Rounding Up

The Math Learning Center

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 4 | Episode 18 – Dr. Jenny Bay-Williams, Productive Ways to Build Fluency with Basic Facts (Rerun)

    Dr. Jenny Bay-Williams, Productive Ways to Build Fluency with Basic Facts ROUNDING UP: SEASON 4 | EPISODE 18 This summer we're replaying favorite listener episodes from the first four seasons of Rounding Up—like this one from Season 1. We'll return with all new episodes in early September. Ensuring students master their basic facts remains a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to their students' math identities. Today on the podcast, Jenny Bay-Williams talks about how to meet that goal and shares a set of productive practices that also support student reasoning and sensemaking.  BIOGRAPHY Jennifer Bay-Williams is a professor of mathematics education at the University of Louisville. She has authored over 40 books and 100 journal articles and book chapters that focus on making mathematics meaningful to all students. She is an international leader in the field of mathematics education, frequently speaking at state, national, and international conferences and serving on national boards.  RESOURCES "Eight Unproductive Practices in Developing Fact Fluency" article by Gina Kling and Jennifer M. Bay-Williams Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention book by Jennifer M. Bay-Williams and Gina Kling Math Fact Fluency companion website by Kentucky Center for Mathematics TRANSCRIPT Mike Wallus: Welcome to the podcast, Jenny. We are excited to have you.  Jennifer Bay-Williams: Well, thank you for inviting me. I'm thrilled to be here and excited to be talking about basic facts.  Mike: Awesome. Let's jump in. So, your recommendations start with an emphasis on reasoning. I wonder if we could start by just having you talk about the why behind your recommendation and a little bit about what an emphasis on reasoning looks like in an elementary classroom when you're thinking about basic facts.  Jenny: All right, well, I'm going to start with a little bit of a snarky response: that the non-reasoning approach doesn't work.  Mike and Jenny: (laugh)  Jenny: OK. So, one reason to move to reasoning is that memorization doesn't work. Drill doesn't work for most people. But the reason to focus on reasoning with basic facts beyond that fact, is that the reasoning strategies grow to strategies that can be used beyond basic facts. So, if you take something like the making 10 idea—that 9 plus 6, you can move one over and you have 10 plus 5—is a beautiful strategy for a 99 plus 35. So, you teach the reasoning upfront from the beginning, and it sets students up for success later on.  Mike: That absolutely makes sense. So, you talk about the difference between telling a strategy and explicit instruction. And I raise this because I suspect that some people might struggle to think about how those are different. Could you describe what explicit instruction looks like and maybe share an example with listeners?  Jenny: Absolutely. First of all, I like to use the whole phrase: "explicit strategy instruction." So, what you're trying to do is have that strategy be explicit, noticeable, visible. So, for example, if you're going to do the making 10 strategy we just talked about, you might have two 10-frames. One of them is filled with nine counters, and one of them is filled with six counters. And students can see that moving one counter over is the same quantity. So, they're seeing this flexibility that you can move numbers around, and you end up with the same sum. So, you're just making that idea explicit and then helping them generalize. You change the problems up and then they come back and they're like, "Oh, hey, we can always move some over to make a ten"—or a twenty, or a thirty, or whatever you're working on. And so, I feel like, in using the counters, or they could be stacking Unifix cubes or things like that. That's the explicit instruction. It's concrete.  And then, if you need to be even more explicit, you ask students in the end to summarize the pattern that they noticed across the three or four problems that they solved. "Oh, that you take the bigger number, and then you go ahead and complete a ten to make it easier to add." And then, that's how you're really bringing those ideas out into the community to talk about.  For multiplication, I'm just going to contrast. Let's say we're doing [the] add a group strategy with multiplication. If you were going to do direct instruction, and you're doing 6 times 8, you might say, "All right, so when you see a six," then a direct instruction would be like, "Take that first number and just assume it's a five." So then, "Five eights is how much? Write that down." That's direct instruction. You're like, "Here, do this step. Here, do this step. Here, do this step." The explicit strategy instruction would have, for example—I like, for eights, boxes of crayons because they oftentimes come in eights. So, but they'd have five boxes of crayons and then one more box of crayons. So, they could see you've got five boxes of crayons. They know that fact is 40, they—if they're working on their sixes, they should know their fives. And so, then what would one more group be about?  So, just helping them see that with multiplication through visuals, you're adding on one group, not one more, but one group. So, they see that through the visuals that they're doing or through arrays or things like that. So, it's about them seeing the number of relationships and not being told what the steps are.  Mike: And it strikes me, too, Jenny, that the role of the teacher in those two scenarios is pretty different.  Jenny: Very different. Because the teacher is working very hard (chuckles) with the explicit strategy instruction to have the visuals that really highlight the strategy. Maybe it's the colors of the dots or the exact 10-frames they've picked and have they filled them or whether they choose to use the Unifix cubes and how they're going to color them and things like that. So, they're doing a lot of thinking to make that pattern noticeable, visible. As opposed to just saying, "Do this first, do that second, do that third." Mike: I love the way that you said that you're doing a lot of thinking and work as a teacher to make a pattern noticeable. That's powerful, and it really is a stark contrast to, "Let me just tell you what to do."  I'd love to shift a little bit and ask you about another piece of your work. So, you advocate for teaching facts in an order that stresses relationships rather than simply teaching them in order. I'm wondering if you can tell me a little bit more about how relationships-based instruction has an impact on student thinking.  Jenny: So, we want every student to enact the reasoning strategies. So, I'm going to go back to addition, for example. And I'm going to switch over to the strategy that I call "pretend-to-10", also called "use 10" or "compensation." But if you're going to set them up for using that strategy, there's a lot of steps to think through. So, if you're doing 9 plus 5, then in the pretend-to-10 strategy, you just pretend that 9 is a 10. So now you've got 10 plus 5 and then you've got to compensate in the end. You've got to fix your answer because it's 1 too much. And so, you've got to come back 1. That's some thinking. Those are some steps. So, what you want is to have the students automatic with certain things so that they're set up for that task. So, for that strategy, they need to be able to add a number onto 10 without much thought.  Otherwise, the strategy is not useful. The strategy is useful when they already know 10 plus 5. So, you teach them this, you teach them that relationship—10 and some more—and then they know that 9's 1 less than 10. That relationship is hugely important, knowing 9 is 1 less than 10. And so then they know their answer has to be 1 less. 9's 1 less than 10. So, 9 plus a number is 1 less than 10 plus the number. Huge idea.  And there's been a lot of research done in kindergarten on students understanding things like 7's 1 more than 6, 7's 1 less than 8. And they're predictive studies looking at student achievement in first grade, second grade, third grade. And students—it turns out that one of the biggest predictors of success is students understanding those number relationships. That 1 more, 1 less, 2 more, 2 less. Hugely important in doing the number sense.  So that's what the relationship piece is, is sequencing facts so that what is going to be needed for the next thing they're going to do, the thinking that's going to be needed, is there for them. And then build on those relationships to learn the next strategy.  Mike: I mean, it strikes me that there's a little bit of a twofer in that one. The first is this idea that what you're doing is purposely setting up a future idea, right? It's kind of like saying, "I'm going to build this prior knowledge about ten-ness, and then I'm going to have kids think about the relationship between 10 and 9." So, the care in this work is actually really understanding those relationships and how you're going to leverage them.  The other thing that really jumps out from what you said [is] this has long term implications for students' thinking. It's not just fact acquisition; it's what you said: Research shows that this has implications for how kids are thinking further down the road. Am I understanding that right?  Jenny: That's absolutely correct. So just that strategy alone. Let's say they're adding 29 plus 39. And they're like, "Oh hey, both of those numbers are right next to the next benchmark. So instead of 29 plus 39, I'm going to add 30 plus 40, [which equals] 70. And I got, I went up 2, so I'm going to come back down 2. And I know that 2 less than a benchmark's going to land on an 8." So that, again, it's coming back to this relationship of how far apart numbers are, what's right there within a s

    26 min

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    Season 4 | Episode 17 – Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks

    Jana Dean & Heather Byington, Supporting Multilingual Learners During Number Talks ROUNDING UP: SEASON 4 | EPISODE 17 What might it be like to engage in a number talk as a multilingual learner? How would you communicate your ideas, and what scaffolds might support your participation?  Today, we're talking with Jana Dean and Heather Byington about ways educators can support multilingual learners' engagement and participation during number talks.  BIOGRAPHIES Heather Byington has taught all grade levels over the span of her 27-year career as a bilingual public educator. She currently teaches middle school mathematics and English language support classes in Lacey, Washington. She is also a student at Washington State University pursuing a PhD in Mathematics Education.  Jana Dean currently serves as CEO of the Mathematics Education Collaborative and supports a fantastic team of middle school math teachers in North Thurston Public Schools. Her research focuses on the intersection of content learning and language learning.  RESOURCES Judit Moschkovich research  Math Between Us blog "Number Talks: A Whole Class Routine for Learning Language for Learning Mathematics" article  Mathematics Education Collaborative website  jdean@mec-math.org Jana Dean email TRANSCRIPT Mike Wallus: Welcome to the podcast, Jana and Heather. I am so excited to be talking with you both today. Jana Dean: Good morning. Yeah, thanks for having us.  Heather Byington: Thanks so much for having us.  Mike: Absolutely.  Jana, before we begin talking about the ways that teachers can support multilingual learners during number talks, I wonder if you can offer a working definition that would help educators visualize what a number talk actually looks like. Jana: Yeah, I'd be happy to do that. A number talk in terms of how we worked with the routine in this project consisted of the teacher providing some sort of visual prompt, starting either with a visual pattern of dots or a computation problem. And then the students get wait time, time to think about how they might solve that problem. And then as they share their strategies, the teacher records and asks them questions about their reasoning for why they approached the problem in the way that they approached it. The teacher creates what I like to think of as a visual mediator of student ideas. So the students' ideas become visible as they share them. So children who are listening can listen to the dialog or conversation between the person sharing and the teacher, but the ideas actually become visible as they're being shared. And the teacher always verifies with the student whether or not they've been understood. And the goal is not for the student to be right, but for the teacher and student to understand each other.  Mike: That's really helpful. Heather, is there anything else you'd add to that?  Heather: In terms of the way that we worked with it with multilingual learners and increasing their opportunities for engagement in the routine, we always gave them an option of talking to a partner and rehearsing their answer before they volunteered to share with the whole group. We prioritized calling on multilingual learners if they volunteered. And we also did a final reflection at the end. So those were some enhancements that we added onto the routine.  Mike: I think that's really helpful and I'm excited to talk a little bit more about the details of those, Heather.  One of the things that really struck me as we were preparing for this conversation was reading about the ways that some of the multilingual learners you worked with, how they described their experience during number talks. And it helped me to see the experience from their perspective and rethink some of the ways that I'd facilitated number talks in the past. And I'm wondering if you could share a bit about some of the feelings students told you that they were experiencing.  Jana: Yeah. One of the things we suspected before we started was that as a language learner myself, talking about ideas that you're just forming in a language you're in the process of learning can be really intimidating. It's very challenging. So they were nervous. And when I interviewed fourth graders about their experience in number talks, even facilitated with language acquisition in mind, they talked about how much courage it took them to share their ideas.  They also talked about and could very keenly remember moments when they had made a contribution that their teacher made use of or a time when they made a contribution that another student made use of later. So there was a lot of pride they felt in having shared their ideas once they found ways to do that.  They also talked about how much easier it was to share our ideas than it was to share my idea. And so if, for instance, we had given them the opportunity—and like Heather said, we almost always gave them the opportunity to talk with a partner—they would often share using the pronoun "we." "This is how we thought of it." And we picked up on that and began to ask them if it was OK to attribute a group of students with a unique idea rather than an individual. And that was also consistent with many of their home cultures. It's not every culture in which individual contributions are elevated, but rather when you dare to speak, you're definitely speaking for the group, for a collective. So that collective understanding was really important.  There was one child, and I'm really curious about how representative he was of many. He always talked to the same friend, and every time he shared, he, I'm going to say, nailed it. He really had it figured out what it was that he was going to say. And there was one particular day when he did a beautiful job sharing, and I asked him about that day and he said, "To be honest, that day I really didn't want to share, but I knew my teacher wanted to hear my idea, so I did anyway." And so there's that element of love and respect for their teacher that I think was also really motivating for them.  Heather: Yeah. Can I add something quickly to that?  So one aspect of that, I think that idea of a student sharing because it meant a lot to the teacher, we also tried to utilize individual conferring with students as much as possible and gave them opportunities to confer with us, whether it was just checking in briefly before the number talk started, encouraging them or maybe telling them, "Hey, you can share the idea with me after the number talk if that feels more comfortable to you." So it's giving them multiple opportunities to do that and encouraging them to share their thoughts.  Mike: What I appreciate about what you all are doing is even in this initial part of the conversation, really getting specific about the practices and the way that those practices played out for kids. And I think as an educator, one of the things that I've come to over all my years teaching is the need to have humility and also continue to be a learner. And that sometimes really leads me to questions about intent versus impact.  Heather, I wonder if you could talk about the parts of the number talk routine or facilitation practices that may have unintentionally provoked some of the anxiety that kids were experiencing.  Heather: So for multilingual learners, when I think about what they will need, the supports that they may need to be able to engage in a routine like a number talk, I think about first the processing time that they might need to understand and think about different ways of solving that prompt. And then I think about their understanding of the prompt. And then the other thing I think about is their ability to communicate their thoughts and ideas with others. So naturally, if it seems like there's a lot of pressure because of time, if they don't have much time, if they feel that pressure to do that processing and think of those ideas and share them quickly, that may provoke anxiety because this, of course, is still a language that they're still developing. So that ability to share with a partner and rehearse those ideas and process that with a partner, that really becomes, as Jana mentioned, more of a team effort.  And then being able to rehearse the words that they're going to use and the way they're going to convey that message and communicate it to others, that again reduces the anxiety because it's a lot less pressure to share my thoughts and ideas with one person than with a whole group. And if I share those thoughts with one person and they seem to understand what I mean, then now I might feel confident enough to share with more people. So I just think that naturally when it's a time constrained activity, that that naturally can provoke anxiety.  Mike: Yeah. I mean, that absolutely makes sense. I will say as a child who was not quick, even in my first language, the impact of that was profound, let alone trying to both process in a language that I was learning and feel like I was under pressure to produce an idea and describe it. That absolutely makes sense.  Jana: I want to back up a bit and quote something that you said, Heather, partway through our working together, which was that Heather had some familiarity with number talks before we started working together, but had a healthy skepticism as well. And at one point she said that she wondered if we might not actually be hurting students when we are facilitating a routine that they cannot find entry into. And so it became really like a guiding light or principle of our work together to work hard to help them find entry into the routine. And something that I didn't realize until a year after we began working together and I was really closely tracking the experiences of the multilingual learners themselves—and this is kind of back to your question about intent and impact—when we listen to children's mathematical ideas with the intent of not correcting them, trying to figure out

    34 min

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    Season 4 | Episode 16 – Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction

    Kristin Frang, Understanding the Roots of Fluency with Addition & Subtraction ROUNDING UP: SEASON 4 | EPISODE 16 Research suggests that supporting students' fluency with addition and subtraction hinges on understanding how children's mathematical thinking develops. So what are the concepts and ideas that play a part in fluency with combinations to 10, 20, and beyond?  Today, we'll explore this question with Kristin Frang, director of instructional programs at Integrow Numeracy Solutions.  BIOGRAPHY Kristin Frang is the director of instructional programs for Integrow Numeracy Solutions. She designs resources and services that support states, districts, schools, and individuals in transforming numeracy education. RESOURCES "Understanding Units Coordination" Season 4, Episode 11 of the Rounding Up podcast Integrow Numeracy Solutions website blog  email address On Track to Numeracy book by Lucinda "Petey" MacCarty, Kurt Kinsey, David Ellemor-Collins, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Kristin. It is so great to be talking with you today.  Kristin Frang: It's great to be here. I feel so honored to be on this podcast.  Mike: Before we dive into a conversation about addition and subtraction, I'd like to do a bit of grounding. So you're currently the director of instructional programs for Integrow Numeracy Solutions. I wonder if briefly you could tell the listeners: What is Integrow Numeracy Solutions, and what's its mission?  Kristin: Yeah. Integrow Numeracy Solutions' mission is to transform numeracy education by connecting research with practice and empowering educators to advance student mathematical thinking and success. But I really want to bring that mission to life through a story, just a quick story, if I can.  Prior to my role with Integrow, I was a K–12 mathematics consultant. And one of the things that I did was, when the Common Core [State Standards] were released, I worked with teachers to transition to the then-new standards. We studied many documents together, including progression documents that were included in the standards, and teachers were honestly fascinated by this idea of a progression and that they were embedded into the standard. But I remember an instance where we had been studying these progressions and a teacher came up and said to me, "I know where my students are at; I can see them in these progressions. But how do I get them to the next stage?"  And I didn't have an answer (laughs) at that point. I was a former middle school and high school teacher. I was working with elementary teachers. I was studying, just like them, these progression documents, and I could only categorize the reasoning that was in front of us. And so that next step to say, "Oh, this is what I would do and bring into action in the classroom," I didn't have an answer for. And so that's really where I was introduced to Integrow—formerly [the] US Math Recovery Council, but now Integrow Numeracy Solutions. And at the heart of our mission to empower educators is to bring research to the classroom in accessible and practical ways that advance student reasoning. We do this in professional learning, we do it in supplemental resources, and we also hire and train educators to deliver high-dosage tutoring for students to accelerate their learning.  Mike: I want to just linger on something you said, which was—and I really appreciate both the truth of the statement you made and also the vulnerability, which is to say—I think for many teachers, there's this experience of, "I can see my students in these progressions, but I'm not sure what to do when it comes to making moves to shift where they're at or help them move." And I think that's a profound truth for so many teachers. And I think it's really important that folks like you, who are doing this work, acknowledge that that's a place you were in once as well because that's so true for so many of us.  Kristin: Yeah. There's always a new thing where we're watching students, we're thinking about the next steps. And so often it boils down to categorizing the things that students are doing now, but not often figuring out: What are the true actions that we take with real children who are in front of us to get them to progress in their own reasoning? We can tell them the next step, but my belief system that is aligned with Integrow Numeracy Solutions is that the most powerful thing is to help students have those experiences and create that understanding themselves. And to do that, it's more complex than just knowing what the next benchmark is for them.  Mike: I think that's a helpful introduction. And I also find it to be a good segue for all the questions that I wanted to explore today. So let me start here: It feels important to acknowledge that supporting students' addition and subtraction fluency actually hinges on understanding how children's mathematical thinking develops. So I wonder if you can talk about some of the concepts and the ideas that play a part in fluency when it comes to combinations of 10, combinations to 20, and even beyond.  Kristin: Yeah. The words that we hear associated with fluency right now are "flexibility," "efficiency," "accuracy." So we've moved on from just speed, which I think is a really positive place for us to be in education. But at the heart of flexibility, efficiency and accuracy is a quantitative understanding of arithmetic. I'm really glad that you had Amy Hackenberg on [the podcast] recently who discussed this concept of units coordination because throughout what we'll talk about, you'll see units coordination come out, but she's definitely the expert to explain it. Just a nod. Just listen to that episode [Season 4, Episode 11]. It was amazing.  Thinking, though, specifically about fluency—fluency isn't just knowing all of these combinations. In the early stages of counting, students view a number simply as a count or result of a count of single items, and there's this critical shift in developing a unit as a fundamental tool of measurement. And that's the act of unitizing where a student conceives of a collection of items as one unit that's simultaneously made of smaller units.  It is a common progression that once a student counts on, that then we would shift to building strategies to solve addition and subtraction within 20, and then of course with 100, and beyond, and then in other domains. But this is all happening in first and second grade for that addition and subtraction to 20 fluency. So attending to this numerical composite—understanding that when a child says "7" and sees that that represents counting from 1 to 7 without having to count—is a really big cognitive shift in their mathematical understanding and can be undermined with, "Oh, now that they're counting on, we're going to tell them these strategies." And so we really do need to have some intentional instructional strategies to make sure that we're developing that first, that numerical composite, before we try to develop all these strategies for addition and subtraction to 20. Because that is the basis for children to move from a counting-based strategy to compose units.  So when they can use a quantity like, "Oh, 8 plus 5, I can break apart this 5 into smaller parts and I can give some of those parts to the 8." So children at that point have to simultaneously hold 5 as a single unit while recognizing the 2 and the 3 make up the 5, but they can be moved to the 8 as well. That's really sophisticated.  Mike: So I want to mark that because I think the notion that this is really sophisticated is important for folks to understand because I'll be vulnerable and honest: I didn't recognize the complexity of what children were grappling with when I started teaching, particularly as a person who was teaching kindergarten and first grade. I really saw my job as helping to build a set of rote procedures like counting and number sequence and memorizing combinations and the outcome of being able to count and the outcome of being able to quickly recall those. I think that's not in question, but understanding the mechanics and the evolution of kids' thinking that's going on, that's a big deal. This whole notion that you have a unit and the unit is composed of smaller units. And one of the things that you said that feels like a really big deal that could be lost is the idea that shifting from a counting-based strategy to a strategy that depends on this notion of units that have smaller units inside and that are also still a unit—that's such a big deal. In order to go from counting everything to counting on to being able to look at a number like 8 and say that it has a 5 and a 3 inside of it—all of that is connected to this notion of units inside of units. And I'm so glad you mentioned that.  Kristin: Yeah. The mental actions that students are doing, making those visible, when we see children do it developmentally, we just assume it's easy. But the shifts that they're making in their understanding of units to move from that pre-numerical stage of "Everything is a 1 and I have to repeat it" to "Now this word can stand in for the count" to "Now I can embed units inside of other units." There's so much happening, and they're so young at that age; we have to remember that too.  Mike: So let's talk about some other important components of developing fluency. What else is an important primer for how people are thinking about this?  Kristin: Yeah. Another important component is supporting students in developing the cognitive structures that allow students to anchor their understanding and quantitative meaning and develop that sophisticated reasoning. Many researchers, many authors have written in different ways and different names about these structures. So like a "mental structure," "mental residue," "mental tools," "patterns of thought." To name a few people, Zaretta Hammond, Betty [K.]

    34 min

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    Season 4 | Episode 15 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 2

    DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 2 ROUNDING UP: SEASON 4 | EPISODE 15 Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners?  Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools. BIOGRAPHY Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels. Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools.  RESOURCES Learning Trajectories website, featuring the work of Doug Clements and Julie Sarama  Math Trajectories for Young Learners book by DeAnn Huinker and Melissa Hedges TRANSCRIPT Mike Wallus: A note to our listeners: This episode contains the second half of my conversation with DeAnn Huinker and Melissa Hedges about math trajectories for young learners. If you've not already listened to the first half of the conversation, I encourage you to go back and give it a listen. The second half of the conversation begins with DeAnn and Melissa discussing practices that educators can use to provide students a more meaningful experience with skip-counting.  Melissa Hedges: One of the things, Mike, that I would add on that actually I just thought about is when you were talking about the importance of us letting the children figure out how they want to approach that task of organizing their count is it's coming from the child. And Clements and Sarama talk about the beautiful work about the trajectory, [which] is that we see that the mathematics comes from the child and we can nurture that along in developmentally appropriate ways.  The other idea that popped into my mind is it's kind of a parallel to when our children get older and we want to teach them a way to add and a way to subtract, and I'm going to show you how to do it and you follow my procedure. I'm going to show it. You follow my procedure. We know that that's not best practice either. And so we're really looking at, how do we grab onto that idea of number sense and move forward with it in a way that's meaningful with children from as young as 1 and 2 all the way up?  Mike: DeAnn, I was going to ask a question to follow up on something that you said just now when you said even something like skip-counting should be done with quantities. And you, I think, anticipated the question I was going to ask, which is: What are the implications of this idea of connecting number and quantity for processes that we have used in the past, like rote counting or skip-counting? And I think what you're saying is we need to attend to those things that, like the counting sequence, we should not create an artificial barrier between speaking the words in sequence and quantity. Am I reading you right or is there more nuance than I'm describing?  DeAnn Huinker: I think you're right on target, Mike. (laughs) Connecting those things to quantity. And I mean, the one that's always salient for me is skip-counting. Skip-counting is such a rote skill for so many children that they don't realize when they go, "5, 10, 15" that they actually have seen, "Oh, there's five [items], there's five more items, there's five more items." So it's making that connection to quantity for something like skip-counting, but also on the counting trajectory, then we start thinking about, "What's a ten? And what makes a ten?" And, "What is 30? And how many tens are composing or embedded in that number 30?" And again, it's not just to rotely say, "3 tens." No. "Show me those objects. Can you make those tens?" Because sometimes we find disconnects. Kids will tell us things and then we say, "Can you show me?" And it doesn't match. (laughs) So we continually start thinking about quantities and putting [objects] with quantities.  Let me add one more thing. In the counting trajectory—and this was very intentional for Melissa—is when we have kids count, we'd like to give them like 31 or 32 counters to see whether [...] they can actually bridge that decade and to go beyond. The other thing that we did, so getting like beyond a ten, also we find when kids get to the number 100, they stop. They just think that's the end. I got to 100, I'm going to stop. And then we say, "Oh, what would be the next number?" And some will say 110, some will say 200, some will give us something else that we find bridging 100 is on the trajectory. And that's actually a really critical point. And again, we want it with quantities with objects.  Mike: I really appreciate this part of the conversation because I think for a teacher who's listening, it helps really get to the specific types of details that would allow them to create the kind of experiences that we think matter for children.  I do want to take a step back though and talk about what's going on for students under the hood, so to speak. So as they're engaging in meaningful counting, what are the cognitive processes that they're learning to coordinate?  Melissa: This is Melissa. So I'll start that one and then invite DeAnn to jump in as I work my way through my thinking.  One of the pieces that, in addition to everything we talked about with all of the skills and ideas and understanding that comes to bear when little ones count, one of the big pieces that we're starting to talk and learn about a whole lot more is this idea of executive functioning. And so executive functioning are those skills that help us manage our attention, help us manage our behavior. They help us stay focused. They help us complete tasks, keep track of things. So hopefully as I'm saying this, what you have in your mind is a little one counting and you're thinking, "Oh my gosh, how do they know where to start?" "How do they know when to stop?" "How do they know when this has been counted with that hasn't been counted?" "What am I going to say next?" All of that tends to be couched very strongly in this idea of executive functions. So when we watch kids count, we know that they're really drawing on those executive functions. And it's actually a really beautiful marriage. So again, we're looking for kids to—are they able to stay on task? Can they keep track? Do they monitor themselves as they go? If someone—this happens a lot—if someone bumps into their collection and their collection gets a little shaky because their desk got moved or someone kicked a counter across the floor, do they remember where that goes and what that stood for in quantity? And for us, that really kind of comes down to some of those higher order skills and in particular, those ideas of the executive functions.  So part of what we notice is that in particular with counting, though all of mathematics, much of what we do and ask kids to do, it takes planning, it takes self-monitoring, and it takes kind of a sense of control and agency over their work. We've talked a little bit about some of that other stuff in the way that it's the work of the child, and that's why we will always ask teachers to step back and just watch, just watch what they do, just watch what they do, because it gives us insight into so many skills, understandings, and kind of where they're at.  DeAnn: Yeah. This is DeAnn. I was thinking of that same thing, Melissa, about this is the work of the child, right? As adults, we're kind of prone sometimes to say, "Let me show you how to do it." But if we want to develop these executive function skills, these ideas and cognitive abilities under the hood, we have to give children opportunities. They need the time to think about how to organize that collection. That's always a great one to kind of think about. As adults, we're like, "Well, just line them up." And it's like, oh no, that's actually huge for a child to realize lining them up or organizing them in some way is a strategy, just like we do with larger numbers. It's a strategy for little kids. So again, that work needs to come from the child and they need to do some trial and error and adjustments in order to develop those things under the hood. And as adults, we can't take that opportunity away from children. We need to create the opportunities so they can explore more of their world and the quantitative world that we live in.  Mike: Everything that we're talking about has some pretty major implications for instructional practice, but what I find myself thinking about is my own time teaching kindergarten. And when I reflect on that, I sometimes found myself falling into something that I would call a readiness trap. And what I mean by that is I had this notion that kids had to have a certain set of skills in place before they were ready to do something like counting a collection. And I think what you're going to tell me is that perhaps I had it backwards. Am I right?  DeAnn: So this is DeAnn and I'm thinking, well, maybe it's not so much backwards, but it's a different perspective. So Melissa and I really struggle with this concept of readiness, and that's because we really frame our work from a developmental perspective. And as we think about learning trajectories, that's what they are. Learning trajectories is a developmental view of children's learning. So what really changes the question for us. We don't ask the question, "Are children ready?" What we ask is, "Oh, where are children currently in their learning?" And then we can start at that spot and then think about the experiences that would help support the next step in thei

    26 min
  5. Season 4 | Episode 14 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 1

    19 MARS

    Season 4 | Episode 14 – Dr. DeAnn Huinker & Dr. Melissa Hedges, Math Trajectories for Young Learners, Part 1

    DeAnn Huinker & Melissa Hedges, Math Trajectories for Young Learners, Part 1 ROUNDING UP: SEASON 4 | EPISODE 14 Research confirms that early mathematics experiences play a more significant role than we once imagined. Studies suggest that specific number competencies in 4-year-olds are strong predictors of fifth grade mathematics success. So what does it look like to provide meaningful mathematical experiences for our youngest learners?  Today, we'll explore this question with DeAnn Huinker from UW-Milwaukee and Melissa Hedges from the Milwaukee Public Schools.  BIOGRAPHY Dr. DeAnn Huinker is a professor of mathematics education in the Department of Teaching and Learning and directs the University of Wisconsin-Milwaukee Center for Mathematics and Science Education Research. Dr. Huinker teaches courses in mathematics education at the early childhood, elementary, and middle school levels. Dr. Melissa Hedges is a curriculum specialist who supports K–5 and K–8 schools for the Milwaukee Public Schools.  RESOURCES Math Trajectories for Young Learners book by DeAnn Huinker and Melissa Hedges Learning Trajectories website, featuring the work of Doug Clements and Julie Sarama  School Readiness and Later Achievement journal article by Greg Duncan and colleagues  Early Math Trajectories: Low‐Income Children's Mathematics Knowledge From Ages 4 to 11 journal article by Bethany Rittle-Johnson and colleagues TRANSCRIPT Mike Wallus: Welcome back to the podcast, DeAnn and Melissa. You have both been guests previously. It is a pleasure to have both of you back with us again to discuss your new book, Math Trajectories for Young Learners. Melissa Hedges: Thank you for having us. We're both very excited to be here. DeAnn Huinker: Yes, I concur. Good to see you and be here again. Mike: So DeAnn, I think what I'd like to do is just start with an important grounding question. What's a trajectory? DeAnn: That's exactly where we need to start, right? So as I think about, "What are learning trajectories?," I always envision them as these road maps of children's mathematical development. And what makes them so compelling is that these learning pathways are highly predictable. We can see where children are in their learning, and then we can be more intentional in our teaching when we know where they are currently at. But if I kind of think about the development of learning trajectories, they really are based on weaving together insights from research and practice to give us this clear picture of the typical development of children's learning. And as we always think about these learning trajectories, there are three main components.  The first component is a mathematical goal. This is the big ideas of math that children are learning. For example, counting, subitizing, decomposing shapes. The second component of a learning trajectory are developmental progressions. This is really the heart of a trajectory. And the progression lays out a sequence of distinct levels of thinking and reasoning that grow in mathematical sophistication. And then the third component are activities and tasks that align to and support children's movement along that particular trajectory.  Now, it's really important that we point out the learning trajectories that we use in our work with teachers and children were developed by Doug Clements and Julie Sarama. So we have taken their trajectories and worked to make them more usable and applicable for teachers in our area. So what Doug and Julie did is they mapped out children's learning starting at birth—when children are just-borns, 1-year-olds, 2-year-olds—and they mapped it out up till about age 8. And right now, last count, they have about 20 learning trajectories. And they're in different topics like number, operations, geometry, and measurement. And we have to put in a plug. They have a wonderful website. It's learningtrajectories.org. We go there often to learn more about the trajectories and to get ideas for activities and tasks.  Now, we're talking about this new book we have on math trajectories for young children. And in the book, we actually take a deep dive into just four of the trajectories. We look at counting, subitizing, composing numbers, and adding and subtracting. So back to your original question: What are they? Learning trajectories are highly predictable roadmaps of children's math learning that we can use to inform and support developmentally appropriate instruction. Mike: That's an incredibly helpful starting point. And I want to ask a follow-up just to get your thinking on the record. I wonder if you have thoughts about how you imagine educators could or should make use of the trajectories. Melissa: This is Melissa. I'll pick up with that question. So I'll piggyback on DeAnn's response and thinking around this highly predictable nature of a trajectory as a way to ground my first comment and that we want to always look at a trajectory as a tool. So it's really meant as an important tool to help us understand where a child is and their thinking right now, and then what those next steps might be to push for some deeper mathematical understanding.  So the first thing that when we work with teachers that we like to keep in mind, and one of the things that actually draw teachers to the trajectories is that they're strength-based. So it's not what a child can't do. It's what a child can do right now based off of experience and opportunity that they've had. We also really caution against using our trajectories as a way to kind of pigeonhole kids or rank kids or label kids because what we know is that as children have more experience and opportunity, they grow and they learn and they advance along that trajectory. So really it's a tool that's incredibly powerful when in the hands of a teacher that understands how they work to be able to think about where are the children right now in their classroom and what can they do to advance them.  And I think the other point that I would emphasize other than what moves children along is experience and opportunity. Children are going to be all over on the trajectory—that's been our experience—and they're in the same classroom. And it's not that some can't and some won't and some can; it's just some need more experience and some need more opportunity. So it's really opened up the door many ways to view a more equitable approach to mathematics instruction.  The other thing that I would say is, and DeAnn and I had big conversations about this when we were first using the trajectories, is: Do we look at the ages? So the trajectories that Clements and Sarama develop do have age markers on them. And we were a bit back and forth on, "Do we use them?," "Do we not?," knowing that mathematical growth is meant to be viewed through a developmental lens. So we had them on and then we had them off and then we shared them with teachers and many of our projects and the teachers were like, "No, no, no, put the ages back on. Trust us. We'll use them well." (laughs) And so the ages are back onto the trajectories. And what we've noticed is that they really do help us understand how to take either intentional steps forward or intentional steps back, depending on what kids are showing us on that trajectory.  The other spot that I would maybe put a plugin for on where we could use a trajectory and what would be an appropriate use for it would be for our special educators out there and to really start to use them to support clear, measurable IEP goals grounded in a developmental progress. So that's kind of what our rule of thumb would be around a "should" and "shouldn't" with the trajectories. Mike: That's really helpful. You mentioned the notion of experiences and opportunities being critical. So I wanted to take perhaps a bit of a detour and talk about what research tells us about the impact of early mathematics experiences, what impact that has on children. I wonder if you could share some of the research that you cite in the book with our listeners. DeAnn: Sure. This is DeAnn, and in the book we cite research throughout all of the chapters and aligned to all of the different trajectories. But as we think about our work, there really are a few studies that we anchor in, always, as we think about children's learning. And the research evidence is really clear that early mathematics matters. The math that children learn in these early years in prekindergarten, kindergarten, first grade—I mean, we're talking 4-, 5-, 6-year-olds, 7-year-olds—that their math learning is really more important than a lot of people think it is. OK? So as we think about these kind of anchor studies that we look at, one of the major studies in this area is from Greg Duncan and his colleagues, and there was a study published in 2007. And what they did is they examined data from thousands of children drawing information from six large-scale studies, and they found that the math knowledge and abilities of 4- and 5-year-olds was the strongest predictor of later achievement. I mean, 4- and 5-year-olds, that's just as they're starting school. Mike: Wow. DeAnn: Yeah. One of the surprising findings was that they found early math knowledge and abilities was a stronger predictor than social emotional skills, stronger than family background, and stronger than family income. That it was the math knowledge that was predictive. Mike: That's incredible. DeAnn: Yes. A couple other surprising things from this study was that early math was a stronger predictor than early reading. Now, we know reading is really important, and we know reading gets a lot of emphasis in the early grades, but math is a stronger predictor than reading. And then one last thing I'll say about this study is that early math not only predicts later math achievement, it also predicts later reading achievement. So that is always a surprise as we share that information with teachers, that early mat

    25 min
  6. Season 4 | Episode 13 – Dr. Mike Steele, Pacing Discourse-Rich Lessons

    5 MARS

    Season 4 | Episode 13 – Dr. Mike Steele, Pacing Discourse-Rich Lessons

    Mike Steele, Pacing Discourse-Rich Lessons ROUNDING UP: SEASON 4 | EPISODE 13 As a classroom teacher, pacing lessons was often my Achilles' heel. If my students were sharing their thinking or working on a task, I sometimes struggled to decide when to move on to the next phase of a lesson.  Today we're talking with Mike Steele from Ball State University about several high-leverage practices that educators can use to plan and pace their lessons.  BIOGRAPHY Mike Steele is a math education researcher focused on teacher knowledge and teacher learning. He is the past president of the Association of Mathematics Teacher Educators, editor in chief of the Mathematics Teacher Educator journal, and member of the NCTM board of directors.  RESOURCES Journal Article "Pacing a Discourse-Rich Lesson: When to Move On" Books 5 Practices for Orchestrating Productive Mathematics Discussions  The 5 Practices in Practice [Elementary]  The 5 Practices in Practice [Middle School]  The 5 Practices in Practice [High School] Coaching the 5 Practices  TRANSCRIPT Mike Wallus: Well, hi, Mike. Welcome to the podcast. I'm excited to talk with you about discourse-rich lessons and what it looks like to pace them. Mike Steele: Well, I'm excited to talk with you too about this, Mike. This has been a real focus and interest, and I'm so excited that this article grabbed your attention. Mike Wallus: I suppose the first question I should ask for the audience is: What do you mean when you're talking about a discourse-rich lesson? What does that term mean about the lesson and perhaps also about the role of the teacher? Mike Steele: Yeah, I think that's a great question to start with. So when we're talking about a discourse-rich lesson, we're talking about one that has some mathematics that's worth talking about in it. So opportunities for thinking, reasoning, problem solving, in-progress thinking that leads to new mathematical understandings. And that kind of implicit in that discourse-rich lesson is student discourse-rich lesson. That we want not just teachers talking about sharing their own thinking about the mathematics, but opportunities for students to share their own thinking, to shape that thinking, to talk with each other, to see each other as intellectual resources in mathematics.  And so to have a lesson like that, you've got to have a number of things in place. You've got to have a mathematical task that's worth talking about. So something that's not just a calculation and we end up at an answer and that the discourse isn't just, "Let me relay to you as a student the steps I took to do this." Because a lot of times when students are just starting to experience discourse-rich lessons, that's kind of mode one that they engage in is, "Let me recite for you the things that I did." But really opportunities to go beyond that and get into the reasoning and the why of the mathematics. And hopefully to explore some approaches or perspectives or representations that they may not have defaulted to in their first run-through or their first experience digging into a mathematical task.  So the task has to have those opportunities and then we have to create learning environments that really foster those opportunities and students as the creators of mathematics and the teacher as the person who's shaping and guiding that discussion in a mathematically productive way. Mike Wallus: One of the things that struck me is there is likely a problem of practice that you're trying to solve in publishing this article, and I wonder if we could pull the curtain back and have you talk a bit about what was the genesis of this article for you? Mike Steele: Absolutely. So let me take us back about 20 or 25 years, and I'll take you back to some early work that went on around these sorts of rich tasks and discourse-rich lessons. So a lot of this legacy comes out of research or a project in the late nineties called the Quasar Project that helped identify: What is a rich task? What is a task, as the researchers described it, of high cognitive demand that has those opportunities for thinking and reasoning? The next question that that line of research brought forward is, "OK, so we know what a task looks like that gives these opportunities. How does this change what teachers do in the classroom? How they plan for lessons, how they make those moment-to-moment decisions as they're engaged in the teaching of that lesson?" Because it's very different than actually when I started teaching middle school in the nineties, where my preparation was: I looked at the content I had for that day, I wrote three example problems I wanted to write on the board that I very carefully got all the steps right and put those up and explained them and answered some questions. "Alright, everybody understand that? OK, great, moving on." And then the students went and reproduced that. That's fine for some procedural things, but if I really wanted them to engage in thinking and reasoning, I had to start changing my whole practice.  So this bubbles up out of the original work of the 5 Practices for Orchestrating Productive Discussions [book] from Peg Smith and Mary Kay Stein. I had the opportunity actually to work with them both in the early two thousands at the University of Pittsburgh. And so as we were working on this five-practices framework that was supposed to help teachers think about, "What does a different conceptualization of planning and teaching look like that really gets us to this discourse-rich classroom environment where students are making sense of and grappling with mathematics and talking to each other in a meaningful way about it?" We worked with teachers around that and the five-practices [framework] is certainly helpful, but then as teachers were working with the five practices and they were anticipating student thinking, they were writing questions that assess and advance student thinking, some of the things that came up were, "OK, what are the moment-to-moment decisions and challenges related to that as we start planning and teaching in this way?"  And a number of common challenges came up. A lot of times when we were using a five-practice lesson, we were doing kind of a launch, explore, share, and discuss sort of format where we've got the teacher who's getting us started on a task, but we're not giving the farm away on that task. We're not saying too much and guiding their thinking. And then we let students have some time individually and in small groups to start messing around with the mathematics, working, talking. And then at some point we're going to call everybody together and we're going to share what the different ways of thinking were. We're going to try to draw that together. Peg Smith likes to talk about this as being more than a show-and-tell. So it's not just, "We stand up, we give our answer, we do that. Great." Next group, doing the same thing, and oftentimes they start to look alike. But there's some really meaningful thinking that goes on in that whole-class discussion. So one of the really pragmatic concerns here is, "How do I know when to move?" So I've got students working individually, and maybe I gave them 3 minutes to get started. Was that enough? What can I see in the work they're doing? What questions am I going to hear to tell me, "OK, now it's a good moment to move to small groups." And then similarly, when you've got small groups working, they're cranking away on a task. There might be multiple subquestions in that task. What's my cue that we're ready to go on to that whole-class discussion?  We were in so many classrooms where teachers were really working hard to do this work, and this happens to me all the time. I have somehow miscalculated what students are going to be able to do—either how quickly they're going to be able to do it, or I expected them to draw on this piece of prior knowledge and it took us a while to get there, or they've flown through something that I didn't expect them to fly through. So I'm having to make some choice in a moment, saying, "This isn't exactly how I imagined it, so what do I do here?" And frequently with teachers that get caught in that dilemma, the first response is to take control back, [to] say, "OK, you're all struggling with this. Let's come back together and let me show you what you should have figured out here." And it's done with the best of intentions. We need to get some closure on the mathematical ideas. But then it takes us right away from what we were trying to do, which was have our students grapple with the mathematics. And so we do this lovely polished job of putting that together and maybe students took the important things away from that, that they wanted to, maybe they didn't, but they didn't get all the way they were on their own. So that's really the problem of practice that this helps us to solve is, when we get in those positions of, "OK, I've got to make a call. I've got this much time left. I've got this sort of work that I see going on in the classroom. Am I ready? What can I do next?" That really keeps that ownership of the mathematics with our students but still gives me some ability to orchestrate, to shape that discussion in a way that's mathematically meaningful and that gets at the goals I had for the lesson. Mike Wallus: Yeah, I appreciated that part of the article and even just hearing you describe that so much, Mike, because you gave words to I think what sat behind the dilemma that I found myself in so often, which was: I was either trying to gauge whether there was enough—and I think the challenge is we're going to get into, what "enough" actually might mean—but given enough time, whether I was confident that there was understanding, how much understanding was necessary. And what that translates into is a lack of clarity around "How do I use my time? How do I gauge when it's worth expending some of the time that I maybe h

    35 min
  7. Season 4 | Episode 12 – Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks

    19 FÉVR.

    Season 4 | Episode 12 – Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks

    Kyndall Thomas, Building a Meaningful Understanding of Properties Through Fact Fluency Tasks ROUNDING UP: SEASON 4 | EPISODE 12 Building fluency with multiplication and division is essential for students in the upper elementary grades. This work also presents opportunities to build students' understanding of the algebraic properties that become increasingly important in secondary mathematics.  In this episode, we're talking with Kyndall Thomas about practical ways educators can support fluency development and build students' understanding of algebraic properties.  BIOGRAPHY Kyndall Thomas serves as a math interventionist and resource teacher with the Oregon Trail School District, focusing on data-driven support and empowering teachers to spark a love of numbers in their students. TRANSCRIPT Mike Wallus: Hi, Kyndall. Welcome to the podcast. I'm really excited to be talking with you today. Kyndall Thomas: Hi, Mike. Thanks for having me. I'm excited to dive into some math talk with you also. Mike: Kyndall, tell us a little bit about your background. What brought you to this work? Kyndall: Yeah. I started in the classroom. I was in upper elementary. I served fifth grade students, and I taught specifically math and science. And then I moved into a more interventionist role where I was a specialist that worked with teachers and also worked with small groups, intervention students. And through that I was able for the first time to really develop an understanding of that mathematical progression that happens at each grade level and the formative things that are introduced at the lower elementary [grades] and then kind of fade out, but still need to be brought back at the upper elementary. Mike: So I've heard other folks talk about the ways students can learn about the algebraic properties as they're building fluency, but I feel like you've taken this a step further. You have some ideas around how we can use visual models to make those properties visible. And I wonder if you could talk a little bit about what you mean by making properties visible and maybe why you think this is an opportunity that's too good to pass up? Kyndall: My thought is bringing visual models back into the classroom with our higher upper elementary students so that they can use those models to build a natural immersion of some of the algebraic properties so that they can emerge rather than just be rules that we are teaching. By supporting students' learning through building models with manipulatives, we're able to build a bridge in a student's mind between their experience with those models and then their mental capacity to visualize those models. This is where the opportunity to bring properties to life is too good to pass up. Mike: OK, so let's get specific. Where would you start? Which of the properties do you see as an opportunity to help students understand as they're building an understanding of fluency? Kyndall: So, when I begin laying the foundation for understanding of the operations and multiplication and division, I intentionally layer in two other major algebraic properties for discovery: the commutative property and the distributive property. We're not setting our students up for success when we simply introduce these properties as abstract rules to memorize. Strong visual models allow students to discover the why behind the rules. They're able to see these properties in action before I even spend any time naming them.  For example, they get to witness or discover how factors can switch order without changing the product, how grouping affects computation, and how numbers can be broken apart and recombined for efficient counting and solving strategies. By teaching basic facts in this structured and intentional way through the behavior of numbers and the authentic discovery of properties, we're not only building fluency, but we're also developing deep conceptual understanding. Students begin to recognize patterns, understand rules, make connections, and rely on reasoning instead of rote memorization. That approach supports long-term mathematical flexibility, which is exactly what we want our students to be able to do. Mike: I want to ask you about two particular tools: the number rack and the 10-frame. Tell me a little bit about what's powerful about the way the [10-frame] is set up that helps students make sense of multiplication. What is it about the way it's designed that you love? Kyndall: The [10-frame] is so powerful because it's set up in our base ten system already. It introduces the tens in a way that is two rows of 5, which is going to lead into properties being identified. So, let me break that up into each individual thing that I love about it.  First, the [10-frame] being broken up into the two rows of 5. That's going to allow students to be able to see that distributive property happening, where we're counting our 5s first and then adding some more into each group. So, when we're seeing a factor like 8 times 2, we're seeing that as two groups of 5 and two groups of 3. Mike: I think what you're making me remember is how it's difficult to help kids visualize that, right? It's a challenge. You can say "'4 times 4' is the same as '4 times 2 plus 4 times 2,'" but that's still an abstraction of what's happening, right? The visual really brings it to life in a way that—even if you're representing that with an equation and doing a true-false equation where it's 4 times 4 is the same as 4 times 2 plus 4 times 2—that's still at a level of abstraction that's not necessarily accessible for children. Kyndall: And as we're talking through this, if I see students and they're working on four groups of 3 and they're seeing those 3s as a double fact plus one more group, I'm on the board writing out the equation, and I'm using the parentheses as that introduction to what this looks like abstractly. They're building it, and they're building those visuals both with their hands and with their minds, and then I'm bringing it to life in the equation on the board. Mike: So, I think what I see in my mind as I hear you describe that is, you have kids with a set of materials. You're doing, for lack of a better word, a translation into a more abstract version of that, and you're helping kids connect the physical materials that they have in front of them to that abstraction and really kind of drawing the connection between the two. Am I getting that right? Kyndall: Yeah. As the students are doing the physical work of math, I'm translating it into its own language up on the board. Absolutely. Mike: I think what's clear to me from this conversation is the way that the tools can illuminate the property, and I think this also helps me think about what my role is as a teacher in terms of building a bridge to an abstraction. Do you actually feel like there's a point where you do introduce the formal language of it? And if you do, how do you decide when? Kyndall: So, the vocabulary kind of comes after the concept has been discovered. But I don't like to introduce the vocabulary first as a rote memorization tool because that has no meaning to it. Mike: I think if I were to summarize this, you're giving them a physical experience with the properties. You're translating that into an abstraction. And then once they've got an experience that they can hang those ideas on top of, then you might decide to introduce the formal language to them at some point. Kyndall: Yeah, absolutely. Mike: So, just as a refresher, for folks who might teach upper elementary and don't have a lot of lived experiences with the number rack—be it the ten or twenty or the hundred—can you describe a little bit about the structure, and maybe what about the structure in particular is important? Kyndall: The structure of a number rack has rows, and each row has 10 beads in it. And typically those beads are divided into two sets of 5: five red beads and five white beads. Then we typically move into a number rack that has two rows so that we're working within 20.  Now, my thought is to take that [to] our third, fourth, and fifth grade, our upper elementary students, and use the hundreds rekenrek [i.e., number rack], where now we have 10 rows and we have 10 beads in each row—still split up into five red [beads] and five white—so that we can use that to teach things. If we're looking at the zero property, students are starting to notice that the rows represent the groups—the rows with the beads on it, that's one group. And so, if we're building zero groups of 3, we don't have a group that we can access to put three beads in. If we're looking at it with the commutative property, students are able to say, "One group of 3. We have one row and we're putting three beads in it."  But what happens when we switch those factors? Now we're utilizing three of our rows, but we're only sliding over one bead. The number rack is also so important when we get to the distributive property because of the way that they have separated those colors. So when we're looking at a factor like 7 times 6—seven groups of 6—then we're gonna be accessing seven rows with six beads in each. That is already set up in the structure of the tool to have five red beads and one white bead showing seven groups of 5 and seven groups of 1 put together. Mike: That is super powerful. One of the things that really jumped out that I want to mark is: If I treat the rows like the groups and then I treat the beads like the number of things in each group, I can model one group with three inside of it, or I can model three groups with one inside of it, and I can really make the difference between those things clear, but also [I can make] the way that the product is still the same clear, right? So, I've got an actual physical model that helps kids understand what was often a rule that was just like 1 times 3 is the same as 3 times 1, because it is. But you're actually saying this is a tool that

    12 min

  8. 5 FÉVR.

    Season 4 | Episode 11 – Dr. Amy Hackenberg, Understanding Units Coordination

    Amy Hackenberg, Understanding Units Coordination ROUNDING UP: SEASON 4 | EPISODE 11 Units coordination describes the ways students understand the organization of units (or a unit structure) when approaching problem-solving situations—and how students' understanding influences their problem-solving strategies. In this episode, we're talking with Amy Hackenberg from the University of Indiana about how educators can recognize and support students at different stages of units coordination. BIOGRAPHY Dr. Amy Hackenberg taught mathematics to middle and high school students for nine years in Los Angeles and Chicago, and is currently a professor of mathematics education at Indiana University-Bloomington. She conducts research on how students construct fractions knowledge and algebraic reasoning. She is the proud coauthor of the Math Recovery series book, Developing Fractions Knowledge. RESOURCES Integrow Numeracy Solutions Developing Fractions Knowledge by Amy J. Hackenberg, Anderson Norton, and Robert J. Wright TRANSCRIPT Mike Wallus: Welcome to the podcast, Amy. I'm excited to be chatting with you today about units coordination. Amy Hackenberg: Well, thank you for having me. I'm very excited to be here, Mike, and to talk with you. Mike: Fantastic. So we've had previous guests come on the podcast and they've talked about the importance of unitizing, but for guests who haven't heard those episodes, I'm wondering if we could start by offering a definition for unitizing, but then follow that up with an explanation of what units coordination is. Amy: Yeah, sure. So unitizing basically means to take a segment of experience as one thing, which we do all the time in order to even just relate to each other and tell stories about our day. I think of my morning as a segment of experience and can tell someone else about it. And we also do it mathematically when we construct number. And it's a very long process, but children began by compounding sensory experiences like sounds and rhythms as well as visual and tactical experiences of objects into experiential units—experiential segments of experience that they can think about, like hearing bells ringing could be an impetus to take a single bong as a unit. And later, people construct units from what they imagine and even later on, abstract units that aren't tied to any particular sensory material. It's again, a long process, but once we start to do that, we construct arithmetical units, which we can think of as discrete 1s. So, it all starts with unitizing segments of experience to create arithmetical items that we might count with whole numbers. Mike: What's really interesting about that is this notion of unitizing grows out of our lived experiences in a way that I think I hadn't thought about—this notion that a unit of experience might be something like a morning or lunchtime. That's a fascinating way to think about even before we get to, say, composing sets of 10 into a unit, that these notions of a unit [exist] in our daily lives. Amy: Yeah, and we make them out of our daily lives. That's how we make units. And what you said about a ten is also important because as we progress onward, we do take more than 1 one as a unit—like thinking of 4 flowers in a row in a garden as a single unit, as both 1 unit and as 4 little flowers—means it has a dual meaning, at least; we call it a composite unit at that point. That's a common term for that. So that's another example of unitizing that is of interest to teachers. Mike: Well, I'm excited to shift and talk about units coordination. How would you describe that? Amy: Yeah, so units coordination is a way for teachers and researchers to understand how children create units and organize units to interpret problem situations and to solve problems. So it originated in understanding how children construct whole number multiplication and division, but it has since expanded from just that to be thinking more broadly about units and structuring units and organizing and creating more units and how people do that in solving problems. Mike: Before we dig into the fine-grain details of students' thinking, I wonder if you can explain the role that units coordination plays in students' journey through elementary mathematics and maybe how that matters in middle school and beyond middle school. Amy: So that's where a lot of the research is right now, especially at the middle school level and starting to move into high school. But units coordination was originally about trying to understand how elementary school children construct whole number multiplication and division, but it's also found to greatly influence elementary school children's understanding of fractions, decimals, measurement and on into middle school students' understanding of those same ideas and topics: fractions ratios and proportional reasoning, rational numbers, writing and transforming algebraic equations, even combinatorial reasoning. So there's a lot of ways in which units coordination influences different aspects of children's thinking and is relevant in lots of different domains in the curriculum. Mike: Part of what's interesting for me is that I don't think I'm alone in saying that this big idea around units coordination sounds really new to me. It's not language that I learned in my preservice work[, nor] in my practice. So I think what's coming together for me is there's a larger set of ideas that flow through elementary school and into middle school and high school mathematics. And it's helpful to hear you talk about that, from the youngest children who are thinking about the notion of units in their daily lives to the way that this notion of units and units coordination continues to play through elementary school into middle school and high school. Amy: Yeah, it's nice that you're noticing that because I do think that's something that's a strength of units coordination in [that] it can be this unifying idea, although there's lots of variation and lots of variation in what you see with elementary students versus middle school students versus high school students versus even college students. Some of the research is on college students' unit coordination these days, but it is an interesting thread that can be helpful to think about in that way. Mike: OK. With that in mind, let's introduce a context for units coordination and talk a little bit about the stages of student thinking. Amy: Yeah. So, one way to understand some differences in how children up through, say, middle school students might coordinate units and engage in units coordination is to think about a problem and describe how solving it might happen.  Here's a garden problem: "Amaya is planting 4 pansies in a row. She plants 15 rows. How many pansies has she planted?" There are three stages of units coordination, broadly speaking—we've begun to understand more about the nuances there. But a stage refers to a set of ways of thinking that tend to fit together in how students understand and solve problems with whole numbers, fractions, quantities, and multiplicative relationships. It's sort of about a nexus of ideas, and—that we tend to see coming together and students don't usually think in a way that's characteristic of a different stage until they've made a significant change in their thinking, like a big reorganization happens for them to move from one stage to the next. So students at stage 1 of units coordination are primarily in a 1s world and their number sequence is not multiplicative. That's going to be hard to imagine. But they can take a group of 1s as one thing. So, they can make a composite unit and that means in the garden problem, they can take a row of pansies as 1 row as well as 4 little ones, and they can continue to do that over and over again. And so they can amass rows of 4 pansies and keep going. And what it usually looks like for them to solve the problem is they'll count by 1s after any known skip-counting patterns. So, in this case they might be like, "Oh, I know 4 and 8; that's two rows. 9, 10, 11, 12; that's three rows." Often using fingers or something to keep track, or in some way to keep track, and continuing to go up and get all the way, barring counting errors, to 60 pansies. And so for them the result, 60 pansies, is a composite unit. It's a unit of 60 units, but they don't maintain the structure that we see at all of the units of 60 as 15 fours. That's not something—even though they did track it in their thinking—they don't maintain that once they get to the 60, it's really just only a big composite unit of 60. So their view of the result is very different than an adult view might be.  So, the students at stage 1 can solve division problems, which means if they give some number of pansies and they're supposed to make rows of 4, they can definitely do it, they can solve that. But they don't think of multiplication and division as inverses. So let me say what I mean by that. If they had this problem next, so: "Amaya's mom gave her 28 pansies. How many rows of 4 can she make?" A student at stage 1 could solve that problem, and they would be able to track 4s over and over again and figure out that they got to 7 fours once they get to 28. But then if immediately afterwards a teacher said, "Well, so, how many pansies are there in 7 rows of 4?," the student at stage 1 would start over and solve the problem from the beginning. They wouldn't think that they had already solved it. And that's one telling sign of a student operating at stage 1. And the reason is that the mental actions they engage in to do the segmenting or the tracking off of the 4s and the 28 pansies are really different to them than what they use then the ways of thinking they use to create the 7 rows of 4 and make the 28 that way. And so they don't recognize them as similar, so they feel like they have to engage in new problem solving to solve that problem. So, to get back to the g

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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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