need to randomize again so that the roles are dynamic and that the students aren't falling into sort of predefined patterns of nonthinking behavior. Mike: I think this is fascinating because we've been doing some work internally at MLC around this idea of status or the way that, the stories that kids tell about one another or the labels that kids carry either from school systems or from the community that they come from, and how those things are subtle. They're unspoken, but they often play a role in classroom dynamics in who gets called on. What value kids place on a...
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Season 2 | Episode 18 - The Promise of Counting Collections - Guest: Danielle Robinson and Dr. Melissa Hedges
05/23/2024
Season 2 | Episode 18 - The Promise of Counting Collections - Guest: Danielle Robinson and Dr. Melissa Hedges
ROUNDING UP: SEASON 2 | EPISODE 18 Earlier this season, we released an episode focused on the complex and interconnected set of concepts that students engage with when they learn to count. In this follow-up episode, we’re going to examine a powerful routine called counting collections. We’ll be talking with Danielle Robinson and Dr. Melissa Hedges from Milwaukee Public Schools about counting collections and the impact this routine can have on students. RESOURCES TRANSCRIPT Mike Wallus: Earlier this season, we released an episode focused on the complex and interconnected set of concepts that students engage with as they learn to count. In this follow-up episode, we're going to examine a powerful routine called “counting collections.” We'll be talking with Danielle Robinson and Dr. Melissa Hedges from the Milwaukee Public Schools about counting collections and the impact that this routine can have on student thinking. Well, welcome to the podcast, Danielle and Melissa. I can't tell you how excited I am to talk with y'all about the practice of counting collections. Melissa Hedges: Thanks for having us. Danielle Robinson: Yes, we're so excited to be here. Mike: I want to start this conversation by acknowledging that the two of you are actually part of a larger team of educators who really took this work on counting collections. You introduced it in the Milwaukee Public Schools. And, Melissa, I think I'll start with you. Can you take a moment to recognize the collaborators who have been a part of this work? Melissa: Absolutely. In addition to Danielle and myself, we are fortunate to work with three other colleagues: Lakesha King, Krista Beal, and Claire Madden. All three are early childhood coaches that actively support this work as well. Mike: So, Danielle, I wonder for some folks if we can help them see this practice more clearly. Can you spend time unpacking: What does counting collections look like in a classroom? If I walked in, what are some of the things that I might see? Danielle: Yeah. I think what's really amazing about counting collections is there might be some different ways that you might see counting collections happening in the classroom. When you walk into a classroom, you might see some students all over. Maybe they're sitting at tables, maybe they're on the carpet. And what they're doing is they're actually counting a baggie of objects. And really their job is to answer this question, this very simple but complicated question of, “How many?” And they get to decide how they want to count. Not only do they get to pick what they want to count, but they also get to pick their strategy of how they actually want to count that collection. They can use different tools. They might be using bowls or plates. They might be using 10-frames. They might be using number paths. You might see kiddos who are counting by 1s. You might see kids who are making different groupings. At times, you might also see kiddos [who] are in stations, and you might see a small group where a teacher is doing counting collections with a few kiddos. You might see them working with partners. And I think the beautiful piece of this and the unique part of counting collections within Milwaukee Public Schools is that we've been able to actually pair the counting trajectory from Doug Clements and Julie Sarama with counting collections, where teachers are able to do an interview with their students, really see where they're at in their counting so that the kids are counting a just-right collection for them—something that's not too easy, something that's not too hard, but something that is available for them to really push them in their understanding of counting. So, you're going to see kids counting different sizes. And we always tell the teachers it's a really beautiful moment when you're looking across the classroom and as a teacher, you can actually step back and know that every one of your kids [is] getting what they need in that moment. Because I think oftentimes, we really don't ever get to feel like that, where we feel like, “Wow, all my kids are getting what they need right now, and I know that I am providing the scaffolds that they need.” Mike: So I want to ask you a few follow-ups, if I might, Danielle. Danielle: Yeah, of course. Mike: There's a bit of language that you used initially, where — I'm paraphrasing, and tell me where I get this wrong. You use the language “simple yet complicated,” I think. Am I hearing that right? Danielle: I did. I did, yeah. Mike: Tell me about that. Danielle: I think it's so interesting because a lot of times when we introduce this idea of counting collections with our teachers, they're like, “Wait a minute. So I'm supposed to give this baggie of a bunch of things to my students, and they just get to go decide how they want to count it?” And we're like, “Yeah, that is absolutely what we're asking you to do.” And they feel nervous because [of] this idea of the kids, they're answering “how many?”, but then there's all these beautiful pieces [that are] a part of it. Maybe kids are counting by 1s. Maybe they're deciding that they want to make groups. Maybe they're working with a partner. Maybe they're using tools. It's kind of opened up this really big, amazing idea of the simple question of “how many?”, but there's just so many things that can happen with it. Mike: There's two words that kept just flashing in front of my eyes as I was listening to you talk. And the words were “access” and “differentiation.” And I think you didn't explicitly say those things, but they really jump out for me in the structure of the task and the way that a teacher could take it up. Can you talk about the way that you think this both creates access and also the places where you see there's possibility for differentiation? Danielle: For sure. I'm thinking about a couple classrooms that I was in this week and thinking about once we've done the counting trajectory interview with our kiddos, you might have little ones who are still really working with counting to 10. So they have collections that they can choose that are just at that amount of about 10. We might have some kiddos who are really working kind of in that range of 20 to 40. And so we have collections that children can choose from there. And we have collections all the way up to about 180 in some cases. So we kind of have this really nice, natural scaffold within there where children are told, “Hey, you can go get this just-right color for you.” We have red collections, blue collections, green, and yellow. Within that, also, the children get to decide how they want to count. So if they are still really working on that verbal count sequence, then we allow them to choose to count by 1s. We have tools for them, like number paths, to help do that. Maybe we've got our kiddos who are starting to really think about this idea of unitizing and making groups of 10s. So then what they might do is they might take a 10-frame and they might fill their 10-frame and then actually pour that 10-frame into a bowl, so they know that that bowl now is a collection of 10. And so it's this really nice idea of helping them really start to unitize and to make different groupings. And I think the other beautiful piece too is that you can also partner. Students can work together and actually talk about counting together. And we found that that really supports them too of just that collaboration piece too. Mike: So you kind of started poking around the question that I was going to ask Melissa. Danielle and Melissa: [laugh] Mike: You said the word “unitizing,” which is the other thing that was really jumping out because I taught kindergarten and first grade for about eight years. And in my head, immediately all of the different trajectories that kids are on when it comes to counting, unitizing, combining — those things start to pop out. But, Melissa, I think what you would say is there is a lot of mathematics that we can build for kids beyond, say, K–2, and I'm wondering if you could talk a little bit about that. Melissa: Absolutely. So before I jump to our older kids, I'm just going to step back for a moment with our kindergarten, first- and second graders, and even our younger ones. So the mathematics that we know that they need to be able to count collections, that idea of cardinality, one-to-one correspondence, organization — Danielle did a beautiful job explaining how the kids are going to grab a bag, figure out how to count, it's up to them — as well as this idea of producing a set, thinking about how many, being able to name how many. The reason why I wanted to go back and touch on those is that we know that as children get older and they move into third, fourth, and fifth grade, those are understandings that they must carry with them. And sometimes those ideas aren't addressed well in our instructional materials. So the idea of asking a first- and second grader to learn how to construct a unit of 10 and know that 10 ones is 1 ten is key because when we look at where place value tends to fall apart in our upper grades. My experience has been, it's fifth grade, where all of a sudden we're dealing with big numbers, we're moving into decimals, we're thinking about different sized units, we've got fractions. There's all kinds of things happening. So the idea of counting collections in the early elementary grades helps build kids' number sense, provides them with that confidence of magnitude of number. And then as they move into those either larger collections or different ways to count, we can make beautiful connections to larger place values — so hundreds, thousands, ten thousands. Sometimes those collections will get big. All those early number relationships also build. So those early number relationships, part-whole reasoning that numbers are composed and decomposed of parts. And then we've just seen lots of really, really fun work about additive and multiplicative thinking. So in a third-, fourth-, fifth grade classroom, what I used to do is dump a cup full of lima beans in the middle of the table and say, “How many are there?” And there's a bunch there. So they can count by 1s. It's going to take a long time. And then once they start to figure out, “Oh wait, I can group these.” “Well, how many groups of 5 do you have?” And how we can extend to that from that additive thinking of 5 plus 5 plus 5 plus 5 to then thinking about and extending it to multiplicative thinking. So I think the extensions are numerous. Mike: There's a lot there that you said, and I think I want to ask a couple follow-ups. First thing that comes to mind is, we've been interviewing a guest for a different podcast, and this idea that unitizing is kind of a central theme that runs really all the way through elementary mathematics and certainly beyond that. But I really am struck by the way that this idea of unitizing and not only being able to unitize, but I think you can physically touch the units, and you can physically reunitize when you pour those things into the cup. And it's giving kids a bit more space with the physical materials themselves before you step into something that might be more abstract. I'm wondering if that's something that you see as valuable for kids and maybe how you see that play out? Melissa: Yes, it's a great question. I will always say when we take a look at our standard base ten blocks, “The person [who] really understands the construction of those base ten blocks is likely the person [who] invented them.” They know that one little cube means 1, and that all of a sudden these 10 cubes are fused together and we hold it up and we say, “Everybody, this is 10 ones. Repeat, 1 ten.” What we find is that until kids have multiple experiences and opportunities over time to construct units beyond 1, they really won't do it with deep understanding. And again, that's where we see it fall apart when they're in the fourth and fifth grade. And they're struggling just to kind of understand quantity and magnitude. So the idea and the intentionality behind counting collections and the idea of unitizing is to give kids those opportunities that — to be quite honest, and no disrespect to the hardworking curriculum writers out there — it is a tricky, tricky, tricky idea to develop in children through paper and pencil and workbook pages. I think we have found over time that it's the importance of going, grabbing, counting, figuring it out. So if my collection is bears, does that collection of 10 bears look the same as 10 little sharks look the same as 10 spiders? So what is this idea of 10? And that they do it over and over and over and over again. And once they crack the code—that's the way I look at it—once our first- and second graders crack the code of counting collections, they're like, “Oh, this is not hard at all.” And then they start to play with larger units. So then they'll go, “Oh, wait, I can combine two groups of 10. I just found out that's 20. Can I make more 20s?” So, then we're thinking about counting not just by 1s, not just by 10s, but by larger units. And I think that we've seen that pay off in so many tremendous ways. And certainly on the affective side, when kids understand what's happening, there's just this sense of joy and excitement and interest in the work that they do, and I actually think they see themselves learning. Mike: Danielle, do you want to jump in here? Danielle: I think to echo that, I just recently was speaking with some teachers. And the principal was finally able to come and actually see counting collections happening. And what was so amazing is these were K–5 kiddos, five-year-olds, who were teaching the principal about what they were doing. This was that example where we want people to come in, and the idea is, what are you learning? How do you know you've learned it, thinking about that work of Hattie? And these five-year-olds were telling him exactly what they were learning and how they were learning it and talking about their strategies. And I just felt so proud of the K–5 teacher who shared that with me because her principal was blown away and was seeing just the beauty that comes from this routine. Mike: We did an episode earlier this year on place value, and the speaker did a really nice job of unpacking the ideas around it. I think what strikes me, and at this point I might be sounding a bit like a broken record, is the extent to which this practice makes place value feel real. These abstract ideas around reunitizing. And I think, Melissa, I'm going back to something you said earlier where you're like, “The ability to do this in an abstract space where you potentially are relying on paper and pencil or even drawing, that's challenging.” Whereas this puts it in kids' hands, and you physically reunitize something, which is such a massive deal, this idea that 1 ten and 10 ones have the same value even though we're looking at them differently, simultaneously. That's such a big deal for kids, and it just really stands out for me as I hear you all talk. Melissa: I had the pleasure of working with a group of first grade teachers the other day, and we were looking at student work for a simple task that the kids were asked to do. I think it was 24 plus 7, and so it was just a very quick PLC: “Look at this work. Let's think about what they're doing.” And many of the children had drawn what the teachers referred to as sticks and circles or sticks and dots. And I said, “Well, what do those sticks and dots mean?” Right? Well, of course the stick is the ten and the dot is the one. And I said, “There's lots of this happening,” I said, “Let's pause for a minute and think, ‘To what degree do you think your children understand that that line means 10 and that dot means 1? And that there’s some kind of a connection, meaningful connection for them just in that drawing.’” It got kind of quiet, and they're like, “Well, yep, you're right. You're right. They probably don't understand what that is.” And then one of the teachers very beautifully said, “This is where I see counting collections helping.” It was fantastic. Mike: Danielle, I want to shift and ask you a little bit about representation. Just talk a bit about the role of representing the collection once the counting process and that work has happened. What do you all ask kids to do in terms of representation and can you talk a little bit about the value of that? Danielle: Right, absolutely. I think one thing that as we continue to go through in thinking about this routine and the importance of really helping our students make sense and count meaningfully. I think we will always go back to our math teaching framework that's been laid out for us through Taking Action , Principles to Action, Catalyzing Change . And really thinking about the power of using multiple representations. And how, just like you said, we want our students to be able to be physically unitizing, so we have that aspect of working with our actual collections. And then how do we help our students understand that “You have counted your collection. Now what I want you to do is, I want you to actually visually represent this. I want you to draw how you counted.”? And so what we talk about with the kids is, “Hey, how you have counted, if you have counted by 1s, I should be able to see that on your paper. I should be able to look at your paper, not see your collection and know exactly how you counted. If you counted by 10s, I should be able to see, ‘Oh my gosh, look, that's their bowl. I see their bowls, I see their plates, I see their 10s inside of there.’” And to really help them make those connections moving back and forth between those representations. And I think that's also that piece too for them that then they can really hang their hat on: “This is how I counted. I can draw a picture of this. I can talk about my strategy. I can share with my friends in my classroom.” And then that's how we like to close with our counting collections routine is really going through and picking a piece of student work and really highlighting a student's particular strategy. Or even just highlighting several and being like, “Look at all this work they did today. Look at all of this mathematical thinking.” So I think it's a really important and powerful piece, especially with our first- and second graders too.; we really bring in this idea of equations too. So this idea of, “If I've counted 73, and I've got my 7 groups of 10, I should have 10 plus 10 plus 10, right, all the way to 70. And then adding my 3.” So I think it's just a continuous idea of having our kids really developing that strong understanding of meaningful counting, diving into place value. Mike: I'm really struck by the way that you described the protocol where you said you're asking kids to really clearly make sure that what they're doing aligns with their drawing. The other piece about that is it feels like, one, that sets kids up to be able to share their thinking in a way where they've got a scaffold that they've created for themself. The other thing that it really makes me think about is how, if I'm a teacher and I'm looking at student work, I can really use that to position that student's idea as valuable or position that student's thinking as something that's important for other people to notice or attend to. So you could use this to really raise a student's idea’s status or raise the student’s status as well. Does that actually play out in a reality? Danielle: It does actually. So a couple of times what I will do is I will go into a classroom. And oftentimes it can be kind of apparent … which students may just not have the strongest...
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Season 2 | Episode 17 – Making Sense of Spatial Reasoning - Guest: Dr. Robyn Pinilla
05/09/2024
Season 2 | Episode 17 – Making Sense of Spatial Reasoning - Guest: Dr. Robyn Pinilla
ROUNDING UP: SEASON 2 | EPISODE 17 Spatial reasoning can be a nebulous concept that is hard for many educators to define. In this episode, we’re talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas at El Paso. We’ll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students. RESOURCES TRANSCRIPT Mike Wallus: Spatial reasoning can be a nebulous concept, and it's often hard for many educators to define. In this episode, we're talking about spatial reasoning with Dr. Robyn Pinilla from the University of Texas [at] El Paso. We'll examine the connections between spatial reasoning and other mathematical concepts and explore different ways that educators can cultivate this type of reasoning with their students. Welcome to the podcast, Robyn. I'm really excited to be talking with you about spatial reasoning. Robyn Pinilla: And I am excited to be here. Mike: Well, let me start with a basic question. So when we're talking about spatial reasoning, is that just another way of saying that we're going to be talking about ideas that are associated with geometry? Or are we talking about something bigger? Robyn: It's funny that you say it in that way, Mike, because geometry is definitely the closest mathematical content that we see in curricula, but it is something much bigger. So I started with the misconception and then I used my own experiences to support that idea, that this was just geometry, because it was my favorite math course in high school because I could see the concepts modeled and I could make things more tangible. Drawing helped me to visualize some of those concepts that I was learning instead of just using a formula that I didn't necessarily understand. So, at that time, direct instruction really ruled, and I'm unsure what the conceptual understandings of my teachers even were because what I recall is doing numbers 3 through 47 odds in the back of the book and just plugging through these formulas. But spatial reasoning allows us to develop our concepts in a way that lead to deeper conceptual understanding. I liked geometry, and it gave me this vehicle for mathematizing the world. But geometry is really only one strand of spatial reasoning. Mike: So you're already kind of poking around the question that I was going to ask next, which is the elevator description of, “What do we mean when we talk about spatial reasoning, and why does it matter? Why is it a big deal for students?” Robyn: So spatial reasoning is a notoriously hard-to-define construct that deals with how things move in space. It's individually how they move in space, in relation to one another. A lot of my ideas come from a network analysis that [Cathy] Bruce and colleagues did back in 2017 that looked at the historical framing of what spatial reasoning is and how we talk about it in different fields. Because psychologists look at spatial reasoning; mathematics educators look at spatial reasoning. There [are] also connections into philosophy, the arts. But when we start moving toward mathematics more specifically, it does deal with how things move in space individually and in relation to one another. So, with geometry, whether the objects are sliding and transforming or we're composing and decomposing to create new shapes, those are the skills in two-dimensional geometry that we do often see in curricula. But the underlying skills are also critical to everyday life, and they can be taught as well. And when we're thinking about the everyday constructs that are being built through our interactions with the world, I like to think about the GPS on our car. So spatial reasoning has a lot of spatiotemporal processes that are going on. It's not just thinking about the ways that things move in relation to one another or the connections to mathematics, but also the way that we move through this world, the way that we navigate through it. So I'll give a little example. Spatiotemporal processes have to do with us running errands, perhaps. How long does it take you to get from work to the store to home? And how many things can you purchase in the store knowing how full your fridge currently is? What pots and pans are you going to use to cook the food that you purchase? And what volume of that food are you and your family going to consume? So all of those daily tasks involve conceptions of how much space things take. And we could call it “capacity,” which situates nicely within the measurement domain of mathematics education. But it's also spatial reasoning, and it extends further than that. Mike: That is helpful. I think you opened up my understanding of what we're actually talking about, and I think the piece that was really interesting is how in that example of “I'm going to the grocery store. How long will it take? How full is my fridge? What are the different tools that I'll use to prepare? What capacity do they have?” I think that really helped me broaden out my own thinking about what spatial reasoning actually is. I wonder if we could shift a bit and you could help unpack, for educators who are listening, a few examples of tasks that kids might encounter that could support the development of spatial reasoning. Robyn: Sure. My research and work [are] primarily focused on early childhood and elementary. So I'm going to focus there but then kind of expand up. Number one, let's play. That's the first thing that I want to walk into a classroom and see: I want to see the kids engaging with blocks, LEGOS, DUPLOs, and building with and without specific intentions. Not everything has to have a preconceived lesson. So one of the activities I've been doing, actually, with teachers and professional development sessions lately is a presentation called “Whosits and Whatsits.” I have the teachers create whatsits that do “thatsits,” meaning, they create something that does something. I don't give them a prompt of what problem they're going to be solving or anything specific for them to build, but rather say, “Here are materials.” We give them large DUPLO blocks; magnet tiles and Magformers; different types of wooden, cardboard, and foam blocks; PVC pipes, which are really interesting in the ways that teachers use them. And have them start thinking as though they're the children in the class, and they're trying to build something that takes space and can be used in different ways. So [in] the session we did a couple of weeks ago, some teachers came up with, first, there was a swing that they had put a little frog in that they controlled with magnets. So they had used the PVC pipe at the top to be the part that the top of the swing connected over and then were using the magnets to guide it back and forth without ever having to touch the swing. And I just thought that was the coolest way for them to be using these materials in really playful, creative ways that could also engender them taking those lessons back into their classroom. I have also recently been reminded of the importance of modeling with fractions. So, are you familiar with the Which One Doesn't Belong? tasks? Mike: Absolutely love them. Robyn: Yes. There's also a website for fraction talks [where] children can look at visual representations of fractions and determine which one doesn't belong for some reason. That helps us to see the ways that children are thinking about the fractional spaces and then justifying their reason around them. With that, we can talk about the spatial positioning of the fractional pieces that are colored in or the ways that they're separated if those colored pieces are in different places on the figure that's being shown. They open up some nice spaces for us to talk about different concepts and use that language of spatial reasoning that is critical for teachers to engage in to show the ways that students can think about those things. Mike: So I want to go back to this notion of play, and what I'm curious about is, why is situating this in play going to help these ideas around spatial reasoning come out, as opposed to, say, situating it in a more controlled structure? Robyn: Well, I think by situating spatial reasoning within play, we do allow teachers to respond in the moment rather than having these lesson plans that they are required to plan out from the beginning. A lot of the ideas within spatial reasoning, because it's a nebulous construct and it's learned through our everyday experiences and interactions with the world, they are harder to plan. And so when children are engaged in play in the classroom, teachers can respond very naturally so that they're incorporating the mathematizing of the world into what the students are already doing. So if you take, for example, one of my old teachers used to do a treasure hunt — [a] great way to incorporate spatial reasoning with early childhood elementary classrooms — where she would set up a mapping task, is really what it was. But it was introducing the children to the school itself and navigating that environment, which is critical for spatial-reasoning skills. And they would play this gingerbread man-type game of, she would read the book and then everybody would be involved with this treasure hunt where the kiddos would start out in the classroom, and they would get a clue to help them navigate toward the cafeteria. When they got to the cafeteria, the gingerbread man would already be gone. He would've already run off. So they would get their next clue to help them navigate to the playground, [and] so on and so forth. They would go to the nurse's office, the principal[’s office], the library, all of the critical places that they would be going through on a daily basis or when they needed to within the school. And it reminds me that there was also a teacher I once interviewed who used orienteering skills with her students. Have you ever heard of orienteering? Mike: The connection I'm making is to something like geocaching, but I think you should help me understand it. Robyn: Yeah, that's really similar. So it's this idea that children would find their way [to] places. Pathfinding and way finding are also spatial-reasoning skills that are applied within our real world. And so while it may not be as scientific or sophisticated as doing geocaching, it has children with the idea of navigating in our real world, helps them start to learn cardinality and the different ways of thinking about traversing to a different location, which, these are all things that might better relate to social studies or technology, other STEM domains specifically, but that are undergirded by the spatial reasoning, which does have those mathematics connections. Mike: I think the first thing that occurred [to me] is, all of the directional language that could emerge from something like trying to find the gingerbread boy. And then the other piece that you made me think about just now is this opportunity to quantify distance in different ways. And I'm sure there are other things that you could draw out, especially in a play setting, where the structure is a little bit looser and it gives you a little bit more space, as you said, to respond to kids rather than feeling like you have to impose the structure. Robyn: Yeah, absolutely. There's an ability when teachers are engaging in authentic ways with the students, that they're able to support language development, support ideation and creation, without necessarily having kids sit down and fill out a worksheet that says, “Where is the ball?” “The ball is sitting on top of the shelf.” Instead, we can be on the floor working with students and providing those directions of, “Oh, hey, I need you to get me those materials from the shelf on the other side of the room,” but thinking about, “How can I say that in a way that better supports children understanding the spatial reasoning that's occurring in our room?” So maybe it's, “Find the pencil inside the blue cup on top of the shelf that's behind the pencil sharpener …” Mike: Oo. Robyn: … getting really specific in the ways that we talk about things so that we're ingraining those ideas in such a way that it becomes part of the way that the kids communicate as well. Mike: You have me thinking that there's an intentionality in language choice that can create that, but then I would imagine as a teacher I could also revoice what students are saying and perhaps introduce language in that way as well. Robyn: Yeah, and now you have me thinking about a really fun routine: number talks, of course. And if we do the idea of a dot talk instead of a number talk, thinking about the spatial structuring of the dots that we're seeing and the different ways that you can see those arrangements and describe the quantification of the arrangement. It's a nice way to introduce educators to spatial reasoning because it might be something that they're already doing in the classroom while also providing an avenue for children to see spatial structuring in a way that they're already accustomed to as well, based on the routines that they're receiving from the teacher. Mike: I think what's really exciting about this, Robyn, is the more that we talk, the more two things jump out. I think one is, my language choices allow me to introduce these ideas in a way that I don't know that I'd thought about as a practitioner. Part two is that we can't really necessarily draw a distinction between work we're doing around numbers and quantity and spatial reasoning, that there are opportunities within our work around number quantity and within math content to inject the language of spatial reasoning and have it become a part of the experience for students. Robyn: Yeah, and that's important that I have conveyed that without explicitly saying it because that's the very work that I'm doing with teachers in their classrooms at this time. So one, as you're talking about language, and I hate to do this, but I'm going to take us a little bit off topic for a moment. So I keep seeing this idea on Twitter, or whatever we call it at this point, that some people actually don't hear music in their heads. This idea is wild to me because I have songs playing in my head all the time. But at the same time, what if we think about the idea that some people don't also visualize things; they don't imagine those movements continuously that I just see. And so, as teachers, we really need to focus on that same idea that children need opportunities to practice what we think they should be able to hear but also practice what we think they should be able to see. I'm not a cognitive scientist. I can't see inside someone's head. But I am a teacher by trade, so I want to emphasize that teachers can do what's within their locus of control so that children can have opportunities to talk about those tasks. One that I recently saw was a lesson on clocks. So while I was sitting there watching her teach — she was using a Judy Clock; she was having fun games with the kids to do a little competition where they could read the clock and tell her what time it was. But I was just starting to think about all of the ways that we could talk about the shorter and longer hands, the minute and hour hands. The ways that we could talk about them rotating around that center point. What shape does the hand make as it goes around that center point? And what happens if it doesn't rotate fully? Now I'm going back to those fractional ideas from earlier with the Which One Doesn't Belong? tasks of having full shapes versus half shapes, and how we see those shapes in our real lives that we can then relate with visualized shapes that some children may or may not be able to see. Mike: You have me thinking about something. First of all, I'm so glad that you mentioned the role of visualization. Robyn: Yeah. Mike: You had me thinking about a conversation I was having with a colleague a while ago, and we had read a text that we were discussing, and the point of conversation came up. I read this and there's a certain image that popped into my head. Robyn: Mm-hmm. Mike: And the joke we were making is, “I'm pretty certain that the image that I saw in my head, having read this text, is not the same as what you saw.” What you said that really struck home for me is, I might be making some real assumptions about the pictures that kids see in their head and helping build those internal images, those mental movies. That's a part of our work as well. Robyn: Absolutely. Because I'm thinking about the way that we have prototypical shapes. So, a few years ago I was working with some assessments, and the children were supposed to be able to recognize an equilateral triangle — whether it was gravity-based or facing another orientation. And there were some children who automatically could see that the triangle was a triangle no matter which direction it was “pointing.” Whereas others only recognized it [as] a triangle if it were gravity-based. And so we need to be teaching the properties of the shapes beyond just that image recognition that oftentimes our younger students come out with. I tend to think of visualization and language as supporting one another with the idea that when we are talking, we're also writing a descriptive essay. Our words are what create the intended picture — can't say that it's always the picture that comes out — but the intended picture for the audience. What we're hopeful for in classrooms is that because we're sharing physical spaces and tangible experiences, that the language used around those experiences could create shared meaning. That's one of the most difficult pieces in talking about spatial reasoning — or, quite frankly, anything else — is that oftentimes our words may have different meanings depending on who the speaker and who the listener are. And so, navigating what those differences are can be quite challenging, which is why spatial reasoning is still so hard to define. Mike: Absolutely. My other follow-up is, if you were to offer people a way to get started, particularly on visualization, is there a kind of task that you imagine might move them along that pathway? Robyn: I think the first thing to do is really grasp an approximation. I'm not going to say, “Figure out what spatial reasoning is,” but just an approximation or a couple of the skills therein that you feel comfortable with. So spatial reasoning is really the set of skills that undergirds almost all of our daily actions, but it also can be inserted into the lessons that teachers are already teaching. I think that we do have to acknowledge that spatial reasoning is hard to define, but the good news is that we do reason spatially all day every day. If I am in a classroom, I want to look first at the teaching that's happening, the routines that are already there, and see where some spatial reasoning might actually fit in. With our young classes, I like to think about calendar math. Every single kindergarten [or] first grade classroom that you walk into, they're going to have that calendar on the wall. So how can you work into the routines that are occurring, that spatial language, to describe the different components of the routine? So as a kiddo is counting on that hundreds chart, talking about the ways in which they're moving the pointer along the numbers. When they're counting by 10s, talk about the ways that they're moving down. When they're finding the patterns that are on the calendar, because all of those little calendar numbers for the day, they wind up having a pattern within them in most of the curricular kits. So thinking about just the ways that we can use language therein. Now, with older students, I think that offering that variety of models or...
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Season 2 | Episode 16 – Strengthening Tasks Through Student Talk - Guest: Drs. Amber Candela and Melissa Boston
04/18/2024
Season 2 | Episode 16 – Strengthening Tasks Through Student Talk - Guest: Drs. Amber Candela and Melissa Boston
ROUNDING UP: SEASON 2 | EPISODE 16 In this episode, we are talking with Drs. Amber Candela and Melissa Boston about powerful but practical strategies for supporting student talk in the elementary math classroom. RESOURCES TRANSCRIPT Mike Wallus: One of the goals I had in mind when we first began recording Rounding Up was to bring to life the best practices that we aspire to in math education and to offer entry points so that educators would feel comfortable trying them out in their classrooms. Today, we're talking with Drs. Amber Candela and Melissa Boston about powerful but practical strategies for supporting student talk in the elementary math classroom. Welcome to the podcast, Amber and Melissa. We're really excited to be talking with you today. Amber Candela: Thank you for having us. Melissa Boston: Yes, thank you. Mike: So we've done previous episodes on the importance of offering kids rich tasks, but one of the things that you two would likely argue is that rich tasks are necessary, but they're not necessarily sufficient, and that talk is actually what makes the learning experience really blossom. Is that a fair representation of where you all are at? Melissa: Yes. I think that sums it up very well. In our work, which we've built on great ideas from , about tasks and the importance of cognitively challenging tasks and work on the importance of talk in the classroom. Historically, it was often referred to as “talk moves.” We've taken up the term “discourse actions” to think about how do the actions a teacher takes around asking questions and positioning students in the classroom—and particularly these talk moves or discourse actions that we've named “linking” and “press”—how those support student learning while students are engaging with a challenging task. Mike: So I wonder if we could take each of the practices separately and talk through them and then talk a little bit about how they work in tandem. And Melissa, I'm wondering if you could start unpacking this whole practice of linking. How would you describe linking and the purpose it plays for someone who, the term is new for them? Melissa: I think as mathematics teachers, when we hear “linking,” we immediately think about the mathematics and linking representations or linking strategies. But we’re using it very specifically here as a discourse action to refer to how a teacher links student talk in the classroom and the explicit moves a teacher makes to link students' ideas. Sometimes a linking move is signaled by the teacher using a student's name, so referring to a strategy or an idea that a student might've offered. Sometimes linking might happen if a teacher revoices a student's idea and puts it back out there for the class to consider. The idea is in the way that we're using linking, that it's links within the learning community, so links between people in the classroom and the ideas offered by those people, of course. But the important thing here that we're looking for is how the links between people are established in the verbal, the explicit talk moves or discourse actions that the teacher's making. Mike: What might that sound like? Melissa: So that might sound like, “Oh, I noticed that Amber used a table. Amber, tell us how you used a table.” And then after Amber would explain her table, I might say, “Mike, can you tell me what this line of Amber's table means?” or “How is her table different from the table you created?” Mike: You're making me think about those two aspects, Melissa, this idea that there's mathematical value for the class, but there's also this connectivity that happens when you're doing linking. And I wonder how you think about the value that that has in a classroom. Melissa: We definitely have talked about that in our work as well. I’m thinking about how a teacher can elevate a student's status in mathematics by using their name or using their idea, just marking or identifying something that the student said is mathematically important that's worthy of the class considering further. Creating these opportunities for student-to-student talk by asking students to compare their strategies or if they have something to add on to what another student said. Sometimes just asking them to repeat what another student said so that there's a different accountability for listening to your peers. If you can count on the teacher to revoice everything, you could tune out what your peers are saying, but if you might be asked to restate what one of your classmates had just said, now there's a bit more of an investment in really listening and understanding and making sense. Mike: Yeah, I really appreciate this idea that there's a way in which that conversation can elevate a student's ideas, but also to raise a student's status by naming their idea and positioning it as important. Melissa: I have a good example from a high school classroom where a student [...] was able to solve the contextual problem about systems of equations, so two equations, and it was important for the story when the two equations or the two lines intersected. And so one student was able to do that very symbolically: they created a graph, they solved the system of equations. Where another student said, “Oh, I see what you did. You found the difference in the cost per minute, and you also found the difference in the starting point, and then one had to catch up to the other.” And so the way that the teacher kind of positioned those two strategies, one had used a sensemaking approach based really in the context. The other had used their knowledge of algebra. And by positioning them together, it was actually the student who had used the algebra had higher academic status, but the student who had reasoned through it had made this breakthrough that was really the aha moment for the class. Mike: That is super cool. Amber, can we shift to “press” and ask you to talk a little bit about what press looks like? Amber: Absolutely. So how Melissa was talking about linking is holding students accountable to the community; press is more around holding students accountable to the mathematics. And so the questions the teacher is going to ask [are] going to be more related specifically to the mathematics. So, “Can you explain your reasoning?” “How did you get that answer?” “What does this x mean?” “What does that intersection point mean?” And so the questions are more targeted at keeping the math conversation in the public space longer. Mike: I thought it was really helpful to just hear the example that Melissa shared. I'm wondering if there's an example that comes to mind that might shed some light on this. Amber: So when I'm in elementary classrooms and teachers are asking their kids about different problems, and kids will be like, “I got 2.” OK, “How did you get 2?” “What operation did you use?” “Why did you use addition when you could have used something else?” So it's really pressing at the, “Yes, you got the answer, but how did you get the answer?” “How does it make sense to you?”, so that you're making the kids rather than the teacher justify the mathematics that's involved. And they're the ones validating their answers and saying, “Yes, this is why I did this because …”. Mike: I think there was a point when I was listening to the two of you speak about this where—and forgive me if I paraphrase this a little bit—but you had an example where a teacher was interacting with a student and the student said something to the effect of, “I get it” or “I understand.” And the teacher came back and she said, “And what do you understand?” And it was really interesting because it threw the justification back to the student. Amber: Right. Really what the linking and press [do, they keep] the math actionable longer to all of the peers in the room. So it's having this discussion out loud publicly. So if you didn't get the problem fully all the way, you can hear your peers through the press moves, talk about the mathematics,. And then you can use the linking moves to think through, “Well, maybe if Mike didn't understand, if he revoices Melissa's comment, he has the opportunity to practice this mathematics speaking it.” And then you might be able to take that and be like, “Oh, wait, I think I know how to finish solving the problem now.” Mike: I think the part that I want to pull back and linger on a little bit is [that] part of the purpose of press is to keep the conversation about the mathematics in the space longer for kids to be able to have access to those ideas. I want y'all to unpack that just a little bit. Amber: Having linking and press at the end is holding the conversation longer in the classroom. And so the teacher is using the press moves to get at the mathematics so the kids can access it more. And then by linking, you're bringing in the community to that space and inviting them to add: “What do you agree [with]?” “Do you disagree?” “Can you revoice what someone said?” “Do you have any questions about what's happening?” Melissa: So when we talk about discourse actions, the initial discourse action would be the questions that the teacher asks. So there's a good task to start with. Students have worked on this task and produced some solution strategies. Now we're ready to discuss them. The teacher asks some questions so that students start to present or share their work and then it's after students' response [that] linking and press come in as these follow-up moves to do what Amber said: to have the mathematics stay in the public space longer, to pull more kids into the public space longer. So we're hoping that by spending more time on the mathematics, and having more kids access the mathematics, that we're bringing more kids along for the ride with whatever mathematics it is that we're learning. Mike: You're putting language to something that I don't know that I had before, which is this idea that the longer we can keep the conversation about the ideas publicly bouncing around—there are some kids who may need to hear an idea or a strategy or a concept articulated in multiple different ways to piece together their understanding. Amber: And like Melissa was saying earlier, the thing that's great about linking is oftentimes in a classroom space, teachers ask a question, kids answer, the teacher moves on. The engagement does drop. But by keeping the conversation going longer, the linking piece of it, you might get called on to revoice, so you need to be actively paying attention to your peers because it's on the kids now. The math authority has been shared, so the kids are the ones also making sense of what's happening. But it's on me to listen to my peers because if I disagree, there's an expectation that I'll say that. Or if I agree or I might want to add on to what someone else is saying. So oftentimes I feel like this pattern of teacher-student-teacher-student-teacher-student happens, and then what can start to happen is teacher-student-student-student-teacher. And so it kind of creates this space where it's not just back and forth; it kind of popcorns more around with the kids. Mike: You are starting to touch on something that I did want to talk about, though. Because I think when I came into this conversation, what was in my head is, like, how this supports kids in terms of their mathematical thinking. And I think where you two have started to go is: What happens to kids who are in a classroom where link and press are a common practice? And what happens to classrooms where you see this being enacted on a consistent basis? What does it mean for kids? What changes about their mathematical learning experience? Melissa: You know, we observe a lot of classrooms, and it's really interesting when you see even primary grade students give an answer and immediately say, you know, “I think it's 5 because …”, and they provide their justification just as naturally as they provide their answer. Or they're listening to their peers and they're very eager to say, “I agree with you; I disagree with you, and here's why” or “I did something similar” or “Here's how my diagram is slightly different.” So to hear children and students taking that up is really great. And it just—[there’s] a big shift in the amount of time that you hear the teacher talking versus the amount of time you hear children talking and what you're able to take away as the teacher or the educator formatively about what they know and understand based on what you're hearing them say. And so [in] classrooms where this has become the norm, you see fewer instances where the teacher has to use linking and press because students are picking this up naturally. Mike: As we were sitting here and I was listening to y'all talk, Amber, the thing that I wanted to come back to is I started reflecting on my own practice and how often, even if I was orchestrating or trying to sequence, it was teacher-student-teacher-student-teacher-student. It bounced back to me. And I'm really kind of intrigued by this idea—teacher-student-student-student-teacher—that the discourse, it's moving from a back and forth between one teacher, one student, rinse and repeat, and more students actually taking up the discourse. Am I getting that right? Amber: Yes. And I think really the thought is we always want to talk about the mathematics, but we also have to have something for the community. And that's why the linking is there because we also need to hold kids accountable to the community that they're in as much as we need to hold them accountable to the mathematics. Mike: So, Amber, I want to think about what does it look like to take this practice up? If you were going to give an educator a little nudge or maybe even just a starting point where teachers could take up linking and press, what might that look like? If you imagined, kind of, that first nudge or that first starting point that starts to build this practice? Amber: We have some checklists with sentence stems in [them]. And I think it's taking those sentence stems and thinking about when I ask questions like, “How did you get that?” and “How do you know this about that answer?” That's when you're asking about the mathematics. And then when you start to ask, “Do you agree with what so-and-so said? Can you revoice what they said in your own words?”, that's holding kids accountable to the community and just really thinking about the purpose of asking this question. Do I want to know about the math or do I want to build the conversation between the students? And then once you realize what you want that to be, you have the stem for the question that you want to ask. Mike: Same question, Melissa. Melissa: I think if you have the teacher who is using good tasks and asking those good initial questions that encourage thinking, reasoning, explanations, even starting by having them try out, once a student gives you a response, asking, “How do you know?” or “How did you get that?” and listening to what the student has to say. And then as the next follow-up, thinking about that linking move coming after that. So even a very formulaic approach where a student gives a response, you use a press move, hear what the student has to say, and then maybe put it back out to the class with a linking move. You know, “Would someone like to repeat what Amber just said?” or “Can someone restate that in their own words?” or whatever the linking move might be. Mike: So if these two practices are new to someone who's listening, are there any particular resources or recommendations that you'd share with someone who wants to keep learning? Amber: [laughs] Why, of course, we absolutely have resources. We wrote an article for the NCTM’s MTLT [Mathematics Teacher: Learning and Teaching PK-12] called “Discourse Actions to Promote Student Access.” And there are some vignettes in there that you can read through and then there [are] checklists with sentence stems for each of the linking and press moves. Melissa: Also, along with that article, we've used a lot of the resources from NCTM’s Principles to Actions [Professional Learning] Toolkit that's online, and some of the resources are free and accessible to everyone. Amber: And if you wanted to dig in a bit more, we do have a book called . And that goes in-depth with all of our rubrics and has other scenarios and videos around the linking and press moves along with other parts of the rubrics that we were talking about earlier. Mike: That's awesome. We will link all of that in our show notes. Thank you both so much for joining us. It was a real pleasure talking with you. Amber: Thanks for having us. Melissa: Thank you. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center |
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Season 2 | Episode 15 – Making Sense of Story Problems - Guest: Drs. Julia Hagge and Aina Appova
04/04/2024
Season 2 | Episode 15 – Making Sense of Story Problems - Guest: Drs. Julia Hagge and Aina Appova
ROUNDING UP: SEASON 2 | EPISODE 15 Contextualized story problems are an important tool that educators use to bring mathematics to life for their students. That said, navigating the meaning and language found in story problems is a challenge for many students. Today we’re talking with Aina Appova and Julia Hagge from The Ohio State University about strategies to help students engage with and make sense of story problems. RESOURCES TRANSCRIPT Mike Wallus: Story problems are an important tool that educators use to bring mathematics to life for their students. That said, navigating the meaning and language found in story problems is a challenge for many students. Today we're talking with Drs. Aina Appova and Julia Hagge from [The] Ohio State University about strategies to help students engage with and make sense of story problems. A note to our listeners: This podcast was recorded outside of our normal recording studio, so you may notice some sound quality differences from our regular podcast.Welcome to the podcast, Aina and Julia. We're excited to be talking to both of you. Aina Appova: Thank you so much for having us. We are very excited as well. Julia Hagge: Yes, thank you. We're looking forward to talking with you today. Mike: So this is a conversation that I've been looking forward to for quite a while, partly because the nature of your collaboration is a little bit unique in ways that I think we'll get into. But I think it's fair to describe your work as multidisciplinary, given your fields of study. Aina: Yes, I would say so. It's kind of a wonderful opportunity to work with a colleague who is in literacy research and helping teachers teach mathematics through reading story problems. Mike: Well, I wonder if you can start by telling us the story of how you all came to work together and describe the work you're doing around helping students make sense of word problems. Aina: I think the work started with me working with fifth grade teachers, for two years now, and the conversations have been around story problems. There's a lot of issues from teaching story problems that teachers are noticing. And so this was a very interesting experience. One of the professional development sessions that we had, teachers were saying, “Can we talk about story problems? It's very difficult.” And so we just looked at a story problem. And the story problem—it was actually a coordinate plane story problem—it included a balance beam. And, you know, you're supposed to read the story problem and locate where this balance beam would be. And I had no idea [where] the balance beam would be. So when I read the story, I thought, “Oh, it must be from the remodeling that I did in my kitchen, and I had to put in a beam, which was structural.” So I'm assuming it's balancing the load. And even that didn't help me. I kept rereading the problem and thinking, “I'm not sure this is on the ceiling, but the teachers told me it's gymnastics.” And so even telling me that it was gymnastics didn't really help me because I couldn't think, in the moment, while I was already in a different context of having the beam, a load-bearing beam. It was very interesting that—and I know I'm an ELL, so English is not my first language—in thinking about a context that you're familiar with by reading a word or this term, “balance beam.” And even if people tell you, “Oh, it's related to gymnastics”—and I've never done gymnastics; I never had gymnastics in my class or in my school where I was. It didn't help. And that's where we started talking about underlying keywords that didn't really help either because it was a coordinate plane problem. So I had to reach out to Julia and say, “I think there's something going on here that is related to reading comprehension. Can you help me?” And that's how this all started. [chuckles] Julia: Well, so Aina came to me regarding her experience. In fact, she sent me the math problem. She says, “Look at this.” And we talked about that. And then she shared [the] frustration of the educators that she had been working with, that despite teaching strategies that are promoted as part of instructional practice, like identifying mathematical keywords, and then also reading strategies have been emphasized, like summarizing or asking questions while you're reading story problems. So her teachers had been using strategies—mathematical and also reading—and their students were still struggling to make sense of and solve mathematical problems. Aina’s experience with this word problem really opened up this thought about the words that are in mathematical story problems. And we came to realize that when we think about making sense of story problems, there are a lot of words that require schema. And schema is the background knowledge that we bring to the text that we interact with. For example, I taught for years in Florida. And we would have students that had never experienced snow. So, as an educator, I would need to do read-alouds and provide that schema for my students so that they had some understanding of snow. So when we think about math story problems, all words matter—not just the mathematical terms, but also the words that require schema. And then when we think about English learners, the implications are especially profound because we know that that vocabulary is one of the biggest challenges for English learners. So when we consider schema-mediated vocabulary and story problems, this really becomes problematic. And so Aina and I analyzed the story problems in the curriculum that Aina’s teachers were using, and we had an amazing discovery. Aina: Just the range of contexts that we came across from construction materials or nuts and bolts and MP3 players that children don't really have anymore, a lot of them have a phone, to making smoothies and blenders, which some households may not have. In addition to that, we started looking at the words that are in the story problems. And like Julia said, there are actually mathematics teachers who are being trained on these strategies that come from literacy research. One of them was rereading the problem. And it didn't matter how many times I reread the problem or somebody reread it to me about the balance beam. I had no kind of understanding of what's going on in the problem. The second one is summarizing. And again, just because you summarized something that I don't understand or read it louder to me, it doesn't help, right? And I think the fundamental difference [when]t we solve problems or the story problems [is that]in … literacy, the purpose of reading a story is very different. In mathematics, the purpose of reading a story is to solve it, making sense of problems for the purpose of solving them. The three different categories of vocabulary we found from reading story problems and analyzing them is there's “technical,” there's “subtechnical,” and there’s “nontechnical.” I was very good at recognizing technical words because that's the strategy that for mathematics teachers, we underline the parallelogram, we underline the integer, we underline the 8 or the square root, even some of the keywords we teach, right? “Total” means some or “more” means addition. Mike: So technical, they're the language that we would kind of normally associate with the mathematics that are being addressed in the problem. Let's talk about subtechnical because I remember from our pre-podcast conversation, this is where some light bulbs really started to go off, and you all started to really think about the impact of subtechnical language. Julia: Subtechnical includes words that have multiple meanings that intersect mathematically and other contexts. So, for example, “yard.” Yard can be a unit of measurement. However, I have a patio in my backyard. So it's those words that have that duality. And then when we put that in the context of making sense of a story problem, it's understanding, “What is the context for that word, and which meaning applies to that?” Other examples of subtechnical would be “table” or “volume.” And so it's important when making sense of a story problem to understand which meaning is being applied here. And then we have nontechnical, which is words that are used in everyday language that are necessary for making sense of or solving problems. So, for example, “more.” More is more. So more has that mathematical implication. However, it would be considered nontechnical because it doesn't have dual meanings. So … categorizing vocabulary into these three different types helped us to be able to analyze the word problems. So we worked together to categorize. And then Aina was really helpful in understanding which words were integral to solving those math problems. And what we discovered is that often words that made the difference in the mathematical process were falling within the subtechnical and nontechnical. And that was really eye-opening for us. Mike: So, Aina, this is fascinating to me. And what I'm thinking about right now is the story that you told at the very beginning of this podcast, where you described your own experience with the word problem that contained the language “balance.” And I'm wondering if you applied the analysis that you all just described with technical and subtechnical and the nontechnical. When you view your own experience with that story problem through that lens, what jumps out? What was happening for you that aligns or doesn't align with your analysis? Aina: I think one of the things that was eye-opening to me is, we have been doing it wrong. That's how I felt. And the teachers felt the same way. They're saying, “Well, we always underline the math words because we assume those are the words that are confusing to them. And then we underline the words that would help them solve the problem.” So it was a very good conversation with teachers to really, completely think about story problems differently. It's all about the context; it's all about the schema. And my teachers realize that I, as an adult who engages in mathematics regularly, have this issue with schema. I don't understand the context of the problem, so therefore I cannot move forward in solving it. And we started looking at math problems very differently from the language perspective, from the schema perspective, from the context perspective, rather than from underlining the technical and mathematical words first. That was very eye-opening to me. Mike: How do you think their process or their perspective on the problems changed, either when they were preparing to teach them or in the process of working with children? Aina: I know the teachers reread a problem out loud and then typically ask for a volunteer to read the problem. And it was very interesting; some of the conversations were [about] how different the reading is. When the teacher reads the problem, there is where you put the emotion, where the certain specific things in the problem are. Prosody? Julia: Yes, prosody is reading with appropriate expression, intonation, phrasing. Aina: So when the teacher reads the problem, the prosody is present in that reading. When the child is reading the problems, it's very interesting how it sounds. It just sounds [like the] word and the next word and the next word and the next word, right? So that was kind of a discussion too. The next strategy the math teachers are being taught is summarizing. I guess discussing the problem and then summarizing the problem. So we kind of went through that. And once they helped me to understand in gymnastics what it is, looking up the picture, what it looks like, how long it is, and where it typically is located and there's a mat next to it, that was very helpful. And then I could then summarize, or they could summarize, the problem. But even [the] summarizing piece is now me interpreting it and telling you how I understand the context and the mathematics in the problem by doing the summary. So even that process is very different. And the teacher said that's very different. We never really experience that. Mike: Julia, do you want to jump in? Julia: And another area where math and reading intersect is the use of visualization. So visualization is a reading strategy, and I've noticed that visualization has become a really strong strategy to teach for mathematics as well. We encourage students to draw pictures as part of that solving process. However, if we go back to the gymnastics example, visualizing and drawing is not going to be helpful for that problem because you are needing a schema to be able to understand how a balance beam would situate within that context and whether that's relevant to solving that word problem. So even though we are encouraging educators to use these strategies, when we think about schema-mediated vocabulary, we need to take that a step further to consider how schema comes into play and who has access to the schema needed, and who needs that additional support to be able to negotiate that schema-mediated vocabulary. Mike: I was thinking the same thing, how we often take for granted that everyone has the same schema. The picture I see in my head when we talk about balance is the same as the picture you see in your head around balance. And that's the part where, when I think about some of those subtechnical words, we really have to kind of take a step back and say, “Is there the opportunity here for someone to be profoundly confused because their schema is different than mine?” And I keep thinking about that lived experience that you had where, in my head I can see a balance beam, but in your head you're seeing the structural beam that sits on the top of your ceiling or runs across the top of your ceiling. Aina: Oh, yeah. And at first, I thought, the word “beam” typically, in my mind for some reason, is vertical. Mike: Yeah. Aina: It's not horizontal. And then when I looked at the word “balance,” I thought, “Well, it could balance vertically.” And immediately what I think about is, you have a porch, then you see a lot of porches that balance the roof, and so they have the two beams … Mike: Yes! Aina: … or sometimes more than that. So at no point did I think about gymnastics. But that's because of my lack of experience in gymnastics, and my school didn't have the program. As a math person, you start thinking about it and you think, “If it's vertically, this doesn't make any sense because we're on a coordinate plane.” So I started thinking about [it] mathematically and then I thought, “Oh, maybe they did renovations to the gymnasium, and they needed a balance beam. So I guess that's the beam that carries the load.” So that's how I flipped, in my mind, the image of the beam to be horizontal. Then the teachers, when they told me it's gymnastics, that really threw me off, and it didn't help. And I totally agree with Julia. You know when we do mathematics with children, we tell them, “Can you draw me a picture?” Mike: Mm-hmm. Aina: And what we mean is, “Can you draw me a mathematical picture to support your problem solving or the strategies you used?” But the piece that was missing for me is an actual picture of what the balance beam is in gymnastics and how it's located, how long it is. So yeah, yeah, that was eye-opening to me. Mike: It's almost like you put on a different pair of glasses that allow you to see the language of story problems differently and how that was starting to play out with teachers. I wonder, could you talk about some of the things that they started to do when they were actually with kids in the moment that you looked at and you were like, “Gosh, this is actually accounting for some of the understanding we have about schema and the different types of words.”? Aina: So the teacher would read a problem, which I think is a good strategy. But then it was very open-ended. “How do you understand what I just read to you?” “What's going on in the story problem?” “Turn to your partner.” “Can you envision? Can you think of it? Do you have a picture in your mind?” So we don't jump into mathematics anymore. We kind of talk about the context, the schema. “Can you position yourself in it?” “Do you understand what's going on?” “Can you retell the story to your partner the way you understand it?” And then we talk about, “So how can we solve this problem?” “What do you think is happening?” based on their understanding. That really helped, I think, a lot of teachers also to see that sometimes interpretations lead to different solutions, and children pay attention to certain words that may take them to a different mathematical solution. It became really about how language affects our thinking, our schema, our image in the head, and then based on all of that, where do we go mathematically in terms of solving the problem? Mike: So there are two pieces that really stuck out for me in what you said. I want to come back to both of them. The first one was, you were describing that set of choices that teachers made about being really open-ended about asking kids, “How do you understand this? Talk to your neighbor about your understanding about this.” And it strikes me that the point you made earlier when you said context has really become an important part of some of the mathematics tasks and the problems we create. This is a strategy that has value not solely for multilingual learners, but really for all learners because context and schema matter a lot. Aina: Yes. Mike: Yeah. And I think the other thing that really hits me, Aina, is when you said, “We don't immediately go to the mathematics, we actually try to help kids situate and make sense of the problem.” There's something about that that seems really obvious. When I think back to my own practice as a teacher, I often wonder how I was trying to kind of quickly get kids into the mathematics without giving kids enough time to really make meaning of the situation or the context that we were going to delve into. Aina: Exactly. Mike, to go back to your question, what teachers can do, because it was such an eye-opening experience that, it's really about the language; don't jump into mathematics. The mathematics and the problem actually is situated around the schema, around the context. And so children have to understand that first before they get into math. I have a couple of examples if you don't mind, just to kind of help the teachers who are listening to this podcast to have an idea of what we're talking about. One of the things that Julia and I were thinking about is, when you start with a story problem, you have three different categories of vocabulary. You have technical, subtechnical, nontechnical. If you have a story problem, how do you parse it apart? OK, in the math story problems we teach to children, it's typically a number and operations. Let's say we have a story problem like this: “Mrs. Tatum needs to share 3 grams of glitter equally among 8 art students. How many grams of glitter will each student get?” So if the teacher is looking at this, technical would definitely be grams: 3, 8, and that is it. Subtechnical, we said “equally,” because equally has that kind of meaning here. It's very precise; it has to be [an] exact amount. But a lot of children sometimes say, “Well that's equally interesting.” That means it's similarly, or kind of, or like, but not exact. So subtechnical might qualify as “equally.” Everything else in the story problem is nontechnical: sharing and glitter, art students, each student, how much they would get. I want the teachers to go through and ask a few questions here that we have. So, for example, the teacher can think about starting with subtechnical and nontechnical, right? Do students understand the meaning of each...
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Season 2 | Episode 14 – Mathematizing and Modeling The World Around Us - Guest: Erin Turner, Ph.D.
03/21/2024
Season 2 | Episode 14 – Mathematizing and Modeling The World Around Us - Guest: Erin Turner, Ph.D.
ROUNDING UP: SEASON 2 | EPISODE 14 Many resources for supporting multilingual learners are included with curriculum materials. What's too often missing, though, is clear guidance for how to use them. In this episode, we're going to talk with Dr. Erin Turner about three resources that are often recommended for supporting multilingual learners. We'll unpack the purpose for each resource and offer a vision for how to put them to good use with your students. GUEST BIOGRAPHY Erin Turner is an assistant professor in the department of Teaching, Learning & Sociocultural Studies at The University of Arizona. She obtained a bachelor of arts In elementary education with a bilingual/ESL specialization as well as a master of arts in curriculum and instruction with a focus on mathematics education, both from Arizona State University. She completed her doctoral studies in mathematics education at The University of Texas at Austin. She has also served as a 4th/5th grade dual-language (Spanish/English) classroom teacher in the Phoenix urban core. Dr. Turner’s scholarship examines the critically important field of equity and social justice in mathematics education. Her work focuses on how mathematics instruction can draw upon children’s multiple mathematical funds of knowledge (for example, their mathematical thinking, as well as their cultural, linguistic and/or community-based knowledge and experiences) in ways that support mathematical understanding and a sense of agency. RESOURCES TRANSCRIPT Mike Wallus: Many resources for supporting multilingual learners are included with curriculum materials. What's too often missing, though, is clear guidance for how to use them. In this episode, we're going to talk with Dr. Erin Turner about three resources that are often recommended for supporting multilingual learners. We'll unpack the purpose for each resource and offer a vision for how to put them to good use with your students. Well, welcome to the podcast, Erin. We are excited to be chatting with you today. Erin Turner: Thank you so much for inviting me. Mike: So, for our listeners, the starting point for this episode was a conversation that you and I had not too long ago, and we were talking about the difference between having a set of resources, which might come with a curriculum, and having a sense of how to use them. And in this case, we were talking about resources designed to support multilingual learners. So today, we're going to talk through three resources that are often recommended for supporting multilingual learners, and we're going to really dig in and try to unpack the purpose and offer a vision for how to put them to use with students. What do you think? Are you ready to get started, Erin? Erin: I am. Mike: Well, one of the resources that often shows up in curriculum are what are often referred to as “sentence frames” or “sentence stems.” So let's start by talking about what these resources are and what purpose they might serve for multilingual learners. Erin: Great. So a sentence stem, or sometimes it's called a “sentence starter,” this is a phrase that gives students a starting place for an explanation. So often it includes three or four words that are the beginning part of a sentence, and it's followed by a blank that students can complete with their own ideas. And a sentence frame is really similar. A sentence frame just typically is a complete sentence that includes one or more blanks that, again, students can fill in with their ideas. And in both cases, these resources are most effective for all students who are working on explaining their ideas when they're flexible and open-ended. So you always want to ensure that a sentence stem or a sentence frame has multiple possible ways that students could insert their own ideas, their own phrasing, their own solutions to complete the sentence. The goal is always for the sentence frame to be generative and to support students' production and use of language—and never to be constraining. So students shouldn't feel like there's one word or one answer or one correct or even intended way to complete the frame. It should always feel more open-ended and flexible and generative. For multilingual learners, one of the goals of sentence stems is that the tool puts into place for students some of the grammatical and linguistic structures that can get them started in their talk so that students don't have to worry so much about, “What do I say first?” or “What grammatical structures should I use?” and they can focus more on the content of the idea that they want to communicate. So the sentence starter is just getting the child talking. It gives them the first three words that they can use to start explaining their idea, and then they can finish using their own insights, their own strategies, their own retellings of a solution, for example. Mike: Can you share an example of a sentence frame or a sentence stem to help people understand them if this is new to folks? Erin: Absolutely. So let's say that we're doing number talks with young children, and in this particular number talk, children are adding 2-digit numbers. And so they're describing the different strategies that they might use to do either a mental math addition of 2-digit numbers or perhaps they've done a strategy on paper. You might think about the potential strategies that students would want to explain and think about sentence frames that would mirror or support the language that children might use. So a frame that includes blanks might be something like, “I broke apart ‘blank’ into ‘blank’ and ‘blank.’” if you think students are using 10s and 1s strategies, where they're decomposing numbers into 10s and 1s. Or if you think students might be working with open number lines and making jumps, you might offer a frame like, “I started at ‘blank’, then I ‘blank,’” which is a really flexible frame and could allow children to describe ways that they counted on on a number line or made jumps of a particular increment or something else. The idea, again, is for the sentence frame to be as flexible as possible. You can even have more flexible frames that imply a sequence of steps but don't necessarily frame a specific strategy. So something like, “First I ‘blank,’ then I ‘blank’” or “I got my answer by ‘blank.’” Those can be frames that children can use for all different kinds of operations or work with tools or representations. Mike: OK, that sets up my next question. What I think is interesting about what you shared is there might be some created sentence frames or sentence stems that show up with the curricular materials I have, but as an educator, I could actually create my own sentence frames or sentence stems that align with either the strategies that my kids are investigating or would support some of the ideas that I'm trying to draw out in the work that we're doing. Am I making sense of that correctly? Erin: Absolutely. So, many curricula do include sample sentence frames, and they may support your students. But you can always create your own. And one place that I really like to start is by listening to the language that children are already using in the classroom because you want the sentence starters or the sentence frames to feel familiar to students. And by that, I mean you want them to be able to see their own ideas populating the sentence frames so that they can own the language and start to take it up as part of the repertoire of how they speak and communicate their ideas. So if you have a practice in your classroom, for example, where children share ideas and maybe on chart paper or on the whiteboard you note down phrases from their explanations—perhaps labeled with their name so that we can keep track of who's sharing which idea—you could look across those notations and just start to notice the language that children are already using to explain their strategies and take that as a starting point for the sentence frames that you create. And that really honors children's contributions. It honors their natural ways of talking, and it makes it more likely that children will take up the frames as a tool or a resource. Mike: Again, I just want to say, I'm so glad you mentioned this. In my mind, a sentence frame or a sentence stem was a tool that came to me with my curriculum materials, and I don't know that I understood that I have agency and that I could listen to kids’ thinking and use that to help design my own sentence frames. One question that comes to mind is: Do you have any guardrails or cautions in terms of creating them that would either support kids' language or that could inadvertently make it more challenging? Erin: So I'll start with some cautions. One way that I really like to think about sentence frames is that they are resources that we offer children, and I'm using “offer” here really strategically. They're designed to support children's use of language. And when they're not supportive, when children feel like it's harder to use the frame to explain their idea because the way they want to communicate something, the way they want to phrase something doesn't fit into the frame that we've offered, then it's not a useful support. And then it can become a frustrating experience for the child as the child's trying to morph or shape their ideas—which make sense to them—into a structure that may not make sense. And so I really think we want to take this idea of offering, and not requiring, frames really seriously. The other caution that I would offer is that frames are not overly complex. And by that I mean, if we start to construct frames with multiple blanks where it becomes more about trying to figure out the teacher's intention and children are thinking, “What word would I put here?” “What should I insert into this blank?”, then we've lost the purpose. The purpose is to support generative language and to help children communicate their ideas, not to play guessing games with children where they're trying to figure out what we intend for them to fill in. This isn't necessarily a caution, but maybe just a strategy for thinking about whether or not sentence frames could be productive for students in your classroom—particularly for multilingual learners—is to think about multiple ways that they might complete the sentence stem or that they might fill in the sentence frame. And if, as a teacher, we can't readily come up with four or five different ways that they could populate that frame, chances are it's too constraining and it's not open-ended enough, and you might want to take a step back toward a more open-ended or flexible frame. Because you want it to be something that the children can readily complete in varied ways using a range of ideas or strategies. So something that I think can be really powerful about sentence frames is the way that they position students. For example, when we offer frames like, “I discovered that …” or “I knew my answer was reasonable because …” or “A connection I can make is … .”—those are all sentence starters—the language in those sentence starters communicates something really powerful to multilingual learners and to any student in our classroom. And that's that we assume as a teacher that they're capable of making connections, that they're capable of deciding for themselves if their answer is reasonable, that they're capable of making discoveries. So the verbs we choose in our sentence frames are really important because of how they position children as competent, as mathematical thinkers, as people with mathematical agency. So sometimes we want to be really purposeful in the language that we choose because of the way that it positions students. Another kind of positioning to think about is that multilingual learners may have questions about things in math class. They may not have clarity about the meaning of a phrase or the meaning of a concept, and that's really true of all students. But we can use sentence frames to normalize those moments of uncertainty or struggle for students. So at the end of a number talk or at the end of a strategy-sharing session, we can offer a sentence frame like, “I had a question about …” or “Something I'm still not sure of is … .” And we can invite children to turn and talk to a partner and to finish that sentence frame. That's offering students language to talk about things that they might have questions about, that they might be uncertain about. And it's communicating to all kids that that's an important part of mathematics learning—that everyone has questions. It's not just particular students in the classroom. Everyone has moments of uncertainty. And so I think it's really important that when we offer these frames to students in our classrooms, they're not positioned as something that some students might need but they're positioned as tools and resources that all students can benefit from. We all can benefit from an example of a reflection. We all can learn new ways to talk about our ideas. We all can learn new ways to talk about our confusions, and that's not limited to the children that are learning the language of instruction. Otherwise, sentence frames become something that has low status in the classroom or is associated with students [who] might need extra help., and they aren't taken up by children if they're positioned in that way, at least not as effectively. Mike: The comparison that comes to mind is the ways that in the past, manipulatives have been positioned as something that's lower status, right? If you're using them, it means something. Typically, at least in the past, it was something not good. Whereas I hope as a field we've gotten to the place where we think about manipulatives as a tool for kids to help express their thinking and understand and make meaning, and that we're communicating that in our classrooms as well. So I'm wondering if you can spend just a few minutes, Erin, talking about how an educator might introduce sentence frames or sentence stems and perhaps a little bit about the types of routines that keep them alive in the classroom. Erin: Yes, thanks for this question. One thing that I found to be really flexible is to start with open-ended sentence frames or sentence stems that can be useful as an attachment or as an enhancement to a routine that children already know. So, just as an example, many teachers use an “I notice, I wonder” or “We know, we wonder” type of routine. Those naturally lend themselves to sentence starters: “I notice ‘blank,’ I wonder ‘blank.’” Similarly, teachers may be already using a same and different routine in their classroom. You can add or layer a sentence frame onto that routine and then that frame becomes a tool that can support students' communication in that routine. So “These are the same because …”, “These are different because … .” And once students are comfortable and they're using sentence frames in those sorts of familiar routines, a next step can be introducing sentence frames that allow children to explain their own thinking or their own strategies. And so we can introduce sentence frames that map onto the strategies that children might use in number talks. We can introduce sentence frames that can support communication around problem-solving strategies. And those can be either really open-ended like, “First …, then I …”-type frames or frames that sort of reflect or represent particular strategies. In every case, it's really important that the teacher introduces the frame or the sentence starter in a whole group. And this can be done in a couple of ways. You can [chorally] read the frame so that all children have a chance to hear what it sounds like to say that frame. And as a teacher, you can model using the frame to describe a particular idea. One thing that I've seen teachers do really effectively is when children are sharing their strategy, teachers often revoice or restate children's strategies sometimes, just to amplify it for the rest of the class or to clarify a particular idea. As part of that revoicing, as teachers we can model using a sentence frame to describe the idea. So we could say something like, “Oh, Julio just told us that he decomposed ‘blank’ into two 10s and three 1s,” and we can reference the sentence frame on the board or in another visible place in the classroom so that children are connecting that mathematical idea to potential language that might help them communicate that idea. And that may or may not benefit Julio, the child [who] just shared. But it can benefit other children in the classroom [who] might have solved the problem or have thought about the problem in a similar way but may not yet be connecting their strategy with possible language to describe their strategy. So by modeling those connections as a teacher, we can help children see how their own ideas might fit into some of these sentence frames. We also can pose sentence frames as [a] tool to practice in a partner conversation. So for example, if children are turning and talking during a number talk and they're sharing their strategy, we can invite children to practice using one of two sentence frames to explain their ideas to a partner. And after that turn-and-talk moment, we can have a couple of children in the class volunteer their possible ways to complete the sentence frame for the whole group. So it just gives us examples of what a sentence frame might sound like in relation to an authentic activity—in this case, explaining our thinking about a number talk. And that sort of partner practice or partner rehearsal is really, really important because it gives children the chance to try out a new frame or a new sentence starter in a really low-stress context, just sharing their idea with one other peer, before they might try that out in a whole-class discussion. Mike: That's really helpful, Erin. I think one of the things that jumps out for me is, when you initially started talking about this, you talked about attaching it to a routine that kids already have a sense of, like “I notice” or “I wonder” or “What's the same?” or “What's different?” And what strikes me is that those are routines that all kids participate in. So again, we're not positioning the resource or the tool of the sentence frame or the sentence starter as only for a particular group of children. They actually benefit all kids. It's positioned as a normal practice that makes sense for everybody to take up. Erin: Absolutely. And I think we need to position them as ways to enhance things in classrooms for all students. And partner talk is another good example. We often send students off to talk with a partner and give them instructions like, “Go tell your partner how you solved the problem.” And many children aren't quite sure what that conversation looks like or sounds like, even children for whom English is their first language. And so, when we offer sentence frames to guide those interactions, we're offering a support, or a potential support, for all students. So for partner talk, we often not only ask kids to explain their thinking, but we say things like, “Oh, and ask your partner questions.” “Find out more about your partner's ideas.” And that can be challenging for 7- and 8-year-olds. So, if we offer sentence frames that are in the form of questions, we can help scaffold those conversations. So things like, “Can you say more about … ?” or “I have a question about …” or “How did you know to … ?” If we want children asking each other questions, we need to often offer them supports or give them tools to support...
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Season 2 | Episode 13 – Rough Draft Math - Guest: Dr. Amanda Jansen
03/07/2024
Season 2 | Episode 13 – Rough Draft Math - Guest: Dr. Amanda Jansen
ROUNDING UP: SEASON 2 | EPISODE 13 What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up. GUEST BIOGRAPHY Dr. Amanda Jansen is a professor in the School of Education at the University of Delaware. She is a mathematics educator who conducts research on students’ engagement in mathematics classrooms and teachers’ learning from their reflections on their own practice. She is committed to honoring students’ voices through her research on students’ motivation and engagement. Her most recent book, “Rough Draft Math: Revising to Learn,” was published by Stenhouse Publishers in March 2020. Before working in academia, Dr. Jansen was a junior high mathematics teacher. At UD, she is invested in continually improving UD’s elementary mathematics teacher education courses through research and development work. RESOURCES TRANSCRIPT Mike Wallus: What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up. Mike: Well, welcome to the podcast, Mandy. We are excited to be talking with you. Mandy Jansen: Thanks, Mike. I'm happy to be here. Mike: So, I'd like to start by asking you where the ideas involved in “Rough Draft Math” originated. What drove you and your collaborators to explore these ideas in the first place? Mandy: So, I work in the state of Delaware. And there's an organization called the Delaware Math Coalition, and I was working in a teacher study group where we were all puzzling together—secondary math teachers—thinking about how we could create more productive classroom discussions. And so, by productive, one of the ways we thought about that was creating classrooms where students felt safe to take intellectual risks, to share their thinking when they weren't sure, just to elicit more student participation in the discussions. One way we went about that was, we were reading chapters from a book called “Exploring Talk in School” that was dedicated to the work of Doug Barnes. And one of the ideas in that book was, we could think about fostering classroom talk in a way that was more exploratory. Exploratory talk, where you learn through interaction. Students often experience classroom discussions as an opportunity to perform. "I want to show you what I know.” And that can kind of feel more like a final draft. And the teachers thought, “Well, we want students to share their thinking in ways that they're more open to continue to grow their thinking.” So, in contrast to final draft talk, maybe we want to call this rough draft talk because the idea of exploratory talk felt like, maybe kind of vague, maybe hard for students to understand. And so, the term “rough draft talk” emerged from the teachers trying to think of a way to frame this for students. Mike: You're making me think about the different ways that people perceive a rough draft. So, for example, I can imagine that someone might think about a rough draft as something that needs to be corrected. But based on what you just said, I don't think that's how you and your collaborators thought about it, nor do I think that probably is the way that you framed it for kids. So how did you invite kids to think about a rough draft as you were introducing this idea? Mandy: Yeah, so we thought that the term “rough draft” would be useful for students if they have ever thought about rough drafts in maybe language arts. And so, we thought, “Oh, let's introduce this to kids by asking, ‘Well, what do you know about rough drafts already? Let's think about what a rough draft is.’” And then we could ask them, “Why do you think this might be useful for math?” So, students will brainstorm, “Oh yeah, rough draft, that's like my first version” or “That's something I get the chance to correct and fix.” But also, sometimes kids would say, “Oh, rough drafts … like the bad version. It's the one that needs to be fixed.” And we wanted students to think about rough drafts more like, just your initial thinking, your first ideas; thinking that we think of as in progress that can be adjusted and improved. And we want to share that idea with students because sometimes people have the perception that math is, like, you're either right or you're wrong, as opposed to something that there's gradients of different levels of understanding associated with mathematical thinking. And we want math to be more than correct answers, but about what makes sense to you and why this makes sense. So, we wanted to shift that thinking from rough drafts being the bad version that you have to fix to be more like it's OK just to share your in-progress ideas, your initial thinking. And then you're going to have a chance to keep improving those ideas. Mike: I'm really curious, when you shared that with kids, how did they react? Maybe at first, and then over time? Mandy: So, one thing that teachers have shared that's helpful is that during a class discussion where you might put out an idea for students to think about, and it's kind of silent, you get crickets. If teachers would say, “Well, remember it's OK to just share your rough drafts.” It's kind of like letting the pressure out. And they don't feel like, “Oh wait, I can't share unless I totally know I'm correct. Oh, I can just share my rough drafts?” And then the ideas sort of start popping out onto the floor like popcorn, and it really kind of opens up and frees people up. “I can just share whatever's on my mind.” So that's one thing that starts happening right away, and it's kind of magical that you could just say a few words and students would be like, “Oh, right, it's fine. I can just share whatever I'm thinking about.” Mike: So, when we were preparing for this interview, you said something that has really stuck with me and that I've found myself thinking about ever since. And I'm going to paraphrase a little bit, but I think what you had said at that point in time was that a rough draft is something that you revise. And that leads into a second set of practices that we could take up for the benefit of our students. Can you talk a little bit about the ideas for revising rough drafts in a math classroom? Mandy: Yes. I think when we think about rough drafts in math, it's important to interact with people thinking by first, assuming those initial ideas are going to have some merit, some strength. There's going to be value in those initial ideas. And then once those ideas are elicited, we have that initial thinking out on the floor. And so, then we want to think about, “How can we not only honor the strengths in those ideas, but we want to keep refining and improving?” So inviting revision or structuring revision opportunities is one way that we then can respond to students’ thinking when they share their drafts. So, we want to workshop those drafts. We want to work to revise them. Maybe it's peer-to-peer workshops. Maybe it's whole-class situation where you may get out maybe an anonymous solution. Or a solution that you strategically selected. And then work to workshop that idea first on their strengths, what's making sense, what's working about this draft, and then how can we extend it? How can we correct it, sure. But grow it, improve it. And promoting this idea that everyone's thinking can be revised. It's not just about your work needs to be corrected, and your work is fine. But if we're always trying to grow in our mathematical thinking, you could even drop the idea of correct and incorrect. But everyone can keep revising. You can develop a new strategy. You can think about connections between representations or connections between strategies. You can develop a new visual representation to represent what makes sense to you. And so, just really promoting this idea that our thinking can always keep growing. That's sort of how we feel when we teach something, right? Maybe we have a task that we've taught multiple times in a row, and every year that we teach it we may be surprised by a new strategy. We know how to solve the problem—but we don't have to necessarily just think about revising our work but revising our thinking about the ideas underlying that problem. So really promoting that sense of wonder, that sense of curiosity, and this idea that we can keep growing our thinking all the time. Mike: Yeah, there's a few things that popped out when you were talking that I want to explore just a little bit. I think when we were initially planning this conversation, what intrigued me was the idea that this is a way to help loosen up that fear that kids sometimes feel when it does feel like there's a right or a wrong answer, and this is a performance. And so, I think I was attracted to the idea of a rough draft as a vehicle to build student participation. I wonder if you could talk a little bit about the impact on their mathematical thinking, not only the way that you've seen participation grow, but also the impact on the depth of kids' mathematical thinking as well. Mandy: Yes, and also I think there's impact on students' identities and sense of self, too. So, if we first start with the mathematical thinking. If we're trying to work on revising—and one of the lenses we bring to revising, some people talk about lenses of revising as accuracy and precision. I think, “Sure.” But I also think about connectedness and building a larger network or web of how ideas relate to one another. So, I think it can change our view of what it means to know and do math, but also extending that thinking over time and seeing relationships. Like relationships between all the different aspects of rational number, right? Fractions, decimals, percents, and how these are all part of one larger set of ideas. So, I think that you can look at revision in a number of different grain sizes. You can revise your thinking about a specific problem. You can revise your thinking about a specific concept. You can revise your thinking across a network of concepts. So, there's lots of different dimensions that you could go down with revising. But then this idea that we can see all these relationships with math … then students start to wonder about what other relationships exist that they hadn't thought of and seen before. And I think it can also change the idea of, “What does it mean to be smart in math?” Because I think math is often treated as this right or wrong idea, and the smart people are the ones that get the right idea correct, quickly. But we could reframe smartness to be somebody who is willing to take risk and put their initial thinking out there. Or someone who's really good at seeing connections between people's thinking. Or someone who persists in continuing to try to revise. And just knowing math and being smart in math is so much more than this speed idea, and it can give lots of different ways to show people's competencies and to honor different strengths that students have. Mike: Yeah, there are a few words that you said that keep resonating for me. One is this idea of connections. And the other word that I think popped into my head was “insights.” The idea that what's powerful is that these relationships, connections, patterns, that those are things that can be become clearer or that one could build insights around. And then, I'm really interested in this idea of shifting kids' understanding of what mathematics is away from answer-getting and speed into, “Do I really understand this interconnected bundle of relationships about how numbers work or how patterns play out?” It's really interesting to think about all of the ramifications of a process like rough draft work and how that could have an impact on multiple levels. Mandy: I also think that it changes what the classroom space is in the first place. So, if the classroom space is now always looking for new connections, people are going to be spending more time thinking about, “Well, what do these symbols even mean?” As opposed to pushing the symbols around to get the answer that the book is looking for. Mike: Amen. Mandy: And I think it's more fun. There are all kinds of possible ways to understand things. And then I also think it can improve the social dimension of the classroom, too. So, if there's lots of possible connections to notice or lots of different ways to relationships, then I can try to learn about someone else's thinking. And then I learn more about them. And they might try to learn about my thinking and learn more about me. And then we feel, like, this greater connection to one another by trying to see the world through their eyes. And so, if the classroom environment is a space where we're trying to constantly see through other people's eyes, but also let them try to see through our eyes, we're this community of people that is just constantly in awe of one another. Like, “Oh, I never thought to see things that way.” And so, people feel more appreciated and valued. Mike: So, I'm wondering if we could spend a little bit of time trying to bring these ideas to life for folks who are listening. You already started to unpack what it might look like to initially introduce this idea, and you've led me to see the ways that a teacher might introduce or remind kids about the fact that we're thinking about this in terms of a rough draft. But I'm wondering if you can talk a little bit about, how have you seen educators bring these ideas to life? How have you seen them introduce rough draft thinking or sustain rough draft thinking? Are there any examples that you think might highlight some of the practices teachers could take up? Mandy: Yeah, definitely. So, I think along the lines of, “How do we create that culture where drafting and revising is welcome in addition to asking students about rough drafts and why they might make sense of math?” Another approach that people have found valuable is talking with students about … instead of rules in the classroom, more like their rights. What are your rights as a learner in this space? And drawing from the work of an elementary teacher in Tucson, Arizona, Olga Torres, thinking about students having rights in the classroom, it's a democratic space. You have these rights to be confused, the right to say what makes sense to you, and represent your thinking in ways that make sense to you right now. If you honor these rights and name these rights, it really just changes students' roles in that space. And drafting and revising is just a part of that. Mandy: So different culture-building experiences. And so, with the rights of a learner brainstorming new rights that students want to have, reflecting on how they saw those rights in action today, and setting goals for yourself about what rights you want to claim in that space. So then, in addition to culture building and sustaining that culture, it has to do—right, like Math Learning Center thinks about this all the time—like, rich tasks that students would work on. Where students have the opportunity to express their reasoning and maybe multiple strategies because that richness gives us so much to think about. And drafts would a part of that. But also, there's something to revise if you're working on your reasoning or multiple strategies or multiple representations. So, the tasks that you work on make a difference in that space. And then of course, in that space, often we're inviting peer collaboration. So, those are kinds of things that a lot of teachers are trying to do already with productive practices. But I think the piece with rough draft math then, is “How are you going to integrate revising into that space?” So eliciting students' reasoning and strategies—but honoring that as a draft. But then, maybe if you're having a classroom discussion anyway, with the five practices where you're selecting and sequencing student strategies to build up to larger connections, at the end of that conversation, you can add in this moment where, “OK, we've had this discussion. Now write down individually or turn and talk. How did your thinking get revised after this discussion? What's a new idea you didn't have before? Or what is a strategy you want to try to remember?” So, adding in that revision moment after the class discussion you may have already wanted to have, helps students get more out of the discussion, helps them remember and honor how their thinking grew and changed, and giving them that opportunity to reflect on those conversations that maybe you're trying to already have anyway, gives you a little more value added to that discussion. It doesn't take that much time, but making sure you take a moment to journal about it or talk to a peer about it, to kind of integrate that more into your thought process. And we see revising happening with routines that teachers often use, like, math language routines such as stronger and clearer each time where you have the opportunity to share your draft with someone and try to understand their draft, and then make that draft stronger or clearer. Or people have talked about routines, like, there's this one called “My Favorite No,” where you get out of student strategy and talk about what's working and then why maybe a mistake is a productive thing to think about, try to make sense out of. But teachers have changed that to be “My Favorite Rough Draft.” So, then you're workshopping reasoning or a strategy, something like that. And so, I think sometimes teachers are doing things already that are in the spirit of this drafting, revising idea. But having the lens of rough drafts and revising can add a degree of intentionality to what you already value. And then making that explicit to students helps them engage in the process and hopefully get more out of it. Mike: It strikes me that that piece that you were talking about where you're already likely doing things like sequencing student work to help tell a story, to help expose a connection. The power of that add-on where you ask the question, “How has your thinking shifted? How have you revised your thinking?” And doing the turn and talk or the reflection. It's kind of like a marking event, right? You're marking that one, it's normal, that your ideas are likely going to be refined or revised. And two, it sets a point in time for kids to say, “Oh yes, they have changed.” And you're helping them capture that moment and notice the changes that have already occurred even if they happened in their head. Mandy: I think it can help you internalize those changes. I think it can also, like you said, kind of normalize and honor the fact that the thinking is continually growing and changing. I think we can also celebrate, “Oh my gosh, I hadn't thought about that before, and I want to kind of celebrate that moment.” And I think in terms of the social dimension of the classroom, you can honor and get excited about, “If I hadn't had the opportunity to hear from my friend in the room, I wouldn't have learned this.” And so, it helps us see how much we need one another, and they need us. We wouldn't understand as much as we're understanding if we weren't all together in this space on this day and this time working on this task. And so, I love experiences that help us both develop our mathematical...
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Season 2 | Episode 12 – Counting - Guest: Dr. Kim Hartweg
02/22/2024
Season 2 | Episode 12 – Counting - Guest: Dr. Kim Hartweg
ROUNDING UP: SEASON 2 | EPISODE 12 Counting is a process that involves a complex and interconnected set of concepts and skills. This means that for most children, the path to counting proficiency is not linear. Today we are talking with Dr. Kim Hartweg from Western Illinois University about the big ideas and skills that are a part of counting and ways educators can support their students on this important part of their math journey. GUEST BIOGRAPHY Professor Kim Hartweg teaches mathematics education at Western Illinois University. RESOURCES TRANSCRIPT Mike Wallus: Counting is a process that involves a complex and interconnected set of concepts and skills. This means that for most children, the path to counting proficiency is not a linear process. Today we're talking with Dr. Kim Hartweg from Western Illinois University about the big ideas and skills that are a part of counting and the ways educators can support their students on this important part of their math journey. Well, hey, Kim, welcome to the podcast. We're excited to be talking with you about counting. Kim Hartweg: Ah, thanks for having me. I'm excited too. Mike: So I'm fascinated by all of the things that we're learning about how young kids count, or at least the way that they attend to quantities. Kim: Yeah, it's exciting what all is taking place, with the research and everything going on with early childhood education, especially in regards to number and number sense. And I think back to an article I read about a 6-month-old baby who's in a crib and there's three pictures in this crib. One of them has two dots on it, another one has one dot, and then a third one has three dots. And a drum sounds, and it goes, “boom, boom, boom.” And the 6-month-old baby turns their head and eyes and they look at the picture with three dots on it. And I just think that's exciting that even at that age they're recognizing that three dots [go] with three drum beats. So it's just exciting. Mike: So you're actually taking us to a place that I was hoping we could go to, which is, there are some ideas and some concepts that we associate with counting. And I'm wondering if we could start the podcast by naming and unpacking a few of the really important ones. Kim: OK, sure. I think of the fundamental counting principles, three different areas. And for me, the first one is that counting sequence, or just learning the language and that we count 1, 2, 3, 4, 5. However, in the English language, it's much more difficult [than] in other languages when we get beyond 10 because we have numbers like 11, 12, 13 that we never hear again. Like, you hear 21, 31, 41, but you don't hear 11 again. It's the only time it's ever mentioned. So I think it's harder for students to get that counting sequence for those who speak English. Mike: I appreciate you saying that because I remember reading at one point that in certain Asian languages, the number 11, the translation is essentially “10 and 1,” as opposed to for English speakers where it really is 11, which doesn't really follow the cadence of the number sequence that kids are learning: 1, 2, 3, 4, and so on. Kim: Exactly. Yes. Mike: It picks up again at 21, but this interim space where the teen numbers show up and we're first talking about a 10 and however many more, it's not a great thing about the English language that suddenly we decided to call those things that don't have that same cadence. Kim: Yeah, after you get past 20, yes. And if you think of kids when they hear the number 16, a lot of times they'll say, “A 1 and a 6 or a 6 and a 1?” Because they hear “sixteen,” so you hear the 6 first. But like you said, in other languages, it's “ten-six,” “ten-seven,” “ten-eight.” So, it kind of fits more naturally with the way we talk and the language. Mike: So there's the language of the counting sequence. Let's talk about a couple of the other things. Kim: OK. One-to-one correspondence is a key idea, and I think of this when I was first starting to teach undergraduate students about early math education. I had kids at the same age, so at a restaurant or wherever we happened to be, I'd get out the sugar packets and I would have them count. And at first, when they're maybe 2 years old or so, and they're just learning the language, they may count those sugar packets as “1, 2, 3.” There may be two packets. There may be five packets. But everything is “1, 2, 3,” whether there's again, five packets or two packets. So once they get that idea that each time they say a number word that it counts for an actual object and they can match them up, that's that idea of one-to-one correspondence to where they say a number and they either point or move the object so you can tell they're matching those up. Mike: OK, let's talk about cardinality because this is one that I think when I first started teaching kindergarten, I took for granted how big of a leap this one is. Kim: Yeah, that's interesting. So once they can count out and you have five sugar packets and they count “1, 2, 3, 4, 5,” and you ask, “How many are there?”, they should be able to say, “five.” That's cardinality of number. If they have to count again—“1, 2, 3, 4, 5”—then they don't have cardinality of number, where whatever number they count last is how many is in that set. Mike: Which is kind of amazing actually. We're asking kids to decide that “I've figured out this idea that when I say a number name, I'm talking about an individual part of the count—until I say the last one, and then I'm actually talking about the entire set.” That's a pretty big leap for kids to start to make sense of. Kim: It is, and it's fun to watch because hear some of them say, “1, 2, 3, 4, 5. Five, there's five.” [laughs] So they kind of get that idea. But yeah, that cardinality of number is a key principle and leads into the conservation of numbers. Mike: Let's talk about conservation of number. What I'm loving about this conversation is the way that you're using these concrete examples from your own children, from sugar packets, to help us make sense of something that we might be seeing, but we might not have a name for. Kim: Yeah, so the conservation of number, this is my favorite task when I have young kids around. I want to see if they can serve number or not. So I might first do the sugar [packet] thing or whatever and see if they can tell you how many there are. But the real fun is, do they conserve that number? So I think back to a friend of mine who brought her daughter over one time, and I had these toy matchbox cards on my table, and her name was Katie. And I said, “Katie, how many cars are there?” And she counted, “1, 2, 3, 4, 5. There's five toy cars.” And I moved them around and I said, “Now how many cars are there?” And she counted, “1, 2, 3, 4, 5. There's five toy cars.” So she has cardinality of number. However, I kept moving those cars into different positions, never adding or taking any away. That's all that were there the whole time. And after about seven or eight times, I said, “Now how many cars are there?” And her mom finally jumped in and said, “Katie, you've counted those already. There's five cars.” [laughs] And I said, “No, no, no. This is just whether she conserves number or not, it's a developmental-type thing.” But you know they conserve number when you ask them, “Well, now how many cars are there?” And they look at you and like, “Well, why would you ask that again? There's five.” [chuckles] So then they can conserve number. So It's real fun to do that with elementary students who are getting their number sense going and even before they enter school. However, there will be some that may not get that conservation of number until they're 5 or 6 years old. Mike: Let's talk about something you named earlier. I've heard people pronounce this as “soo-bi-ty-zing” or “su-bi-ty-zing” but in any case, it's really an important idea for people, especially if you're teaching young children to make sense of this. Can you talk about what that means? Kim: Yeah, so subitizing, I think that's interesting. You know, we work so hard getting kids to count and learn the language and have one-to-one correspondence, and then be able to eventually conserve number. But then we want them to just be able to recognize a set of numbers without counting. And that's when they're really starting to develop some number sense. I think of dice. And if you roll a single [die], we want students to just know that when there's an arrangement of four dice, they know it's four without having to count “1, 2, 3, 4.” So the subitizing idea, a lot of dice games, maybe some 10-frame cards, dot cards, lots of things like that can help students develop a little bit more of that subitizing, or recognizing a set of items without having to count those. Mike: So when I look at a set of three dots, I can just say, “That's three,” as opposed to an earlier point where a child might actually say, “1, 2, 3. That's three.” Kim: Exactly. So that would be subitizing—just instantly knowing what's there without having to count. Mike: So I wonder if we could unpack two other counting behaviors that sometimes pop up with kids when they're combining or separating quantities. And what I'm thinking about is the difference between the child who counts everything and the child who either counts on from a number or counts back from a number. And I'm wondering if you can talk about what these two behaviors can tell us about how kids are thinking about the numbers that they're operating on. Kim: Yeah, it's so interesting when you have activities like a cup, and maybe you have eight counters and you put three under the cup and you say, “How many are here?” “Three.” And then you cover those up and you ask, “Well, how many are altogether?” There are some kids who don't have any trouble with counting on “4, 5, 6, 7, 8,” but there's other kids who have to lift up the cup and start again at 1. So they don't have that idea that there's three items under that cup whether you can see them or not. So it's difficult for them to be able to count on, and we shouldn't as teachers force that on them until they're ready to do it. So it's a hard concept for kids to get, but especially if they're not developmentally ready for it. Mike: I think that's a really nice caution because I think you could accidentally potentially get kids to mimic a practice that you're trying to show them, but without understanding there's some real danger that you're just causing confusion. Kim: Yeah, and we want to give kids the idea that counting collections and things, it's a fun thing to do. And I know there may be teachers that have seashells or rocks or different types of collections they might count, and we want students to count those and then discuss how they counted them, arrange them. And I'm thinking of this little girl that I saw on a video where she was counting eight bears, and she arranged them first by color, then counted how many there were. And the teacher then went on to use that and make a problem-solving task for her, such as, “Well, how many green bears do you have?” And she would count them. “Well, what if you gave me those green bears? Do you know what you would have left?” And she said, “Well, I don't know. Let's try it.” And I love that because I think that's the kind of idea we want students to have. They're counting, and “I don't know, let's try it.” They're excited about it. They're not afraid to take chances, and we don't want them to think that “Oh, this is difficult to do.” It's just, “Hey, let's try it. Give it a try here.” Mike: Well, I've heard people talking a lot about this idea of counting collections lately. It seems like we are almost rediscovering the value of a routine like that. I'm wondering if you could talk about the value that can come out of an experience of counting collections and help bring that idea to life for people. Kim: The idea here is that we want students to get good at counting. And the research is showing that students who maybe don't show one-to-one correspondence when they count out, maybe, eight counters, might show one-to-one correspondence when they count out 31 pennies, which seems like it shouldn't happen. But there's research out there that over 70 percent of them did better counting 31 pennies than they did with eight counters. So, I think what you count makes a big difference for kids—and to not hold them back, to not think that “OK, we've got to get one-to-one correspondence before we count this collection of 50 items.” I don't think that's the case. And the research is even showing that these ideas that we've talked about all develop concurrently. It's not a linear process. But this counting collections is kind of a big deal with that. And having students count, again, collections that they're interested in, writing number sentences about their collections, comparing what they counted with another partner, and then turning it into problem-solving questions where they're actually doing, “What happens if you lost five of yours?” or “What happens if you combined your collection with somebody else?” And turning it into where they're actually doing addition and subtraction, but not actually the formal process of that. Mike: The other thing that you made me think about is, I would imagine you could also have kids finish counting a collection and then you could ask them to represent it either on paper or in some other way. Kim: Exactly. And writing out those number sentences or even creating their own word problems so that they can ask a friend or a partner, it makes it fun. And it relates to what they've done. And let's face it, once you've taken that time to count those collections, you may as well get as much use out of it as you can. [chuckles] Mike: Kim, you're making me think of something that I don't know that I had words for when I was teaching kindergarten, which is, when I look back now, I was looking to see that kids could do a particular thing like one-to-one correspondence or that they had cardinality before I would give them access to a task like counting collections. And I think what you're making me think is that those things shouldn't be a gatekeeper; that they actually develop by doing those things. Am I making sense to you? Kim: Yes. I always thought you had to have the language first. You had to be able to do one-to-one correspondence before you could get cardinality of number, and you needed cardinality of number before you could do conservation of number. But what the research is showing is, it develops concurrently with students; that it's not something that is a linear process by any means. So, when we have these activities, it's OK if they don't have one-to-one correspondence and you're doing problem-solving tasks with counters. We need to be planning these activities so they're getting all of this, and they will develop it as it fits in the schema of what they're working on and thinking of in their minds. Mike: So I want to bring up a set of manipulatives that are actually attached to our bodies, particularly when it comes to counting. I'm thinking about fingers. And part of what's on my mind is, again, to go back to my practice, there was a point in time where I was really hung up on whether kids should make use of their fingers when they're counting or when they're operating on numbers. And I'm wondering if you could just offer some guidance around that. Kim: Yes. I think again, it goes to that idea that we have these 10 fingers that are great manipulatives, that we shouldn't stop students from doing that. And I know there was a time when teachers would say, “Don't use your fingers,” “Don't count on your fingers.” And I get the idea that we want students to start to subitize eventually and make combinations and not have to count on their fingers, but to stop them from doing it when they need that would be very detrimental to them. And I actually have a story. When I was supervising student teachers, one teacher was telling a student, “Don't count with their fingers.” And I saw them nodding their head, and I went over and I said, “What are you doing?” He said, “Well, I can't count my fingers, so I'm using my tongue, and I'm counting my teeth.” [laughs] So coming up with a problem that way, still using a manipulative, but it wasn't their fingers. Mike: That's pretty creative. Kim: [laughs] Yeah. Mike: Well, part of what strikes me about it, too, is our entire number system is based on tens and ones, and we've got a set of them right in front of us, right? We're trying to get kids to make sense of shifting from units of one to units of ten or maybe units of five. So these tools that are attached to our bodies are actually pretty helpful because they're really the basis for our number system in a lot of ways. Kim: Yes, exactly. And being able to come up with even using your fingers to answer questions. I'm thinking, we want students to subitize. So even something to where there's a dot card that a teacher flashes for 3 seconds, and it's in the formation of maybe a five on a [die]. And you could have students, you know, hold up how many there are. And you could do that five or 10 times, with dot flashes. Or you could hold up one more than what you see on the [die]. So, they only see it for 5 seconds and the number's five, but they hold up six. So just uses of fingers to kind of make those connections can be very helpful. Mike: So before we go, you mentioned that you work with preservice teachers, folks who are getting ready to go into the field and work with elementary children in the area of mathematics. I'm wondering if there are any particular resources that really help your students, and perhaps teachers who are already in the field, just make sense of counting and number to really understand some of the ideas that we've been talking about today. Do you have anything in particular that you would recommend to teachers? Kim: Yeah, I'll just mention a few that we use a lot of. We do the two-color counters a lot where one side is yellow and one is red. But we do a lot of dot cards, where again, there are arrangements of dots on a card that you could just flash to a student, kind of like I've already explained. There's lots of resources on the website. That has 10-frame activities. And if you haven't used rekenreks before, I think those are pretty amazing as well—along with hundreds charts. And just being able to have students create some of their own manipulatives and their own numbers makes a huge difference for kids. Mike: I think that's a great place to close the conversation. Thank you so much for joining us, Kim. It's really been a pleasure chatting with you. Kim: Hey, thanks so much. It's been fun, Mike. Thanks for asking me. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability. © 2024 The Math Learning Center |
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Season 2 | Episode 11 – Translanguaging - Guest: Tatyana Kleyn, Ed.D.
02/08/2024
Season 2 | Episode 11 – Translanguaging - Guest: Tatyana Kleyn, Ed.D.
ROUNDING UP: SEASON 2 | EPISODE 11 Over the past two years, we’ve done several episodes on supporting multilingual learners in math classrooms. In this episode, we’re going back to the topic of support for multilingual learners to talk about translanguaging, an asset-focused approach that invites students to bring their full language repertoire into the classroom. We’ll talk about what translanguaging looks like and how all teachers can integrate the practice into their classrooms. GUEST BIOGRAPHY Tatyana Kleyn is a professor in the Bilingual Education and TESOL programs and founding faculty advisor to the Dream Team at The City College of New York. Her dissertation,focused on the intersections of bilingual and multicultural education, earned an Outstanding Dissertation Award from the National Association for Bilingual Education (NABE). RESOURCES TRANSCRIPT Mike Wallus: Over the past two years, we've done several episodes on supporting multilingual learners in math classrooms. Today, we're going back to this topic to talk about translanguaging, an asset-focused approach that invites students to bring their full language repertoire into the classroom. We'll talk with Tatyana Kleyn about what translanguaging looks like and how all teachers can integrate this practice into their classrooms. Well, welcome to the podcast, Tatyana. We're excited to be talking with you today. Tatyana Kleyn: Thank you. This is very exciting. Mike: So, your background with the topic of multilingual learners and translanguaging, it's not only academic. It's also personal. I'm wondering if you might share a bit of your own background as a starting point for this conversation. Tatyana: Yes, absolutely. I think for many of us in education, we don't randomly end up teaching in the areas that we're teaching in or doing the work that we're doing. So I always like to share my story so people know why I'm doing this work and where I'm coming from. So my personal story, I work a lot at the intersection of language, migration, and education, and those are all three aspects that have been critical in bringing me here. So, I was actually born in what was the Soviet Union many, many years ago, and my family immigrated to the United States as political refugees, and I was just 5½ years old. So I actually never went to school in the Soviet Union. Russian was my home language, and I quickly started speaking English, but my literacy was not quick at all, and it was quite painful because I never learned to read in my home language. I never had that foundation. So, when I was learning to read in English, it wasn't meaning making, it was just making sounds. It was kind of painful. I once heard somebody say, “For some people, reading is like this escape and this pure joy, and for other people it's like cleaning the toilet. You get in and you get out.” And I was like, “That's me. I'm the toilet cleaner.” [laughs] So, that was how reading was for me. I always left my home language at the door when I came into school, and I wanted it that way because I, as a young child, got this strong message that English was the language that mattered in this country. So, for example, instead of going by Tatyana, I went by Tanya. So, I always kind of kept this secret that I spoke this other language. I had this other culture, and it wasn't until sixth grade where my sixth-grade teacher, Ms. Chang, invited my mom to speak about our immigration history. And I don't know why, but I thought that was so embarrassing. I think in middle school, it's not really cool to have your parents around. So, I was like, “Oh my God, this is going to be horrible.” But then I realized my peers were really interested—and in a good way—and I was like, “Wait, this is a good thing?” So, I started thinking, “OK, we should be proud of who we are and let just people be who they are.” And when you let people be who they are, they thrive in math, in science, in social studies, instead of trying so hard to be someone they're not and then focusing on that instead of everything else that they should be focusing on as students. Mike: So, there's a lot there. And I think I want to dig into what you talked about over the course of the interview. I want to zero in a little bit on translanguaging, though, because for me, at least until quite recently, this idea of translanguaging was really a new concept, a new idea for me. And I'm going to guess that that's the case for a lot of the people who are listening to this as well. So, just to begin, would you talk briefly about what translanguaging is and your sense of the impact that it can have on learners? Tatyana: Sure. Well, I'm so glad to be talking about translanguaging in this space specifically, because often when we talk about translanguaging, it's in bilingual education or English as a second language or is a new language, and it's important in those settings, right? But it's important in all settings. So, I think you're not the only one, especially if we're talking about math educators or general elementary educators, it's like, “Oh, translanguaging, I haven't heard of that,” right? So, it is not something brand new, but it is a concept that Ofelia García and some of her colleagues really brought forth to the field in the early 2000s—around 2009. And what it does is instead of saying, “English should be the center of everything, and everyone who doesn't just speak English is peripheral,” it's saying, “Instead of putting English at the center, let's put our students' home language practices at the center. And what would that look like?” So, that wouldn't mean everything has to be in English. It wouldn't mean the teacher's language practices are front and center, and the students have to adapt to that. But it's about centering the students and then the teacher adapting to the languages and the language practices that the students bring. Teachers are there to have students use all the languages at their resource—whatever language it is, whatever variety it is. And all those resources will help them learn. The more you can use, when we're talking about math, well, if we're teaching a concept and there are manipulatives there that will help students use them, why should we hide them? Why not bring them in and say, “OK, use this.” And once you have that concept, we can now scaffold and take things away little by little until you have it on your own. And the same thing with sometimes learning English. We should allow students to learn English as a new language using their home language resources. But one thing I will say is we should never take away their home language practices from the classroom. Even when they're fully bilingual, fully biliterate, it's still about, “How can we use these resources? How can they use that in their classroom?” Because we know in the world, speaking English is not enough. We're becoming more globalized, so let's have our students grow their language practices and then students are allowed and proud of the language practices they bring. They teach their language practices to their peers, to their teachers. So it's really hard to say it all in a couple of minutes, but I think the essence of translanguaging is centering students' language practices and then using that as a resource for them to learn and to grow, to learn languages and to learn content as well. Mike: How do you think that shifts the experience for a child? Tatyana: Well, if I think about my own experiences, you don't have to leave who you are at the door. We are not saying, “Home language is here, school language is there, and neither shall the two meet.” We're saying, “language,” and in the sense that it's a verb. And when you can be your whole self, it allows you to have a stronger sense of who you are in order to really grow and learn and be proud of who you are. And I think that's a big part of it. I think when kids are bashful about who they are, thinking who they are isn't good enough, that has ripple effects in so many ways for them. So I think we have to bring a lens of critical consciousness into these kind[s] of spaces and make sure that our immigrant-origin students, their language practices are centered through a translanguaging lens. Mike: It strikes me that it matters a lot how we as educators—internally, in the way that we think and externally, in the things that we do and the things that we say—how we position the child's home language, whether we think of it as an asset that is something to draw upon or a deficit or a barrier. That the way that we're thinking about it makes a really big difference in the child's experience. Tatyana: Yes, absolutely. Ofelia García, Kate Seltzer and Susana Johnson talk about a translanguaging stance. So translanguaging is not just a practice or a pedagogy like, “Oh, let me switch this up,” or “Let me say this in this language.” Yes, that's helpful, but it's how you approach who students are and what they bring. So if you don't come from a stance of valuing multilingualism, it's not really going to cut it, right? It's something, but it's really about the stance. So something that's really important is to change the culture of classrooms. So just because you tell somebody like, “Oh, you can say this in your home language” or “You can read this book side by side in Spanish and in English if it'll help you understand it.” Some students may not want to because they will think their peers will look down on them for doing it, or they'll think it means they're not smart enough. So, it's really about centering multilingualism in your classroom and celebrating it. And then as that stance changes the culture of the classroom, I can see students just saying, “Ah, no, no, no, I'm good in English.” Even though they may not fully feel comfortable in English yet, but because of the perception of what it means to be bilingual. Mike: I'm thinking even about the example that you shared earlier where you said that an educator might say, “You can read this in Spanish side by side with English if you need to or if you want to.” But even that language of “You can ” implies that, potentially, this is a remedy for a deficit as opposed to the ability to read in multiple languages as a huge asset. And it makes me think even our language choices sometimes will be a tell to kids about how we think about them as a learner and how we think about their language. Tatyana: That's so true, and how do we reframe that? “Let's read this in two languages. Who wants to try a new language?” right? Making this something exciting as opposed to framing it in a deficit way. So that's something that's so important that you picked up on. Yeah. Mike: Well, I think we're probably at the point in the conversation where there’s a lot of folks who are monolingual who might be listening and they're thinking to themselves, “This stance that we're talking about is something that I want to step into.” And now they're wondering what might it actually look like to put this into practice? Can we talk about what it would look like, particularly for someone who might be monolingual, to both step into the stance and then also step into the practice a bit? Tatyana: Yes. I think the stance is really doing some internal reflection, questioning about, “What do I believe about multilingualism?” “What do I believe about people who come here, to come to the United States?” In New York City, about half of our multilingual learners are U.S. born. So, it's not just immigrant students, but their parents, or they're often children of immigrants. So really looking closely and saying, “How am I including, respecting, valuing the languages of students regardless of where they come from?” And then I think for the practice, it's about letting go of some control. As teachers, we are kind of control freaks. I can just speak for myself [laughs], right? I like to know everything that's going on. Mike: I will add myself to that list, Tatyana. Tatyana: It's a long list. It's a long list. [laughs] But I think first of all, as educators, we have a sense when a kid is on task, and you can tell when a kid is not on task. You may not know exactly what they're saying. So, I think it's letting go of that control and letting the students, for example, when you are giving directions. I think one of the most dangerous things we do is we give directions in English when we have multilingual students in our classrooms, and we assume they understood it. If you don't understand the directions, the next 40 minutes will be a waste of time because you will have no idea what's happening. So, what does that mean? It means perhaps putting the directions into Google Translate and having it translate the different languages of your students. Will it be perfect? No. But will it be better than just being in English? A million times yes, right? Sometimes it's about putting students in same-language groups. If there are enough—two or three or four students—that speak the same home language, and having them discuss something in their home language or multilingually before actually starting to do the work to make sure they're all on the same page. Sometimes it can mean, if, asking students—if they do come from other countries, sometimes I'm thinking of math, math is done differently in different countries. So, we teach one approach, but what is another approach? Let's share that. Instead of having kids think like, “Oh, I came here, now this is the bad way.” Or, “When I go home and I ask my family to help me, they're telling me all wrong.” No, again, these are the strengths of the families, and let's put them side by side and see how they go together. And I think what it's ultimately about is thinking about your classroom, not as a monolingual classroom, but as a multilingual classroom. And really taking stock of, “Who are your students? Where are they and their families coming from, and what languages do they speak?” And really centering that. Sometimes you may have students that may not tell you because they may feel like it's shameful to share that we speak a language that maybe other people haven't heard of. I'm thinking of Indigenous languages from Honduras, like Garífuna, Miskito, right? Of course, Spanish, everyone knows that. But really excavating the languages of the students, the home language practices, and then thinking about giving them opportunities to translate if they need to translate. I'm not saying everything should be translated. I think word problems, having problems side by side, is really important. Because sometimes what students know is they know the math terms in English, but the other terms, they may not know those yet. And I'll give you one really powerful example. This is a million years ago, but it stays with me from my dissertation. It was in a Haitian Creole bilingual classroom. They were taking a standardized test, and the word problem was where it was like three gumballs, two gumballs, this color, what [is] the probability of a blue gumball coming out of this gumball machine? And this student just got stuck on “gumball machine” because in Haiti people sell gum, not machines, and it was irrelevant to the whole problem. So language matters, but culture matters, too, right? So giving students the opportunity to see things side by side and thinking about, “Are there any things here that might trip them up that I could explain to them?” So I think it's starting small. It's taking risks. It's letting go of control and centering the students. Mike: So from one recovering control freak to another, there are a couple of things that I'm thinking about. One is expanding a little bit on this idea of having two kids who might speak to one another in their home language, even if you are a monolingual speaker and you speak English and you don't necessarily have access to the language that they're using. Can you talk a little bit about that practice and how you see it and any guidance that you might offer around that? Tatyana: Yeah, I mean, it may not work the first time or the second time because kids may feel a little bit shy to do that. So maybe it's, “I want to try out something new in our class. I really am trying to make this a multilingual class. Who speaks another language here? Let's try—I am going to put you in a group and you're going to talk about this, and let's come back. And how did you feel? How was it for you? Let me tell you how I felt about it.” And it may be trying over a couple times because kids have learned that in most school settings, English is a language you should be using. And to the extent that some have been told not to speak any other language, I think it's just about setting it up and, “Oh, you two spoke, which language? Wow, can you teach us how to say this math term in this language?” “Oh, wow, isn't this interesting? This is a cognate, which means it sounds the same as the English word. And let's see if this language and this language, if the word means the same thing.” Getting everyone involved in centering this multilingualism. And language is fun. We can play with language, we can put language side by side. So then if you're labeling or if you have a math word wall, why not put key terms in all the languages that the students speak in the class and then they could teach each other those languages? So I think you have to start little. You have to expect some resistance. But over time, if you keep pushing away at this, I think it will be good for not only your multilingual students, but all your students to say, like, “Oh, wait a minute, there's all these languages in the world, but they're not just in the world. They're right here by my friend to the left and my friend to the right” and open up that space. Mike: So I want to ask another question. What I'm thinking about is participation. And we've done an episode in the past around not privileging verbal communication as the only way that kids can communicate their ideas. We were speaking to someone who—their focus really was elementary years mathematics, but specifically, with multilingual learners. And the point that they were making was, kids’ gestures, the way that they use their hands, the way that they move manipulatives, their drawings—all of those things are sources of communication that we don't have to only say, “Kids understand things if they can articulate it in a particular way.” That there are other things that they do that are legitimate forms of participation. The thing that was in my head was, it seems really reasonable to say that if you have kids who could share an explanation or a strategy that they've come up with or a solution to a problem in their home language in front of the group, that would be perfectly legitimate. Having them actually explain their thinking in their home language is accomplishing the goal that we're after, which is, “Can you justify your mathematical thinking?” I guess I just wanted to check in and say, does that actually seem like a reasonable logic to follow? That that's actually a productive practice for a teacher, but also a productive practice for a kid to engage in? Tatyana: That makes a lot of sense. So I would say for every lesson you, you may have a math objective, you may have a language objective, and you may have both. If your objective is to get kids to understand a concept in math or to explain something in math, who cares what language they do it in? It's about learning math. And if you're only allowing them to do it in a language that they are still developing in, they will always be about English and not about math. ...
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Season 2 | Episode 10 – The Big Place Value Episode - Guest: Eric Sisofo, Ed.D
01/18/2024
Season 2 | Episode 10 – The Big Place Value Episode - Guest: Eric Sisofo, Ed.D
ROUNDING UP: SEASON 2 | EPISODE 10 If you ask an educator to share some of the most important ideas in elementary mathematics, most would include place value on that list. But what does it mean to understand place value, really? And what types of language, practices, and tools support students as they build their understanding? On this week’s podcast, we discuss these questions and more with Dr. Eric Sisofo from the University of Delaware. GUEST BIOGRAPHY Dr. Eric Sisofo is an assistant professor specializing in mathematics education in the School of Education at the University of Delaware. He earned his EdD in educational leadership, curriculum, and instruction; his MEd in curriculum and instruction with a concentration in mathematics education; and his BS in mathematical sciences from the University of Delaware. With a focus on mathematics education, Dr. Sisofo’s education leadership portfolio, the capstone project for his doctoral degree, investigated the effects of lesson study on teacher candidates’ use of student thinking when planning and reflecting on their teaching. RESOURCES TRANSCRIPT Mike Wallus: If you ask an educator to share some of the most important ideas in elementary mathematics, I'm willing to bet that most would include place value on that list. But what does it mean to understand place value really? And what types of language practices and tools support students as they build their understanding? Today, we're digging deep into the topic of place value with Dr. Eric Sisofo from the University of Delaware. Mike: Welcome to the podcast, Eric. We're glad to have you with us. Eric Sisofo: Thanks for having me, Mike. Really excited to be here with you today. Mike: I'm pretty excited to talk about place value. One of the things that's interesting is part of your work is preparing pre-service [teachers] to become classroom elementary teachers. And one of the things that I was thinking about is what do you want educators preparing to teach to understand about place value as they're getting ready to enter the field? Eric: Yeah, that's a really great question. In our math content courses at the University of Delaware, we focus on three big ideas about place value with our novice teachers. The first big idea is that place value is based on the idea of grouping a total amount of stuff or bundling a total amount of stuff into different size units. So, as you know, we use groups of ones, tens, hundreds, thousands, and so on—not just ones—in our base ten system to count or measure a total amount of stuff. And we write a numeral using the digits 0 through 9 to represent the amount of stuff that we measured. So interestingly, our novice teachers come to us with a really good understanding of this idea for whole numbers, but it's not as obvious to them for decimal quantities. So we spend a lot of time with our novice teachers helping them think conceptually about the different groupings, or bundlings, that they're using to measure a decimal amount of stuff. In particular, getting them used to using units of size: one-tenth, one-hundredth, one-thousandth, and so on. So, that's one big idea that really shines through whether you're dealing with whole numbers or decimal numbers, is that place value is all about grouping, or bundling, a total amount of stuff with very specific different-size units. The second big idea we help our novice teachers make sense of at UD is that there's a relationship between different place value units. In particular, we want our novice teachers to realize that there's this 10 times relationship between place value units. And this relationship holds true for whole numbers and decimal numbers. So, 10 of one type of grouping will make one of the next larger-sized grouping in our decimal system. And that relationship holds true for all place value units in our place value system. So, there might be some kindergarten and first grade teachers listening who try to help their students realize that 10 ones are needed to make one 10. And some second- and third grade teachers who try to help their students see that 10 tens are needed to make 100. And 10 hundreds are needed to make 1,000, and so on. In fourth and fifth grade, we kind of extend that idea to decimal amounts. So helping our students realize that 10 of these one-tenths will create a one. Or 10 of the one-hundredths are needed to make one-tenth, and so on and so on for smaller and smaller place value units. So, that's the second big idea. And the third big idea that we explicitly discuss with our pre-service teachers is that there's a big difference between the face value of a digit and the place value of a digit. So as you know, there are only 10 digits in our base ten place value system. And we can reuse those digits in different places, and they take on a different value. So, for example, for the number 444, the same digit, 4, shows up three different times in the numeral. So, the face value is four. It's the same each digit in the numeral, but each 4 represents a different place value or a different grouping or an amount of stuff. So, for 444, the 4 in the hundreds place means that you have four groupings of size 100, the 4 in the tens place means you have four groupings of size 10, and the 4 in the ones place means you have four groupings of size one. So this happens with decimal numbers too. With our novice teachers, we spend a lot of time trying to get them to name those units and not just say, for example, “3.4” miles when they're talking about a numeral. We wouldn't want them to say “3.4.” We instead want them to say “three and four-tenths,” or “three ones and four-tenths miles.” So saying the numeral “3.4” focuses mostly just on the face value of those digits and removes some of the mathematics that's embedded in the numeral. So instead saying the numerals “three ones and four-tenths” or “three and four-tenths” really requires you to think about the face value and the place value of each digit. So those are the three big ideas that we discuss often with our novice teachers at the University of Delaware, and we hope that this helps them develop their conceptual understanding of those ideas so that they're better prepared to help their future students make sense of those same ideas. Mike: You said a lot there, Eric. I'm really struck by point two, where you talk about the relationship between units. And I think what's hitting me is that I don't know that when I was a child learning mathematics—but even when I was an adult getting started teaching mathematics—that I really thought about relationships. I think about things like “add a zero” or even the language of “point-something.” And how in some ways some of the procedures or the tricks that we've used have actually obscured the relationship as opposed to shining a light on it. Does that make sense? Eric: I think the same was true when I was growing up. That math was often taught to be a bunch of procedures or memorized kinds of things that my teacher taught me, that I didn't really understand the meaning behind what I was doing. And so, mathematics became more of just doing what I was told and memorizing things and not really understanding the reasoning why I was doing it. Talking about relationships between things I think helps kids develop number sense. And so, when you talk about how 10 tenths are required to make one one, and knowing that that's how many of those one-tenths are needed to make one one, and that same pattern happens for every unit connected to the next larger unit. Seeing that in decimal numbers helps kids develop number sense about place value. And then when they start to need to operate on those numerals or on those numbers, if they need to add two decimal numbers together and they get more than 10 tenths when they add down the columns or something like that in a procedure—if you're doing it vertically. If they have more of a conceptual understanding of the relationship, maybe they'll say, “Oh, I have more than 10 tenths, so 10 of those tenths will allow me to get one one, and I'll leave the others in the tens place,” or something like that. So, it helps you to make sense of the regrouping that's going on and develop number sense so that when you operate and solve problems with these numbers, you actually understand the reasoning behind what you're doing as opposed to just memorizing a bunch of rules or steps. Mike: Yeah. I will also say, just as an aside, I taught kindergarten and first grade for a long time, and just that idea of 10 ones and one ten, simultaneously, is such a big deal. And I think that idea of being able to say, “This unit is comprised of these equal-sized units,” how challenging that can be for educators to help build that understanding. But how rich and how worthwhile the payoff is when kids do understand that level of equivalence between different sets of units. Eric: Absolutely, and it starts at a young age with children. And getting them to visualize those connections and that equivalence that a 10, one ten, can be broken up into these 10 ones or 10 ones can create one ten. And seeing that visually, multiple times, in lots of different situations really does pay off because that pattern will continue to show up throughout the grades. When you're going into second, third grade, like I said before, you’ve got to realize that 10 of these things we call “tens,” then we'll make a new unit called one “hundred.” Or 10 of these one hundreds will then make a unit that is called a “thousand.” And a thousand is equivalent to 10 hundreds. So these ideas are really critical pieces of students’ understanding about place value when they go ahead and try to add or subtract with these using different strategies or the standard algorithm. They're able to break numbers up, or decompose numbers into pieces that make sense to them. And their understanding of the mathematical relationships or ideas can just continue to grow and flourish. Mike: I'm going to stay on this for one more question, Eric, and then I think you're already headed to the place where I want to go next. What you're making me think about is this work with kids not as, “How do I get an answer today?” but “What role is my helping kids understand these place value relationships going to play in their long-term success?” Eric: Yeah, that's a great point. And learning mathematical ideas, it just doesn't happen in one lesson or in one week. When you have a complex idea like place value that, it spans over multiple years. And what kindergarten and first grade teachers are teaching them with respect to the relationship, or the equivalence, between 10 ones and one ten is setting the foundation, setting the stage for the students to start to make sense of a similar idea that happens in second grade. And then another similar idea that happens in third grade where they continue to think about this 10 times relationship between units, but just with larger and larger groupings. And then when you get to fourth, fifth, sixth, seventh grade, you're talking about smaller units, units smaller than one, and seeing that if we're using a decimal place value system, that there's still these relationships that occur. And that 10 times relationship holds true. And so, if we're going to help students make sense of those ideas in fourth and fifth grade with decimal units, we need to start laying that groundwork and helping them make sense of those relationships in the earlier grades as well. Mike: That's a great segue because I suspect there are probably educators who are listening who are curious about the types of learning activities that they could put into place that would help build that deeper understanding of place value. And I'm curious, when you think about learning activities that you think really do help build that understanding, what are some of the things that come to mind for you? Eric: Well, I'll talk about some specific activities in response to this, and thankfully there are some really high-quality instructional materials and math curricula out there that suggest some specific activities for teachers to use to help students make sense of place value. I personally think there are lots of cool instructional routines nowadays that teachers can use to help students make sense of place value ideas too. Actually, some of the math curricula embed these instructional routines within their lesson plans. But what I love about the instructional routines is that they're fairly easy to implement. They usually don't take that much time, and as long as you do them fairly consistently with your students, they can have real benefits for the children's thinking over time. So one of the instructional routines that could really help students develop place value ideas in the younger grades is something called “counting collections.” And with counting collections, students are asked to just count a collection of objects. It could be beans or paper clips or straws or Unifix cubes, whatever you have available in your classroom. And when counting, students are encouraged to make different bundles that help them keep track of the total more efficiently than if they were just counting by ones. So, let's say we asked our first- or second grade class to count a collection of 36 Unifix cubes or something like that. And when counting, students can put every group of 10 cubes into a cup or make stacks of 10 cubes by connecting them together to represent every grouping of 10. And so, if they continue to make stacks of 10 Unifix cubes as they count the total of 36, they'll get three stacks of 10 cubes or three cups of 10 cubes and six singletons. And then teachers can have students represent their count in a place value table where the columns are labeled with tens and ones. So, they would put a 3 in the tens column and a 6 in the ones column to show why the numeral 36 represents the total. So giving students multiple opportunities to make the connection between counting an amount of stuff and using groupings of tens and ones, writing that numeral that corresponds to that quantity in a place value table, let's say, and using words like 3 tens and 6 ones will hopefully help students over time to make sense of that idea. Mike: You're bringing me back to that language you used at the beginning, Eric, where you talked about face value versus place value. What strikes me is that counting collections task, where kids are literally counting physical objects, grouping them into, in the case you used, tens, you actually have a physical representation that they've created themself that helps them think about, “OK, here's the face value. Where do you see this particular chunk of that and what place value does it hold?” That's a lovely, super simple, as you said, but really powerful way to kind of take all those big ideas—like 10 times as many, grouping, place value versus face value—and really touch all of those big ideas for kids in a short amount of time. Eric: Absolutely. What's nice is that this instructional routine, counting collections, can be used with older students too. So, when you're discussing decimal quantities let's say, you just have to make it very clear what represents one. So, suppose we were in a fourth- or fifth grade class, and we still wanted students to count 36 Unifix cubes, but we make it very clear that every cup of 10 cubes, or every stack of 10 cubes, represents, let's say, 1 pound. Then every stack of 10 cubes represents 1 pound. So every cube would represent just one-tenth of a pound. Then as the students count the 36 Unifix cubes, they would still get three stacks of 10 cubes, but this time each stack represents one. And they would get six singleton cubes where each singleton cube represents one-tenth of a pound. So if you have students represent this quantity in a place value table labeled ones and tenths, they still get 3 in the ones place this time and 6 in the tenths place. So over time, students will learn that the face value of a digit tells you how many of a particular-size grouping you need, and the place value tells you the size of the grouping needed to make the total quantity. Mike: That totally makes sense. Eric: I guess another instructional routine that I really like is called “choral counting.” And with choral counting, teachers ask students to count together as a class starting from a particular number and jumping either forward or backward by a particular amount. So for example, suppose we ask students to start at 5 and count by tens together. The teacher would record their counting on the board in several rows. And so as the students count together, saying “5, 15, 25, 35,” and so on, the teacher's writing these numerals across the board. He or she puts 10 numbers in a row. That means that when the students get to 105, the teacher starts a new row beginning at 105 and records all the way to 195, and then the third row would start at 205 and go all the way to 295. And after a few rows are recorded on the board, teachers could ask students to look for any patterns that they see in the numerals on the board and to see if those patterns can help them predict what number might come in the next row. So students might notice that 10 is being added across from one number to the next going across, or 100 is being added down the columns. Or 10 tens are needed to make a hundred. And having students notice those patterns and discuss how they see those patterns and then share their reasoning for how they can use that pattern to predict what's going to happen further down in the rows could be really helpful for them too. Again, this can be used with decimal numbers and even fractional numbers. So this is something that I think can also be really helpful, and it's done in a fun and engaging way. It seems like a puzzle. And I know patterns are a big part of mathematics, and choral counting is just a neat way to incorporate those ideas. Mike: Yeah, I've seen people do things like counting by unit fractions too, and in this case counting by tenths, right? One-tenths, two-tenths, three-tenths, and so on. And then there's a point where the teacher might start a new column, and you could make a strategic choice to say, “I'm going to start a new column when we get to ten-tenths.” Or you could do it at five-tenths. But regardless, one of the things that's lovely is choral counting can really help kids see structure in a way that counting out loud, if it doesn't have the, kind of, written component of building it along rows and columns, it's harder to discern that. You might hear it in the language, but choral accounting really helps kids see that structure in a way that, from my experience at least, is really powerful for them. Eric: And like you said, the teacher, strategically, chooses when to make the new row happen to help students, kind of, see particular patterns or groupings. And like you said, you could do it with fractions too. So even unit fractions: zero, one-seventh, two-sevenths, three-sevenths, four-sevenths all the way to six-sevenths. And then you might start a new row at seven-sevenths, which is the same as one. And so, [students] kind of realize that, “Oh, I get a new one when I regroup seven of these sevenths together. And so with decimal numbers, I need 10 of the one-tenths to get to one.” And so if you help kids, kind of, realize that these numerals that we write down correspond with units and smaller amounts of stuff, and you need a certain amount of those units to make the next-sized unit or something like that, like I said, it can go a long way even into fractional or decimal kinds of quantities. Mike: I think...
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Season 2 | Episode 9 – Responsive Interpretations of Student Thinking - Guest: Kim Morrow Leong, PhD
01/04/2024
Season 2 | Episode 9 – Responsive Interpretations of Student Thinking - Guest: Kim Morrow Leong, PhD
ROUNDING UP: SEASON 2 | EPISODE 9 What are the habits of mind that educators can adopt to be more responsive to our students' thinking? And how can we turn these habits of mind into practical steps that we can take on a regular basis? Dr. Kim Morrow-Leong has some thoughts on this topic. Today, Kim joins the podcast, and we'll talk with her about three mental shifts that can profoundly impact educators’ instructional and assessment practices. GUEST BIOGRAPHY Kim Morrow-Leong is an experienced mathematics education specialist and consultant with a demonstrated history of working productively with teachers in the area of mathematics education leadership. Skilled in lesson planning, educational technology, instructional design, leadership, and curriculum development, Kim holds a PhD and MEd in mathematics education leadership from George Mason University. RESOURCES TRANSCRIPT Mike Wallus: What are the habits of mind that educators can adopt to be more responsive to our students' thinking? And how can we turn these habits of mind into practical steps that we can take on a regular basis? Dr. Kim Morrow-Leong has some thoughts on this topic. Today, Kim joins the podcast, and we'll talk with her about three mental shifts that can profoundly impact educators’ instructional and assessment practices. Kim Morrow-Leong, welcome to the podcast. We're excited to have you. Kim Morrow-Leong: Thank you, Mike. It's nice to be here. Mike: I'm really excited to talk about the shifts educators can make to foster responsive interpretations of student thinking. This is an idea that for me has been near and dear for a long time, and it's fun to be able to have this conversation with you because I think there are some things we're going to get into that are shifts in how people think, but they're also practical. You introduced the shift that you proposed with a series of questions that you suggested that teachers might ask themselves or ask their colleagues. And the first question that you posed was, “What is right?” And I'm wondering, what do you mean when you suggest that teachers might ask themselves or their colleagues this question when they're interpreting student thinking? Kim: So, I'm going to rephrase your question a little bit and change the emphasis to say, “What is right ?” And the reason I want to change the emphasis of that is because we often talk about what is wrong. And so rather than talking about what is wrong, let's talk about what's right. When we look at student work, it's a picture. It's a snapshot of where they are at that particular moment. And the greater honesty that we can bring to that situation to understand what their thinking is, the better off we're going to be. So, there's a lot of talk lately about asset-based instruction, asset-based assessment. And I think it's a great initiative, and it really gets us thinking about how we can think about what students are good at and what they bring to the table or what they bring to the classroom culture. But we don't often talk a lot about how we do that, how we break the mold. Because many of our metaphors and our language about learning are linear, and they indicate that students are moving from somewhere to achieve a goal somewhere down the path, somewhere down the line. How do you switch that around? Well, rather than looking at what they're missing and what part of the path they haven't achieved yet, we can look at where they are at the moment because that reflects everything they've learned up to that moment. So, one of the ways we can do this is to unpack our standards a little more carefully. And I think a lot of people are very good at looking at what the skills are and what our students need to be able to do by the end of the year. But a lot of what's behind a standard are concepts. What are some big ideas that must be in place for students to be successful with the skills? So, I'm going to give a very specific example. This one happens to be about a fourth-grade question that we've asked before in a district I used to work at. The task is to sketch as many rectangles as you can that are 48 square units. There's some skills behind this, but understanding what the concepts are is going to give us a little more insight into student thinking. So, one of the skills is to understand that there are many ways to make 48: to take two factors and multiply them together, and only two factors, and to make a product of 48 or to get the area. But a concept behind that is that 48 is the product of two numbers. It's what happens when you multiply one dimension by the other dimension. It's not the measure of one of the dimensions. That's a huge conceptual idea for students to sort out what area is and what perimeter is, and we want to look for evidence of what they understand about the differences between what the answer to an area problem is and what the answer to, for example, a perimeter problem is. Another concept is that area indicates that a space is covered by squares. The other big concept here is that this particular question is going to have more than one answer. You're going to have 48 as a product, but you could have 6 times 8 and 4 times 12 and many others. So that's a lot of things going into this one, admittedly very rich, task for students to take in. One of the things I've been thinking a lot about lately is this idea of a listening stance. So, a listening stance describes what you're listening for. It describes how you're listening. Are you listening for the right answer? Are you listening to understand students' thinking? Are you listening to respond or are you listening to hear more—and asking for more information from your student or really from any listener? So, one of the ways we could think about that, and perhaps this sounds familiar to you, is you could have what we call an evaluative listening stance. An evaluative listening stance is listening for the right answer. As you listen to what students say, you're listening for the student who gives you the answer that you're looking for. So, here's an example of something you might see. Perhaps a student covers their space and has dimensions for the rectangle of 7 times 6, and they tell you that this is a space that has an area of 48 square units. There's something right about that. They are really close. Because you can look at their paper and you can see squares on their paper and they're arranged in an array. And you can see the dimensions on this side and the dimensions on that side, and you can see that there's almost 48 square units. I know we all can see what's wrong about that answer, but that's not what we're thinking about right now. We're thinking about what's right. And what's right is they covered that space with an area that is something by 6. This is a great place to start with this student to figure out where they got that answer. If you're listening evaluatively, that's a wrong answer and there's nowhere else to go. So, when we look at what is right in student work, we're looking for the starting point. We're looking for what they know so that we can begin there and make a plan to move forward with them. You can't change where students are unless you meet them where they are and help them move forward. Mike: So, the second question that you posed was, “Can you cite evidence for what you're saying?” So again, talk us through what you're asking, when you ask teachers to pose this question to themselves or to their colleagues. Kim: Think about ways that you might be listening to a student's answer and very quickly say, “Oh, they got it,” and you move on. And you grab the next student's paper or the next student comes up to your desk and you take their work and you say, “Tell me what you're thinking.” And they tell you something. You say, “That's good,” and you move to the next one. Sometimes you can take the time to linger and listen and ask for more and ask for more and ask for more information. Teachers are very good at gathering information at a glance. We can look at a stack of papers and in 30 seconds get a good snapshot of what's happening in that classroom. But in that efficiency we lose some details. We lose information about specifics, about what students understand, that we can only get by digging in and asking more questions. Someone once told me that every time a student gives an answer, you should follow it with, “How do you know?” And somebody raised their hand and said, “Well, what if it's the right answer?” And the presenter said, “Oh, you still ask it. As a matter of fact, that's the best one to ask. When you ask, ‘How do you know?’ you don't know what you're going to hear. You have no idea what's going to happen.” And sometimes those are the most delightful surprises, is to hear some fantastical, creative way to solve a problem that you never would've thought about. Unless you ask, you won't hear these wonderful things. Sometimes you find out that a correct answer has some flawed reasoning behind it. Maybe it's reasoning that only works for that particular problem, but it won't work for something else in the future. You definitely want to know that information so that you can help that student rethink their reasoning so that the next time it always works. Sometimes you find out the wrong answers are accidents. They're just a wrong computation. Everything was perfect up until the last moment and they said, “3 times 2 is 5,” and then they have a wrong answer. If you don't ask more, either in writing or verbally, you have incorrect information about that student's progress, their understandings, their conceptual development, and even their skills. That kind of thing happens to everyone because we're human. By asking for more information, you're really getting at what is important in terms of student errors and what is not important, what is just easily fixable. I worked with a group of teachers once to create some open-ended tasks that require extended answers, and we sat down one time to create rubrics. And we did this with student work, so we laid them all out and someone held up a paper and said, “This is it! This student gets it.” And so, we all took a copy of this work and we looked at it. And we were trying to figure out what exactly does this answer communicate that makes sense to us that seems to be an exemplar. And so, what we did was we focused on exactly what the student said. We focused on the evidence in front of us. This one was placing decimal numbers on a number line. We noted that the representation was accurate, that the position of the point on the number line was correct. We noticed that the label on the point matched the numbers in the problem, so that made sense. But then all of a sudden somebody said, “Well, wait a minute. There's an answer here, but I don't know how this answer got here.” Something happened, and there's no evidence on the page that this student added this or subtracted this, but magically the right answer was there. And it really drove home for this group—and for me, it really stuck with me—the idea that you can see a correct answer but not know the thinking behind it. And so, we learned from that point on to always focus on the evidence in front of us and to make declarative statements about what we saw, what we observed, and to hold off on making inferences. We saved our inferences for the end. After I had this experience with the rubric grading and with this group of teachers and coaches, I read something about over-attribution and under-attribution. And it really resonated with me. Over-attribution is when you make the claim that a student understands something when there really isn't enough evidence to make that statement. It doesn't mean that's true or not true; it means that you don't have enough information in front of you. You don't have enough evidence to make that statement. You over-attribute what it is they understand based on what's in front of you. Similarly, you get under-attribution. You have a student who brings to you a drawing or a sketch or a representation of some sort that you don't understand because you've never seen anybody solve a problem this way before. You might come to the assumption that this student doesn't understand the math task at hand. That could be under attribution. It could be that you have never seen this before and you have not yet made sense of it. And so, focusing on evidence really gets us to stop short of making broad, general claims about what students understand, making broad inferences about what we see. It asks us to cite evidence to be grounded in what the student actually put on the paper. For some students, this is challenging because they mechanically have difficulties putting things on paper. But we call a student up to our desk and say, “Can you tell me more about what you've done here? I'm not following your logic.” And that's really the solution is to ask more questions. I know, you can't do this all the time. But you can do it once in a while, and you can check yourself if you are assigning too much credit for understanding to a student without evidence. And you can also check yourself and say, “Hmm, am I not asking enough questions of this student? Is there something here that I don't understand that I need to ask more about?” Mike: This is really an interesting point because what I'm finding myself thinking about is my own practice. What I feel like you're offering is this caution, which says, “You may have a set of cumulative experiences with children that have led you to a set of beliefs about their understanding or how they come to understanding. But if we're not careful—and sometimes even if we are careful—we can bring that in a way that's actually less helpful, less productive,” right? It's important to look at things and actually say, “What's the evidence?” rather than, “What's the body of my memory of this child's previous work?” It's not to say that that might not have value, but at this particular point in time, what's the evidence that I see in front of me? Kim: That's a good point, and it reminds me of a practice that we used to have when we got together and assessed these open-ended tasks. The first thing we would do is we put them all in the middle of the table and we would not look at our own students' work. That's a good strategy if you work with a team of people, to use these extended assessments or extended tasks to understand student thinking, is to share the load. You put them all out there. And the other thing we would do is we would take the papers, turn them over and put a Post-it note on the back. And we would take our own notes on what we saw, the evidence that we saw. We put them on a Post-it note, turn them over and then stick the Post-it note to the back of the work. There are benefits to looking at work fresh without any preconceived notions that you bring to this work. There are other times when you want all that background knowledge. My suggestion is that you try it differently, that you look at students' work for students you don't know and that you not share what you're seeing with your colleagues immediately, is that you hold your opinions on a Post-it to yourself, and then you can share it afterwards. You can bring the whole conversation to the whole table and look at the data in front of you and discuss it as a team afterwards. But to take your initial look as an individual with an unknown student. Mike: Hmm. I'm going to jump to the third shift that you suggest, which is less of a question and more of a challenge. You talk about the idea of moving from anticipating to targeting a learning trajectory, and I'm wondering if you could talk about what that means and why you think it's important? Kim: Earlier we talked about how important it is to understand and unpack our standards that we're teaching so that we know what to look for. And I think the thing that's often missed, particularly in standards in the older grades, is that there are a lot of developmental steps between, for example, a third-grade standard and a fourth-grade standard. There are skills and concepts that need to grow and develop, but we don't talk about those as much as perhaps we should. Each one of those conceptual ideas we talked about with the area problem we discussed may come at different times. It may not come during the unit where you are teaching area versus perimeter versus multiplication. That student may not come to all of those conceptual understandings or acquire all of the skills they need at the same time, even though we are diligently teaching it at the same time. So, it helps to look at third grade to understand, what are these pieces that make up this particular skill? What are the pieces that make up the standard that you're trying to unpack and to understand? So, the third shift in our thinking is to let go of the standard as our goal, but to break apart the standard into manageable pieces that are trackable because really our standards mean “by the end of the year.” They don't mean by December; they mean by the end of the year. So that gives you the opportunity to make choices. What are you going to do with the information you gather? You've asked what is right about student work. You've gathered evidence about what they understand. What are you going to do with that information? That perhaps is the hardest part. There's something out there called a learning trajectory that you've mentioned. A learning trajectory comes out of people who really dig in and understand student thinking on a fine-grain level, how students will learn developmentally. What are some ideas they will develop before they develop other ideas? That's the nature of a learning trajectory. And sometimes those are reflected in our standards. The way that kindergartners are asked to rote count before they're asked to really understand one-to-one correspondence. We only expect one-to-one correspondence up to 20 in kindergarten, but we expect rote counting up to 100 because we acknowledge that that doesn't come at the same time. So, a learning trajectory, to some degree, is built into your standards. But as we talked about earlier, there are pieces and parts that aren't outlined in your standards. One of the things we know about students and their interactions with grids and arrays is that a student might be able to recognize an array that is 6-by-8, but they may not yet be able to draw it. The spatial structuring that's required to create a certain number of lines going vertically and a certain number of lines going horizontally may not be in place. At the same time, they are reading arrays and understanding what they mean. So, the skill of structuring the space around you takes time. The task where we ask them to draw these arrays is asking something that some kids may not yet be able to do, to draw these grids out. If we know that, we can give them practice making arrays, we can give them tools to make arrays, we can give them blocks to make arrays, and we can scaffold this and help them move forward. What we don't want to assume is that a student who cannot yet make a 6-by-8 array can't do any of it because that's not true. There's parts they can do. So, our job as teachers is to look at what they do, look carefully at the evidence of what they do, and then make a plan. Use all of that skill and experience that's on our teams. Even if you're a new teacher, all those people on your teams know a lot more than they're letting on, and then you can make a plan to move forward and help that student make these smaller steps so they can reach the standard. Mike: When we talked earlier, one of the things that you really shifted for me was some of the language that I found myself using. So,...
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Season 2 | Episode 8 – It's a Story, Not a Checklist! - Guest: Dr. John Staley
12/21/2023
Season 2 | Episode 8 – It's a Story, Not a Checklist! - Guest: Dr. John Staley
ROUNDING UP: SEASON 2 | EPISODE 8 There’s something magical about getting lost in a great story. Whether you’re reading a book, watching a movie or listening to a friend, stories impart meaning and capture our imagination. Dr. John Staley thinks a lot about stories. On this episode of Rounding Up, we’ll talk with John about the ways he thinks that the concept of story can impact our approach to the content we teach and the practices we engage in to support our students. GUEST BIOGRAPHY Dr. John Staley is currently in his 27th year with Baltimore County Public Schools where he had the opportunity to teach middle and high school mathematics, serve as the coordinator of secondary mathematics and director of mathematics PreK–12. He currently works on special projects that involve supporting building administrators and their leadership teams and coordinating the work of our system improvement teams to improve student achievement by “raising the bar, closing gaps, and preparing every student for the future.” Throughout his career of more than 30 years, he taught in schools in Pennsylvania, Virginia, and Maryland at the secondary and post-secondary levels. In addition, he has presented at state, national, and international conferences; served on many committees and tasks forces; facilitated workshops and professional development sessions on a variety of topics; received the Presidential Award for Excellence in Teaching Mathematics and Science; and served as President for NCSM, the mathematics education leadership organization (2015–2017), and chair of the U.S. National Commission on Mathematics Instruction (2018–2020). RESOURCES TRANSCRIPT Mike Wallus: There's something magical about getting lost in a great story. Whether you're reading a book, watching a movie, or listening to a friend, stories impart meaning, and they capture our imagination. Dr. John Staley thinks a lot about stories. On this episode of Rounding Up, we'll talk with John about the ways that he thinks that the concept of story can impact our approach to the content we teach and the practices we engage in to support our students. Well, John, welcome to the podcast. We're really excited to talk with you today. John Staley: I'm glad to be here. Thank you for the invitation, and thank you for having me. Mike: So when we spoke earlier this year, you were sharing a story with me that I think really sets up the whole interview. And it was the story of how you and your kids had engaged with the themes and the ideas that lived in the Harry Potter universe. And I'm wondering if you could just start by sharing that story again, this time with the audience. John: OK. When I was preparing to present for a set of students over at Towson University and talking to them about the importance of teaching and it being a story. So the story of Harry Potter really began for me with our family—my wife, Karen, and our three children—back in ’97 when the first book came out. Our son Jonathan was nine at that time and being a reader and us being a reading family, we came together. He would read some, myself and my wife would read some, and our daughter Alexis was five, our daughter Mariah was three. So we began reading Harry Potter. And so that really began our journey into Harry Potter. Then when the movies came out, of course we went to see the movies and watch some of those on TV, and then sometimes we listened to the audio books. And then as our children grew, because Harry Potter took, what, 10 years to develop the actual book series itself, he's 19 now, finally reading the final book. By then our three-year-old has picked them up and she's begun reading them and we're reading. So we're through the cycle of reading with them. But what they actually did with Harry Potter, when you think about it, is really branch it out from just books to more than books. And that right there had me thinking. I was going in to talk to teachers about the importance of the story in the mathematics classroom and what you do there. So that's how Harry Potter came into the math world for me, [chuckles] I guess you can say. Mike: There's a ton about this that I think is going to become clear as we talk a little bit more. One of the things that really struck me was how this experience shaped your thinking about the ways that educators can understand their role when it comes to math content and also instructional practice and then creating equitable systems and structures. I'm wondering if we can start with the way that you think this experience can inform an educator's understanding for content. So in this case, the concepts and ideas in mathematics. Can you talk about that, John? John: Yeah, let's really talk about the idea of what happens in a math classroom being a story. The teaching and learning of mathematics is a story that, what we want to do is connect lesson to lesson and chapter to chapter and year to year. So when you think about students’ stories, and let's start pre-K. When students start coming in pre-K and learning pre-K math, and they're engaging in the work they do in math with counting and cardinality initially, and as they grow across the years, especially in elementary, and they're getting the foundation, it's still about a story. And so how do we help the topics that we're taught, the grade level content become a story? And so that's the connection to Harry Potter for me, and that's what helped me elevate and think about Harry Potter because when you think about what Harry Potter and the whole series did, they've got the written books. So that's one mode of learning for people for engaging in Harry Potter. Then they went from written books to audiobooks, and then they went from audiobooks to movies. And so some of them start to overlap, right? So you got written books, you got audiobooks, you got movies—three modes of input for a learner or for an audience or for me, the individual interested in Harry Potter, that could be interested in it. And then they went to additional podcasts, Harry Potter and the Sacred Text and things like that. And then they went to this one big place called Universal Studios where they have Harry Potter World. That's immersive. That I can step in; I can put on the robes; I can put the wand in my hand. I can ride on, I can taste, so my senses can really come to play because I'm interactive and engaged in this story. When you take that into the math classroom, how do we help that story come to life for our students? Let's talk one grade. So it feels like the content that I'm learning in a grade, especially around number, around algebraic thinking, around geometry, and around measurement and data. Those topics are connected within the grade, how they connect across the grade and how it grows. So the parallel to Harry Potter's story—there's, what, seven books there? And so you have seven books, and they start off with this little young guy called Harry, and he's age 11. By the time the story ends, he's seven years later, 18 years old. So just think about what he has learned across the years and how what they did there at Hogwarts and the educators and all that kind of stuff has some consistency to it. Common courses across grade levels, thinking, in my mind, common sets of core ideas in math: number, algebra, thinking, geometry, measurement of data. They grow across each year. We just keep adding on. So think about number. You're thinking with base ten. You then think about how fractions show up as numbers, and you're thinking about operations with whole numbers, base ten, and fractions. You think about decimals and then in some cases going into, depending if you're K–8 or K–5, you might even think about how this plays into integers. But you think about how that's all connected going across and the idea of, “What's the story that I need to tell you so that you understand how math is a story that's connected?” It's not these individual little pieces that don't connect to each other, but they connect somehow in some manner and build off of each other. Mike: So there are a couple of things I want to pick up on here that are interesting. When you first started talking about this, one of the things that jumped out for me is this idea that there's a story, but we're not necessarily constrained to a particular medium. The story was first articulated via book, but there are all of these ways that you can engage with the story. And you talked about the immersive experience that led to a level of engagement. John: Mm-hmm. Mike: And I think that is helping me make sense of this analogy—that there's not necessarily one mode of building students’ understanding. We actually need to think about multiple modes. Am I picking up on that right? John: That's exactly right. So what do I put in my tool kit as an educator that allows me to help tap into my students’ strengths, to help them understand the content that they need to understand that I'm presenting that day, that week, that month, that I'm helping build their learning around? And in the sense of thinking about the different ways Harry Potter can come at you—with movies, with audio, with video—I think about that from the math perspective. What do I need to have in my tool kit when it comes to my instructional practices, the types of routines I establish in the classroom? Just think about the idea of the mathematical tools you might use. How do the tools that you use play themselves out across the years? So students working with the different manipulatives that they might be using, the different mathematical tools, a tool that they use in first grade, where does that tool go in second grade, third grade, fourth grade, as they continue to work with whole numbers, especially with doing operations, with whatever the tool might be? Then what do you use with fractions? What tools do you use with decimals? We need to think about what we bring into the classroom to help our students understand the story of the mathematics that they're learning and see it as a story. Is my student in a more concrete stage? Do they need to touch it, feel it, move it around? Are they okay visually? They need to see it now, they’re at that stage. They're more representational so they can work with it in a different manner or they're more abstract. Hmm. Oh, OK. And so how do we help put all of that into the setting? And how are we prepared as classroom teachers to have the instructional practices to meet a diverse set of students that are sitting in our classrooms? Mike: You know, the other thing you're making me think about, John, is this idea of concepts and content as a story. And what I'm struck by is how different that is than the way I was taught to think about what I was doing in my classroom, where it felt more like a checklist or a list of things that I was tracking. And oftentimes those things felt disconnected even within the span of a year. But I have to admit, I didn't find myself thinking a lot about what was happening to grade levels beyond mine or really thinking about how what I was doing around building kindergartners’ understanding of the structure of number or ten-ness. John: Mm-hmm. Mike: How that was going to play out in, say, fifth grade or high school or what have you. You're really causing me to think how different it is to think about this work we're doing as story rather than a discrete set of things that are kind of within a grade level. John: When you say that, it also gets me thinking of how we quite often see our content as being this mile-wide set of content that we have to teach for a grade level. And what I would offer in the space is that when you think about the big ideas of what you really need to teach this year, let's just work with number. Number base ten, or, if you're in the upper elementary, number base ten and fractions. If you think about the big ideas that you want students to walk away with that year, those big ideas continue to cycle around, and those are the ones that you're going to spend a chunk of your time on. Those are the ones you're going to keep bringing back. Those are the ones you're going to keep exposing students to in multiple ways to have them make sense of what they're doing. And the key part of all of that is the understanding, the importance of the vertical nature as to what is it I want all of my students sitting in my classroom to know and be able to do, have confidence in, have their sense of agency. Like, “Man, I can show you. I can do it, I can do it.” What do we want them to walk away with that year? So that idea of the vertical nature of it, and understanding your learning progressions, and understanding how number grows for students across the years is important. Why do I build student understanding with a number line early? So that when we get the fractions, they can see fractions as numbers. So later on when we get the decimals, they can see decimals as numbers, and I can work with it. So the vertical nature of where the math is going, the learning progression that sits behind it, helps us tell the story so that students, when they begin and you are thinking about their prior knowledge, activate that prior knowledge and build it, but build it as part of the story. The story piece also helps us think about how we elevate and value our students in the classroom themselves. So that idea of seeing our students as little beings, little people, really, versus just us teaching content. When you think about the story of Harry Potter, I believe he survived across his time at Hogwarts because of relationships. Our students make it through the math journey from year to year to year to year because of relationships. And where they have strong relationships from year to year to year to year, their journey is a whole lot better. Mike: Let's make a small shift in our conversation and talk a little bit about this idea of instructional practice. John: OK. Mike: I'm wondering how this lived experience with your family around the Harry Potter universe, how you think that would inform the way that an educator would think about their own practice? John: I think about it in this way. As I think about myself being in the classroom—and I taught middle school, then high school—I'm always thinking about what's in my tool kit. I think about the tools that I use and the various manipulatives, the various visual representations that I need to have at my fingertips. So part of what my question would be, and I think about it, is what are those instructional strategies that I will be using and how do I fine-tune those? What are my practices I'm using in my routines to help it feel like, “OK, I'm entering into a story”? Harry Potter, when you look at those books, across the books, they had some instructional routines happening, some things that happen every single year. You knew there was going to be a quidditch match. You knew they were going to have some kind of holiday type of gathering or party or something like that. You knew there was going to be some kind of competition that happened within each book that really, that competition required them to apply the knowledge and skills from their various courses that they learned. They had a set of core courses that they took, and so it wasn't like in each individual course that they really got to apply. They did in some cases, they would try it out, they’d mess up and somebody's nose would get big, ears would get big, you know, change a different color. But really, when they went into some of those competitions, that's when the collection of what they were learning from their different courses, that's when the collection of the content. So how do we think about providing space for students to show what they know in new settings, new types of problems? Especially in elementary, maybe it's science application type problems, maybe they're doing something with their social studies and they're learning a little bit about that. As an educator, I'm also thinking about, “Where am I when it comes to my procedural, the conceptual development, and the ability to think through and apply the applications?” And so I say that part because I have to think about students coming in, and how do I really build this? How do I strike this balance of conceptual and procedural? When do I go conceptual? When do I go procedural? How do I value both of them? How do I elevate that? And how do I come to understand it myself? Because quite often the default becomes procedural when my confidence as a teacher is not real deep with building it conceptually. I'm not comfortable, maybe, or I don't have the set of questions that go around the lesson and everything. So I’ve got to really think through how I go about building that out. Mike: That is interesting, John, because I think you put your finger on something. I know there have been points in time during my career when I was teaching even young children where we'd get to a particular idea or concept, and my perception was, “Something's going on here and the kids aren't getting it.” But what you're causing me to think is often in those moments, the thing that had changed is that I didn't have a depth of understanding of what I was trying to do. Not to say that I didn't understand the concept myself or the mathematics, but I didn't have the right questions to draw out the big ideas, or I didn't have a sense of, “How might students initially think about this and how might their thinking progress over time?” So you're making me think about this idea that if I'm having that moment where I'm feeling frustrated, kids aren't understanding, it might be a point in time where I need to think to myself, “OK, where am I in this? How much of this is me wanting to think back and say, what are the big ideas that I'm trying to accomplish? What are the questions that I might need to ask?” And those might be things that I can discover through reflection or trying to make more sense of the mathematics or the concept. But it also might be an opportunity for me to say, “What do my colleagues know? Are there ways that my colleagues are thinking about this that I can draw on rather than feeling like I'm on an island by myself?” John: You just said the key point there. I would encourage you to get connected to someone somehow. As you go through this journey together, there are other teachers out there that are walking through what they're walking through, teaching the grade level content. And that's when you are able to talk deeply about math. Mike: The other thing you're making me think about is that you're suggesting that educators just step back from whether kids are succeeding or partially succeeding or struggling with a task and really step back and saying, like, “OK, what's the larger set of mathematics that we're trying to build here? What are the big ideas?” And then analyzing what's happening through that lens rather than trying to think about, “How do I get kids to success on this particular thing?” Does that make sense? Tell me more about what you're thinking. John: So when I think about that one little thing, I have to step back and ask myself the question, “How and where does that one thing fit in the whole story of the unit?" The whole story of the grade level. And when I say the grade level, I'm thinking about those big ideas that sit into the big content domains, the big idea number. How does this one thing fit into that content domain? Mike: That was lovely. And it really does help me have a clearer picture of the way in which concepts and ideas mirror the structures of stories in that, like, there are threads and connections that...
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Season 2 | Episode 7 – Making Sense of Fractions - Guest: Susan Empson, PhD
12/07/2023
Season 2 | Episode 7 – Making Sense of Fractions - Guest: Susan Empson, PhD
ROUNDING UP: SEASON 2 | EPISODE 7 For quite a few adults, fractions were a stumbling block in their education that caused many to lose their footing and begin to doubt their ability to make sense of math. But this doesn’t have to be the case for our students! Today on the podcast, we’re talking with Dr. Susan Empson about big ideas in fractions and how we can make them more meaningful for our students. GUEST BIOGRAPHY Susan B. Empson is a professor in the Department of Learning, Teaching, and Curriculum and the Richard Miller endowed chair of mathematics education at the University of Missouri. Her research on children’s thinking about fractions is the topic of her 2011 book, Extending Children’s Mathematics: Fractions and Decimals, co-authored with Linda Levi. She has published widely in refereed journals, including Cognition and Instruction, Journal for Research in Mathematics Education , Educational Studies in Mathematics, Teaching Children Mathematics , and Journal of Mathematics Teacher Education. She has been a researcher of Cognitively Guided Instruction since 1989 and is a co-author of Children’s Mathematics: Cognitively Guided Instruction (1st and 2nd editions). RESOURCES TRANSCRIPT Mike Wallus: For quite a few adults, fractions were a stumbling block in their education that caused many to lose their footing and begin to doubt their ability to make sense of math. But this doesn't have to be the case for our students. Today on the podcast, we're talking with Dr. Susan Empson about big ideas and fractions and how we can make them more meaningful for our students. Welcome to the podcast. Susan. Thanks for joining us. Susan Empson: Oh, it's so great to be here. Thank you for having me. Mike: So, your book was a real turning point for me as an educator, and one of the things that it did—for me at least—it exposed how little that I actually understood about the meaning of fractions. And I say this because I don't think that I'm alone in saying that my own elementary school experience was mostly procedural. So rather than attempting to move kids quickly to procedures, what types of experiences can help children build a more meaningful understanding of fractions? Susan: Great question. Before I get started, I just want to acknowledge my collaborators because I've had many people that I've worked with. There's Linda Levi, co-author of the book, and then my current research partner, Vicki Jacobs. And of course, we wouldn't know anything without many classroom teachers we've worked with, andthe current and past graduate students. In terms of the types of experiences that can help children build more meaningful experiences of fractions, the main thing we would say is to offer opportunities that allow children to use what they already understand about fractions to solve and discuss story problems. Children's understandings are often informal and early on, for example, may consist mainly of partitioning things in half. What I mean by informal is that understandings emerge in situations out of school. So, for example, many children have siblings and have experienced situations where they have had to share, let's say, three cookies or slices of pizza between two children. In these kinds of situations, children appreciate the need for equal shares, and they also develop strategies for creating them. So, as children solve and discuss story problems in school, their understandings grow. The important point is that story problems can provide a bridge between children's existing understandings and new understandings of fractions by allowing children to draw on these informal experiences. Generally, we recommend lots of experiences with story problems before moving on to symbolic work to give children plenty of opportunity to develop meaning for fractions. And we also recommend using story problems throughout fraction instruction. Teachers can use different types of story problems and adjust the numbers in those problems to address a range of fraction content. There are ideas that we think are foundational to understanding fractions, and they're all ideas that can be elicited and developed as children engage in solving and discussing story problems. So, one idea is that the size of a piece is determined by its relationship to the whole. What I mean is that it's not necessarily the number of pieces into which a whole is partitioned that determines the size of a piece. Instead, it's how many times the piece fits into the whole. So, in their problem-solving, children create these amounts and eventually name them and symbolize them as unit fractions. That's any fraction with 1 in the numerator. Mike: One of the things that stands out for me in that initial description you offered, is this idea of kids don't just make meaning of fractions at school, that their informal lived experiences are really an asset that we can draw on to help make sense of what a fraction is or how to think about it. Susan: That's a wonderful way to say it. And absolutely! The more teachers get to know the children in their classrooms and the kinds of experiences those children might have outside of school, the more that can be incorporated into experiences like solving story problems in school. Mike: Well, let's dig into this a little bit. Let's talk about the kinds of story problems or the structure that actually provides an entry point and can build understanding of fractions for students. Can you talk a bit about that, Susan? Susan: Yes. So, I'll describe a couple of types of story problems that we have found especially useful to elicit and develop children's fraction understandings. So first, equal sharing story problems are a powerful type of story problem that can be used at the beginning of and even throughout instruction. These problems involve sharing multiple things among multiple sharers. So, for example, four friends equally sharing 10 oranges. How much orange would each friend get? Problems like this one allow children to create fractional amounts by drawing things, partitioning those things, and then attaching fraction names and symbols. So, let's [talk] a little bit about how a child might solve the oranges problem. A child might begin by drawing four friends and then distributing whole oranges one by one until each friend has two whole oranges. Now, there are two oranges left and not enough to give each friend another whole orange. So, they have to think about how to partition the remaining oranges. They might partition each orange in half and give one more piece to each friend, or they might partition each of the remaining oranges into fourths and give two pieces to each friend. Finally, they have to think about how to describe how much each friend gets in terms of the wholes and the pieces. They might simply draw the amount, they might shade it in, or they might attach number names to it. I also want to point out that a problem about four friends equally sharing 10 oranges can be solved by children with no formal understanding of fraction names and symbols because there are no fractions in the story problem. The fractions emerge in children's strategies and are represented by the pieces in the answer. The important thing here is that children are engaged in creating pieces and considering how the pieces are related to the wholes or other pieces. The names and symbols can be attached gradually. Mike: So the question that I wanted to ask is how to deal with this idea of how you name those fractional amounts, because the process that you described to me, what's powerful about it is that I can directly model the situation. I can make sense of partitioning. I think one of the things that I've always wondered about is, do you have a recommendation for how to navigate that naming process? I've got one of something, but it's not really one whole orange. So how do I name that? Susan: That's a great question. Children often know some of the informal names for fractions, and they might understand halves or even fourths. Initially, they may call everything a half or everything a piece or just count everything as one. And so, what teachers can do is have conversations with children about the pieces they've created and how the pieces relate to the whole. A question that we've found to be very helpful is: How many of those pieces fit into the whole? Mike: Got it. Susan: Not a question about how many pieces are there in the whole, but how many of the one piece fit into the whole. Because it then focuses children on thinking about the relationship between the piece and the whole rather than simply counting pieces. Mike: Let's talk about the other problem type that was kind of front and center in your thinking. Susan: Yes. So, another type of story problem that can be used early in fraction instruction involves what we think of as special multiplication and division story problems that have a whole number of groups and a unit fraction amount in each group. So, what do I mean by that? For example, let's say there are six friends and they each will get one-third of a sub sandwich for lunch. So there's a whole number of groups—that's the six friends—and there's a unit fraction amount in each group—that's the one-third of a sandwich that they each get. And then the question is: How many sandwiches will be needed for the friends? So a problem like this one essentially engages children in reasoning about six groups of one third. And again, as with the equal sharing problem about oranges, they can solve it by drawing out things. They might draw each one-third of a sandwich, and then they have to consider how to combine those to make whole sandwiches. An important idea that children work on with this problem, then, is that three groups of one-third of a sandwich can be combined to make one whole sandwich. There are other interesting types of story problems, but teachers have found these two typesin particular effective in developing children's understandings of some of the big ideas in fractions. Mike: I wonder if you have educators who hear you talk about the second type of problem and are a little bit surprised because they perceive it to be multiplication. Susan: Yes, it is surprising. And the key is not that you teach all of multiplying and dividing fractions before adding and subtracting fractions, but that you use these problem types with special number combinations. So, a whole number of groups—for example, the six groups—[and] unit fractions in each group—because those are the earliest fractions children understand. And I think maybe one way to think about it is that fractions come out of multiplying and dividing, kind of in the way that whole numbers come out of adding and counting. And the key is to provide situations—story problems—that have number combinations in them that children are able to work with. Mike: That totally makes sense. Can you say more about the importance of attending to the number combinations? Susan: Yes. Well, I think that the number combinations that you might choose would be the ones that are able to connect with the fraction understandings that children already have. So, for example, if you're working with kindergartners, they might have a sense of what one-half is. So you might choose equal sharing problems that are about sharing things among two children. So for example, three cookies among two children. You could even—once children are able to name the halves they create in a problem like that—you can even pose problems that are about five children who each get half of a sandwich. How many sandwiches is that? But those are all numbers that are chosen to allow children to use what they understand about fractions. And then as their understandings grow and their repertoire of fractions also grows, you can increase the difficulty of the numbers. So, at the other end, let's think about fifth grade and posing equal sharing problems. If we take that problem about four friends sharing 10 oranges, we could change the number just a little bit to make it a lot harder—to four friends sharing 10 and a half oranges. And then fifth graders would be solving a problem that's about finding a fraction of a fraction, sharing the half orange among the four children. Mike: Let me take what you've shared and ask a follow-up question that came to me as you were talking. It strikes me that the design, the number choices that we use in problems matter, but so does the space that the teacher provides for students to develop strategies and also the way that the teacher engages with students around their strategy. Could you talk about that, Susan? Susan: Yes. We think it's important for children to have space to solve problems—fraction story problems—in ways that make sense to them and also space to share their thinking. So, just as teachers might do with whole number problem-solving. In terms of teacher questioning in these spaces, the important thing is for the teacher to be aware of and to appreciate the details of children's thinking. The idea is not to fix children's thinking with questioning, but to understand it or explore it. So one space that we have found to be rich for this kind of questioning is circulating. So that's the time when, as children solve problems, the teacher circulates and has conversations with individual children about their strategies. So follow-up questions that focus on the details of children's strategies help children to both articulate their strategies and to reflect on them, and help teachers to understand what children's strategies are. We've also found that obvious questions are sometimes underappreciated. So, for example, questions about what this child understands about what's happening in a story problem, what the child has done so far in a partial strategy, even questions about marks on a child's paper—shapes or tallies that you as a teacher may not be quite sure about—asking what they mean to the child: “What are those? Why did you make those? How did they connect with the problem?” So in some it benefits children to have the time to articulate the details of what they've done, and it benefits the teacher because they learn about children's understandings. Mike: You're making me think about something that I don't know that I had words for before, which is: I wonder if, as a field, we have made some progress about giving kids the space that you're talking about with whole number operations, especially with addition and subtraction. And you're also making me wonder if we still have a ways to go about not trying to simply funnel kids to—even if it's not algorithms—answer-getting strategies with rational numbers. I'm wondering if that strikes a chord for you or if that feels off base. Susan: It feels totally on base to me. I think that it is as beneficial—perhaps even more beneficial—for children to engage in solving story problems and teachers to have these conversations with them about their strategies. I actually think that fractions provide certain challenges that whole numbers may not. And the kinds of questioning that I'm talking about really depend on the details of what children have done. And so, teachers need to be comfortable with and familiar with children's strategies and how they think about fractions as they solve these problems. And then that understanding, that familiarity, lays the groundwork for teachers to have these conversations. The questions that I'm talking about can't really be planned in advance. Teachers need to be responsive to what the child is doing and saying in the moment. And so that also just adds to the challenge. Mike: I'm wondering if you think that there are ways that educators can draw on the work that students have done composing and decomposing whole numbers to support their understanding of fractions? Susan: Yes. We see lots of parallels. J as children's understandings of whole numbers develop, they’re able to use these understandings to solve multidigit operations problems by composing and decomposing numbers. So, for example—to take an easy addition—to add 37 + 8, , a child might say, “I don't know what that is, but I do know how to get from 37 to 40 with 3.” So, they take 3 from the 8, add it to the 37 get to 40, and then once at 40 they might say, “I know that 40 plus 5 more is 45.” So, in other words, they decompose the 8 in a way that helps them use what they understand about decade numbers. Operations with fractions work similarly, but children often do not think about the similarities because they don't understand fractions are numbers, too—versus two numbers, one on top of the other. If children understand that fractions can be composed and decomposed just as whole numbers can be composed and decomposed, then they can use these understandings to add, subtract, multiply, and divide fractions. For example, to add 1 ⅘ + ⅗ , a child might say, “I know how to get up to 2 from 1 ⅘. I need one more fifth. And then I have two more fifths still to add from the ⅗.So, it's 2 ⅖.” So, in other words, just as they decompose the 8 into 3 and 5 to add 8 to 37, they decompose the ⅗ into ⅕ and ⅖ to add it to 1 ⅘. Mike: I could imagine a problem like 1 ½ + ⅝. I could say, “Well, I know I need to get a half up. Five-eighths is really ⁴⁄₈ and ⅛, and ⁴⁄₈ is a half.” Susan: Yep. Mike: “So, I'm actually going from 1 ½ + ⁴⁄₈. OK. That gets me to 2, and then I've got one more eighth left. So, it's 2 ⅛.” Susan: Nice. Yeah, that's exactly the kind of reasoning this approach can encourage. Mike: Well, I have a final question for you, Susan. came out in 2011, and I'm wondering what you've learned since the book came out. So are there ideas that you feel like have really been affirmed or refined? And what are some of the questions about the ways that students make meaning of fractions that you're exploring right now? Susan: Well, I think, for one, I have a continued appreciation for the power of equal sharing problems. You can use them to elicit children's informal understandings of fractions early in instruction. You can use them to address a range of fraction understandings, and they can be adapted for a variety of fraction content. So, for example, building meaning for fractions, operating with fractions, concepts of equivalence. Vicki and I are currently writing up results from a big research project focused on teachers' responsiveness to children's fraction thinking during instruction. And right now, we're in the process of analyzing data on third, fourth, and fifthgrade children’s strategies for equal sharing problems. We specifically focused on over 1,500 drawing-based strategies used by children in a written assessment at the end of the school year. We've been surprised both by the variety of details in these strategies—so, for example, how children represent items, how they decide to distribute pieces to people—and also by the percentages of children using these drawing-based strategies. For each of grades three, four, and five, over 50 percent of children used a drawing-based strategy. There are also, of course, other kinds of strategies that don't depend on drawings that children use, but by far the majority of children were using these strategies. Mike: That's interesting because I think it implies that we perhaps need to recognize that children actually benefit from time using those strategies as a starting point for making sense of the problems that they're solving. Susan: I think it speaks to the length of time and the number of experiences that children need to really build meaning for fractions that they can then use in more...
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Season 2 | Episode 6 – Making the Shift: Moving From Additive to Multiplicative Thinking - Guest: Anderson Norton, Ph.D.
11/23/2023
Season 2 | Episode 6 – Making the Shift: Moving From Additive to Multiplicative Thinking - Guest: Anderson Norton, Ph.D.
ROUNDING UP: SEASON 2 | EPISODE 6 One of the most important shifts in students’ thinking during their elementary years is also one of the least talked about. I’m talking about the shift from additive to multiplicative thinking. If you’re not sure what I’m talking about, I suspect you’re not alone. Today, we talk with Dr. Anderson Norton about this important but underappreciated shift. GUEST BIOGRAPHY Anderson Norton’s research is driven by a desire to understand how humans have access to knowledge as powerful and reliable as mathematics. Throughout his career, building upon Jean Piaget's genetic epistemology, he has learned that many philosophical questions about the nature of mathematics have psychological answers. He grounds his research in psychological models of students' mathematics, and collaborates with psychologists and neuroscientists to find these answers. In 2022, he authored a related book, published by Routledge: The Psychology of Mathematics: A Journey of Personal Mathematical Power for Educators and Curious Minds . RESOURCES TRANSCRIPT Mike Wallus: One of the most important shifts in students' thinking during their elementary years is also one of the least talked about. I'm talking about the shift from additive to multiplicative thinking. If you're not sure what I'm talking about, I suspect you're not alone. Today we talk with Dr. Anderson Norton about this important but underappreciated shift. Welcome to the podcast, Andy. I'm excited to talk with you about additive and multiplicative thinking. Andy Norton: Oh, thank you. Thanks for inviting me. I love talking about that. Mike: So, I want to start with a basic question. When we're talking about additive and multiplicative thinking, are we just talking about strategies or operations that students would carry out to find a sum or a product of a problem? Or are we talking about something larger? Andy: Yeah, definitely something larger, and it doesn't come down to strategies. Students can solve multiplication tasks—what, to us, look like multiplication tasks—using additive reasoning. And they often do. I think they get through a lot of elementary school using, for example, repeated addition. If I gave a task like, “What is 4 times 5?”, then they might just say, “That's 5 and 5 and 5 and 5,” which is fine. They're solving a multiplication problem, but their method for solving it is repeated addition, so it's basically additive reasoning. But it starts to catch up to them in later grades where that kind of additive reasoning requires them to do more and more sophisticated or complicated strategies that maybe their teachers can teach them, but it starts to add up, especially when they get to fractions or algebra. Mike: So, let's dig into this a little bit deeper. How would you describe the difference between additive and multiplicative thinking? And I'm wondering if there's an example of the differences in how a student might approach a task or a problem that could maybe highlight that distinction. Andy: The main distinction is with additive reasoning, you're working within one level of unit. So, for example, if I want to know—going back to that 4 times 5 example—really what I'm doing is I'm working with ones. So, I say I have 5 ones and 5 ones and 5 ones and 5 ones, and that's 20 ones. But in a multiplication problem, you're really transforming across units. If I want to understand 4 times 5 as a multiplication problem, what I'm saying is, “If I measure a quantity with a unit of 5, the measure is 4.” Just to make it a little more concrete, suppose my unit of measure is like a stick that's 5 feet long, and then I say, “OK, I measured this length, and it was four of these sticks. So, it's four of these 5-foot sticks. But I want to know what it is in just feet.” So, I've changed my unit. I'm saying, “I measured this thing in one unit, this stick length, but I want to understand its measure in a different unit, a unit of ones.” So, you're transforming between this one kind of unit into another kind of unit, and it's a 5-to-1 transformation. So, I'm not just doing 5 plus 5 plus 5 plus 5, I'm saying every one of that stick length contains 5 feet, five of these 1-foot measures. And so, it's a transformation from one unit into another, one unit for measuring into a different unit for measuring. Mike: I mean, that's a really big shift, and I'm glad that you were able to describe that with a practical example that someone could listen to this and visualize. I think understanding that for me clarifies the importance of not thinking about this in terms of just procedural steps that kids would take to either add or multiply; that really there's a transformation in how kids are thinking about what's happening rather than just the steps that they're following. Andy: Yeah, that's right. And a lot of times as teachers, or even as researchers studying children, we're frustrated like the kids are when they're solving tasks, when they're struggling. And so we try to give them those procedures. We might give them a visual model. We might give them an array model for multiplication, which can solve a lot of problems. You just sort of think about things going vertically and things going horizontally, and then you're looking at an area or a number of intersections. So, that makes it possible for them to solve these individual tasks. And there's a lot of pressure on teachers to cover curriculum, so we feel like we have to support them by giving them these strategies. But in the end, it just becomes more and more of these complicated strategies without really necessitating the need for something we might call a “productive struggle”; that is, where students can actually start to go through developmental changes by allowing them to struggle so that they actually develop these kinds of multiplicative structures instead of just giving them a bunch of strategies for dealing with that one task at a time. Mike: I'm wondering if you might share some examples of what multiplicative thinking might look like or sound like in different scenarios. For example, with whole numbers, with fractions or decimals … Andy: Uh-huh. Mike: … and perhaps even in a context like measurement. What might an educator who was listening or observing students' work, what might they see that would indicate to them that multiplicative reasoning or multiplicative thinking was something that was happening for the student? Andy: So, it really is that sort of transformation of units. Like imagine, I know something is nine-fifths, and nine-fifths doesn't make a whole lot of sense unless I can think about it as nine units of one-fifth. We have to think about it as a measure like it's nine of one-fifth. And then I have to somehow compare that to, “OK, it's nine of this one unit, this one-fifth unit, but what is it of a whole unit? A unit of one?” So, having an estimate for how big nine-fifths is, yes, it's nine units of one-fifth. But at the same time, I want to know how big that is relative to a one. So, there's this multiplicative nature kind of built into tasks like that, and it's one explanation for why students struggle so much with improper fractions. Mike: So, I'm going to put my teacher hat on for a second because what you've got me thinking is, what are the types of tasks or experiences or even questions that an educator could put in front of students that would nudge them to make this shift without potentially pushing them to a place where they're not quite ready to go yet? Andy: Hmm. Mike: Could you talk a little bit about what types of tasks or experiences or questions might help provide a little bit of that nudge? Andy: Yeah, that's a really good question, because it goes back to this idea that students are already solving the kinds of tasks that should involve multiplicative reasoning, but they might be using additive strategies to do it. Those strategies get more and more complicated, and we as teachers facilitate students just, sort of, doing something more procedural instead of really struggling with the issue. And what the issue should be is opportunities to work with multiple levels of units and then to reflect on their activity and working with them. So, for example, one task I like to give students is, I'll cut out a piece of construction paper and I'll hand it to the student. And I'll have hidden what I'm going to label a whole, and I'll have hidden what I'm going to label to be the unit fraction that might be appropriate for measuring this thing I gave them. So, I'll give them this piece of construction paper and I'll say, “Hey, this is five-sevenths of my whole.” Now what I've given them as a rectangular strip of paper … Mike: Mm-hmm Andy: …without any partitions in it. I've hidden the whole from which I created this five-sevenths. I've hidden one-seventh, and I've put them away, maybe inside of envelopes. So, it becomes like a game: “Can you guess what I have in this envelope? I just gave you five-sevenths. Can you guess, what is this five of? What is the unit that this is five of?” and “What is the whole this five-sevenths fraction is?” So, it's getting them thinking about two different levels of units at once. They've been given this one measurement, but they don't know the unit in which it's measured, and they don't even have visually present for them what the whole unit would be. So, what they might do, is they might engage in partitioning activity. Sometimes they might partition what I give them into seven equal parts instead of five because I told them “five-sevenths” and five-sevenths to them, that means, “partition it a seventh.” Well, that could lead to problems, and if they see that their unit is smaller than the one I have hidden, they might have to reason through what went wrong, “Why might have you gotten a different answer than I did?” So, it's those kinds of activities—of partitioning or iterating a unit, measuring out with a unit, and then reflecting on that activity—that give them a basis for starting to coordinate these units at higher and higher levels and, therefore, in line with Amy Hackenberg’s framing, develop multiplicative concepts. Mike: I think that example is really helpful. I was picturing it in my head, and I could see the opportunities that that affords for, kind of, pressing on some of those big ideas. One of the things that you made me think about is the idea of manipulatives, or even if we broaden it out a little bit, visual models. Because the question I was going to ask is, “What role might a visual model or a manipulative play in supporting a shift from additive to multiplicative thinking?” I'm curious about how you would respond to that initially. And then I think I have a follow-up question for you as well. Andy: OK. I can think of two important roles for visual models—or at least two for manipulatives—and at least one works with visual models as well. But before answering that, the bigger answer is, no one manipulative is going to be the silver bullet. It's how we use them. We can use manipulatives in ways where students are just following our procedures. We can use visual models where students are just doing what we tell them to do and reading off the answer on paper. That really isn't qualitatively any different than when we just teach them an algorithm. They don't know what they're doing. They get the answer, they read it off the paper. You could consider that to be a visual model, what they're doing on their paper or even a manipulative; they're just following a procedure. What manipulatives should afford is opportunities for students to manipulate. They should be able to carry out their mental actions. So, maybe when they're trying to partition something and then iterate it, or they're thinking about different units. That's too much for them to keep in mind in their visual imagination. So, a visual model or a manipulative gives them a way to carry those actions out to see how they work with each other, to notice the effects of those actions. So, if the manipulative is used truly as a manipulative, then it's an opportunity for them to carry out their mental actions to coordinate them with a physical material and to see what happens. And visual models could be similar, gives them a way to sort of carry out their mental actions, maybe a little more abstractly, because they're just using representations rather than the actual manipulative, but maybe gives them a way to keep track of what would happen if I partitioned this into three parts and then took one of those parts and partitioned into five. How would that compare to the whole? So, it's their actions that have to be afforded by the manipulative or the visual model. And to decide what is an appropriate manipulative or an appropriate task, we need to think about, “OK, what can they already do without it?” And I'm trying to push them to do the next thing where it helps them coordinate at a level they can't just do in their imagination, and then to reflect on that activity by looking at what they wrote or looking at what they did. So, it's always that: Carrying out actions in slightly more powerful ways than they could do in their mind. That's sort of the sense in which mathematics builds on itself. After they've reflected on what they've done and they've seen the results, now maybe that's something that they can take as an object, as something that's just there for them in [their] imagination so they can do the next thing: adding complexity. Mike: OK. So, I take it back. I don't think I have a follow-up question because you answered it in that one. What I was kind of going to dig into is the thing that you said, which is, there's a larger question about the role that a manipulative plays, and I think that your description of a manipulative should be there to manipulate to help kids carry out … Andy: Mm-hmm. Mike: … the mental action and make meaning of that. I think that piece to me is one that I really needed clarified, just to think about my own teaching and the role the manipulatives are going to play when I'm using them to support student thinking. Andy: And I'll just add one thing, not to use too many fractions examples, but that is where most of my empirical research has been, was working with elementary and middle-school children with fractions. But I have to make these decisions based on the child. So, sometimes I'll use these Cuisenaire Rods, the old fraction rods, the colored fraction rods. Sometimes I'll use those with students because then it sort of simplifies the idea. They don't have to wonder whether a piece fits in exactly a certain number of times. The rods are made to fit exactly. And maybe I'm not as concerned about them cutting a construction paper into equal parts or whatever. So, the rods are already formed. But other times I feel like they might be relying too much on the rods, where they start to see the brown rod as a 4. They're not even really comparing the red rod, which fits into it twice. They're just, “Oh, the red is a 2; the brown is a 4. I know it's in there twice because two and two is four.” So, you start to think about them, whole numbers. And so sometimes I'll use the rods because I want them to manipulate them in certain ways, and then other times I'll switch to the construction paper to sort of productively frustrate this idea that they're just going to work with whole numbers. I actually want them to create parts and to see the measurements and actually measure things out. So, it all depends on what kind of mental action I want them to carry out that would determine [which] manipulative as well. Because manipulatives have certain affordances and certain constraints. So, sometimes Cuisenaire Rods have the affordances I want, and other times they have constraints that I want to go beyond with, say, construction paper. Mike: Absolutely. So, there's kind of a running theme that started to develop on the podcast. And one of the themes that comes to mind is this idea that it's important for us to think about what's happening with our students’ thinking as a progression rather than a checklist. What strikes me about this conversation is this shift from additive to multiplicative thinking has really major implications for our students beyond simple calculation. And I'm wondering if you could just afford us a view of, why does this shift in thinking matter for our students both in elementary school, and then also when they move beyond elementary school into middle and high school? Could you just talk about the ramifications of that shift and why it matters so much that we're not just building a set of procedures, we're building growth in the way that kids are thinking? Andy: Yeah. So, one big idea that comes up starting in middle school—but becomes more and more important as they move into algebra and calculus, any kind of engineering problem—is a rate of change. So, a rate of change is describing a relationship between units. It's like, take a simple example of speed. It's taking units of distance and units of time and transforming them into a third level of unit that is speed. So, it's that intensive relationship that's defining a new unit. When I talk about units coordination, I'm not usually talking about physical units like distance, time, and speed. I'm just talking about different numerical units that students might have to coordinate. But to get really practical, when we talk about the sciences, units coordinations have to happen all the time. So, students are able to be successful with their additive reasoning up to a point, and I would argue that point is probably around where they first see improper fractions. (chuckles) They're able to work with them up to a point, and then after that, things [are] going to be less and less sensible if they're just relying on these additive sort of strategies that each have a separate rule for a different task instead of being able to think more generally in terms of multiplicative relationships. Mike: Well, I will say from a former K–12 math curriculum director, thank you for making a very persuasive case for why it's important to … Andy: (chuckles) Mike: … help kids build multiplicative thinking. You certainly hit on some of the things that can be pitfalls for kids who are still thinking in an additive way when they start to move into upper elementary, middle school and beyond. Before we go, Andy, I suspect that this idea of shifting from additive to multiplicative thinking, that it's probably a new idea for our listeners. And you've hinted a bit about some of the folks who have been powerful in the field in terms of articulating some of these ideas. I'm wondering if there are any particular resources that you'd recommend for someone who wants to keep learning about this topic? Andy: Yeah. So, there are a bunch of us developing ideas and trying to even create resources that teachers can pick up and use. Selfishly, I'll mention one called “ ,” used by the U.S. Math Recovery Council in their professional development programs for teacher-leaders across the country. That book is probably, at least as far as fractions, that book is maybe the most comprehensive. But then, beyond that, there are some research articles that people can access, even going in Google Scholar and looking up units coordination and multiplicative reasoning. Maybe put in Steffe's name for good measure, S-T-E-F-F-E. (chuckles) You'll find a lot of papers there. Some of them have been written in teacher journals as well, like journals published by the National Council of Teachers of Mathematics, like materials that are specifically designed for teachers. Mike: Andy, thank you so much for joining us. It's really been a pleasure talking with you. Andy: OK....
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