Rounding Up
The Math Learning Center
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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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Sep 21, 2023
Empathy Interviews - Guest: Kara Imm, PhD.
Mike (00:01):
If there were a list of social skills we hope to foster in children, empathy is likely close to the top. Empathy matters, it helps us understand how others are feeling so we can respond appropriately and it can help teachers understand the way their students are experiencing school. Today on a podcast, we talk with Dr. Karara Imm about a practice referred to as an empathy interview. We'll discuss the ways empathy interviews can help educators understand their students' lived experience with mathematics and make productive adaptations to instructional practice. Well, welcome to the podcast, Karara. We're excited to have you join us.
Kara (00:37):
Thanks, Mike. Happy to be here.
Mike (00:39):
So I have to confess that the language of an empathy interview was new to me when I started reading about this, and I'm wondering if you could just take a moment and unpack what is an empathy interview for folks who are new to the idea?
Kara (00:50):
Yeah, sure. I think I came to understand empathy interviews in my work with design thinking as a former teacher, classroom teacher, and now teacher educator. I've always thought of myself as a designer. So when I came to understand that there was this whole field around design thinking, I got very intrigued. And the central feature of design thinking is that designers who are essentially thinking about creating new products, services, interactions, ways of being for someone else, have to start with empathy because we have to get out of our own minds and our own experiences and make sure we're not making assumptions about somebody else's lived experience. So an empathy interview as I know it now is first and foremost a conversation. It's meant to be as natural a conversation as possible. When I do empathy interviews, I have a set of questions in mind, but I often abandon those questions and follow the child in front of me or the teacher depending on who I'm interviewing.
(01:47):
And the goal of an empathy interview is to elicit stories, really granular, important stories, the kind of stories that we tell ourselves that get reiterated and retold, and the kinds of stories that cumulatively make up our identities. So I'm not trying to get a resume, I'm not interested in the facts of the person, the biography of the person. I'm interested in the stories people tell about themselves, and in my context, the stories that kids tell themselves about their own learning and their own relationship to school, their classrooms, and to mathematics. I'm also trying to elicit emotions. So designers are particularly listening for what they might call unmet needs, where as a designer we would then use the empathy interview to think about the unmet needs of this particular person and think about designing something uniquely and specifically for them with the idea that if I designed something for them, it would probably have utility and purpose for other people who are experiencing that thing. So what happened more recently is that I started to think could empathy interviews change teachers' relationship to their students? Could it change leaders' relationships to the teachers? And so far we're learning that it's a different kind of conversation and it's helping people move out of deficit thinking around children and really asking important questions about what does it mean to be a kid in a math class?
Mike (03:14):
There's some language that you've used that really stands out for me. And I'm wondering if you could talk a little bit more about it. You said the stories that we tell about ourselves, or maybe paraphrase the stories that kids tell themselves. And then you had this other bit of language that I'd like to come back to the cumulative impact of those stories on our identity. Can you unpack those terms of phrase you used and talk a little bit about them specifically as you said, when it comes to children and how they think about their identity with relation to mathematics?
Kara (03:44):
Sure. I love that kind of phrase, the story we tell ourselves. That's been a pivotal phrase for me. I think stories kind of define and refine our existence. Stories capture this relationship between who we are and who we want to become. But when I'm thinking about stories in this way, I imagine as an interviewer that I'm trying to paint a portrait of a child typically. And so I'm trying to interact with this child in such a way that I can elicit these stories, painting a unique picture of this kid, not only as a learner, but also as a human. What inevitably happens when you do these interviews is that I'm interested in their experience in math class. When I listen to kids, they have internalized, I'm good at math and here's why. Or I'm bad at math and here's why. I just know it. But when you dig a little bit deeper, the stories they tell are a little more nuanced and they kind of live in the space of gray. And I'm interested in that space, not the space of testing and measurement that would land you in a particular identity as meant for math or not meant for math.
Mike (04:46):
I think what I was going to suggest is why don't we listen to a few, because you shared a couple clips before we got ready for the interview, and I was fascinated by the approach that you had in chatting with these children and just how much information I could glean from even a minute or two of the interview slices that you shared. Why don't we start and get to know a few of these kiddos and see what we can learn together.
Kara (05:11):
Sounds great.
Mike (05:12):
We've got a clip that I'm going to invite you to set it up and give us as much context as you want to, and then we'll play the clip and then we can talk a little bit about it. I would love to start with our friend Leanna. Great.
Kara (05:24):
Leanna is a third grader. She goes to an all girls school. I've worked in Leanna School over multiple years. I know her teacher well. I'm a part of that community. Leanna was kind of a new mathematician to me earlier in the day. I had been in Leanna's classroom and the interview starts with a moment that really struck me, which I won't say much more about. And I invited Leanna to join me after school so we could talk about this particular moment. And I really wanted to know how she made sense of what happened. So I think we'll leave it at that and we'll listen to what happened.
Mike (05:58):
Alright, let's give it a listen.
Leanna (05:59):
Hi, I'm Leanna and I'm eight years old. Hi,
Kara (06:02):
Leanna. Today when I was in your class, something interesting happened where I think the kids said to me and they said, do you know we have a math genius in our class? Do you remember that moment? Yeah. Tell me what happened in that moment. Mind.
Leanna (06:16):
They said, we have a math genius in our class. And then they all started pointing at me.
Kara (06:24):
And what was that like for you?
Leanna (06:30):
Maybe it was nice, but also it was kind of like all the pressure was on me.
Kara (06:34):
Yeah, I was wondering about that. Why do you think the girls today, I mean, I'm a visitor, right? Why do you think they use the word math genius And why did they choose you? What do you think they think of you?
Leanna (06:47):
A mathematician, because I go to this thing every Wednesday. They ask me what I want to be when I grow up, and I always say a mathematician. So they think that I am a math genius.
Kara (07:02):
Gotcha. Do you think all the girls in your class know that you want to be a mathematician when you grow up? But do they mean something else? They didn't say, we have a mathematician in our class. They said, we have a math genius.
Leanna (07:14):
Maybe
Kara (07:15):
Are you a math genius? Do think, what does that even mean?
Leanna (07:19):
I'm really good at math.
Kara (07:20):
Yeah. Do you think that's a true statement?
Leanna (07:25):
Yeah, a little bit. A
Kara (07:26):
Little bit. Do you love math? Yeah. Yeah.…
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Sep 7, 2023
Practical Ways to Build Strength-Based Math Classroom - Guest: Beth Kobett, EdD
Rounding Up
Season 2 | Episode 1 – Practical Ways to Build Strengths-based Math Classrooms
Guest: Beth Kobett
Mike Wallus: What if it were possible to capture all of the words teachers said or thought about students and put them in word clouds that hovered over each student throughout the day? What impact might the words in the clouds have on students’ learning experience? This is the question that Beth Kobett and Karen Karp pose to start their book about strengths-based teaching and learning. Today on the podcast, we're talking about practices that support strengths-based teaching and learning and ways educators can implement them in their classrooms.
Mike: Hey, Beth, welcome to the podcast.
Beth Kobett: Thank you so much. I'm so excited to be here, Mike.
Mike: So, there's a paragraph at the start of the book that you wrote with Karen Karp. You said: ‘As teachers of mathematics, we've been taught that our role is to diagnose, eradicate, and erase students' misconceptions. We've been taught to focus on the challenges in students' work rather than recognizing the knowledge and expertise that exist within the learner.’ This really stopped me in my tracks, and it had me thinking about how I viewed my role as a classroom teacher and how I saw my students’ work. I think I just want to start with the question, ‘Why start there, Beth?’
Beth: Well, I think it has a lot to do with our identity as teachers, that we are fixers and changers and that students come to us, and we have to do something. And we have to change them and make sure that they learn a body of knowledge, which is absolutely important. But within that, if we dig a little bit deeper, is this notion of fixing this idea that, ‘Oh my goodness, they don't know this.’ And we have to really attend to the ways in which we talk about it, right? For example, ‘My students aren't ready. My students don't know this.’ And what we began noticing was all this deficit language for what was really very normal. When you show up in second grade, guess what? There's lots of things you know, and lots of things you're going to learn. And that's absolutely the job of a teacher and a student to navigate. So, that really helped us think about the ways in which we were entering into conversations with all kinds of people; teachers, families, leadership, and so on, so that we could attend to that. And it would help us think about our teaching in different ways.
Mike: So, let's help listeners build a counter-narrative. How would you describe what it means to take a strengths-based approach to teaching and learning? And what might that mean in someone's daily practice?
Beth: So, we can look at it globally or instructionally. Like, I'm getting ready to teach this particular lesson in this class. And the counter-narrative is, ‘What do they know? What have they been showing me?’ So, for example, I'm getting ready to teach place value to second-graders, and I want to think about all the things that they've already done that I know that they've done. They've been grouping and counting and probably making lots of collections of 10 and so on. And so, I want to think about drawing on their experiences, A. Or B, going in and providing an experience that will reactivate all those prior experiences that they've had and enable students to say, ‘Oh yeah, I've done this before. I've made sets or groups of 10 before.’ So, let's talk about what that is, what the names of it, why it's so important, and let's identify tasks that will just really engage them in ways that help them understand that they do bring a lot of knowledge into it. And sometimes we say things so well intentioned, like, ‘This is going to be hard, and you probably haven't thought about this yet.’ And so, we sort of set everybody on edge in ways that set it's going to be hard, which means, ‘That's bad.’ It's going to be hard, which means, ‘You don't know this yet.’ Well, why don't we turn that on its edge and say, ‘You've done lots of things that are going to help you understand this and make sense of this. And that's what our job is right now, is to make sense of what we're doing.’
Mike: There's a lot there. One of the things that I think is jumping out for me is this idea is multifaceted. And part of what we're asking ourselves is, ‘What do kids know?’ But the other piece that I want to just kind of shine a flashlight on, is there's also this idea of what experiences have they had—either in their home life or in their learning life at school—that can connect to this content or these ideas that you're trying to pull out? That, to me, actually feels like another way to think about this. Like, ‘Oh my gosh, we've done partitioning, we've done grouping,’ and all of those experiences. If we can connect back to them, it can actually build up a kid's sense of, like, ‘Oh, OK.’
Beth: I love that. And I love the way that you just described that. It's almost like positioning the student to make those connections, to be ready to do that, to be thinking about that and providing a task or a lesson that allows them to say, ‘Oh!’ You know, fractions are a perfect example. I mean, we all love to use food, but do we talk about sharing? Do we talk about when we've divided something up? Have we talked about, ‘Hey, you both have to use the same piece of paper, and I need to make sure that you each have an equal space.’ I've seen that many times in a classroom. Just tweak that a little bit. Talk about when you did that, you actually were thinking about equal parts. So, helping students … we don't need to make all those connections all the time because they're there for students and children naturally make connections. That's their job ( chuckles ). It really is their job, and they want to do that.
Mike: So, the other bit that I want to pick up on is the subtle way that language plays into this. And one example that really stood out for me was when you examined the word ‘misconception.’ So, talk about this particular bit of language and how you might tweak it or reframe it when it comes to student learning.
Beth: Well, thank you for bringing this up. This is a conversation that I am having consistently right now. Because this idea of misconception positions the student. ‘You're wrong, you don't understand something.’ And again, let's go back to that again, ‘I've got to fix it.’ But what if learning is pretty natural and normal to, for example, think about Piaget’s conservation ideas, the idea that a young child can or can't conserve based on how the arrangement. So, you put in a, you know, five counters out, they count them and then you move them, spread them out and say, ‘Are they the same, more or less?’ We wouldn't say that that's a misconception of a child because it's developmental. It's where they are in their trajectory of learning. And so, we are using the word misconception for lots of things that are just natural, the natural part of learning. And we're assuming that the student has created a misunderstanding along the way when that misunderstanding or that that idea of that learning is very, very normal.
Beth: Place value is a perfect example of it. Fractions are, too. Let's say they're trying to order fractions on a number line, and they're just looking at the largest value wherever it falls, numerator, denominator, I'm just throwing it down. You know, those are big numbers. So, those are going to go at the end of a number line. But what if we said, ‘Just get some fraction pieces out’? That's not a misconception 'cause that's normal. I'm using what I've already learned about value of number, and I'm throwing it down on a number line ( chuckles ). Um, so it changes the way we think about how we're going to design our instruction when we think about what's the natural way that students do that. So, we also call it fragile understanding. So, fragile understanding is when it'…
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Jul 6, 2023
Building Fluency and Procedural Understanding with Work Places - Guest: Lori Bluemel
Rounding Up
Season 1 | Episode 20 – Work Places
Guest: Lori Bluemel
Mike Wallus: When I meet someone new at a gathering and tell them that I work in math education, one of the most common responses I hear is, “I was never good at math in school.” When I probe a bit further, this belief often originated in the person's experience memorizing basic facts. How can we build students' fluency with facts, encourage flexible thinking, and foster students' confidence? That's the topic we'll explore in this episode of Rounding Up.
Mike: One of the challenges that we face in education can be letting go of a practice—even if the results are questionable—when the alternative is unclear. In elementary math, this challenge often arises around building computational fluency. We know that speed tests, drill and kill, and worksheets, those are all ineffective practices. And even worse, they can impact students' math identity. So, today we're going to spend some time unpacking an alternative, a component of the Bridges in Mathematics curriculum called Work Places. We're doing this not to promote the curriculum, but to articulate an alternative vision for ways that students can develop computational fluency. To do that, we're joined by Lori Bluemel, a curriculum consultant for The Math Learning Center.
Mike: Lori, welcome to the podcast. It's great to have you with us.
Lori: Thank you. It's good to be here.
Mike: Well, let's just start with a basic question: If I'm a listener who's new to the Bridge's curriculum, can you describe what a Work Place is?
Lori: The simple answer would be that it's math activities or games that are directly focusing on the skills or the ideas and concepts that students are working on during Problems & Investigations. The best aspect, or the feature about Work Places, is that teachers have an opportunity to be like a fly on the wall as they're listening into their students and learning about what strategies they're using and the thinking process that they're going through.
Mike: How do you think practicing using a Work Place differs from the version of practice that children have done in the past? What changes for the child or for the learner?
Lori: Well, I always felt like a piece of paper was pretty static. There wasn't a lot of interaction. You could run through it so quickly and be finished with it without really doing a lot of thinking and processing—and with absolutely no talking. Whereas during Work Places, you're discussing what you're doing. You're talking to your partner. You're listening to your partner. You're hearing about what they're doing and the different methods or strategies that they're using. And [there’s] nothing at all static about it because you're actively working together to work through this game or this activity.
Mike: That is so fascinating. It makes me think of a book that I was reading recently about thinking classrooms, and one of the things that they noted was, there's data that suggests that the more talk that's happening in a classroom, the more learning that's actually happening. It really connects me to what you just said about Work Places.
Lori: Yeah, and I feel like that's the big difference between Work Places and doing a worksheet on your own. You can do it completely isolated without any outside interaction, whereas Work Places, it's very interactive, very collaborative.
Mike: Yeah. So, as a former classroom teacher who used Work Places on a daily basis, how did you set up norms and routines to make them successful for students?
Lori: Well, I actually went through several different methods, or routines, before I landed on one that really worked well for me. One that worked best for me is, at the beginning of the year when we first started doing Work Places, I would take that very first Work Place time, and we would just have a class meeting and talk about what we're doing in Work Places. Why would we even have Work Places? We would create an anchor chart, and we'd have one side that would say “Students.” The other side would say “Teachers.” And then we would talk about the expectations. And the students would come up with those. Then we would talk about me as the teacher, what do they think I should be doing? And again, that would come up with all different ideas. And then we always came back to that final thought of,
“We need to be having fun.”
Mike: Hmm.
Lori: Math needs to be fun during Work Places. And then we would start in, and students would go to Work Places. They would choose their partner, and then they would get started. And that first few times we did Work Places, I always just kind of watched and listened and walked around. And if I felt like things needed to be slightly different, maybe they weren't talking about math or they weren't really playing the Work Place, then we would call a class meeting. And everyone would freeze, and we'd go to our meeting spot, and we would talk about what I saw. And we would also talk about what was going well and what they personally could do to improve. And then we'd go back to Work Places and try it again. Needless to say, a lot of times those first few times at Work Places they didn't play the games a lot because we were setting up expectations. But in the long run, it made Work Places run very smoothly throughout the rest of the year.
Mike: Yeah. The word that comes to mind as I listen to you talk, Lori, is investment.
Lori: Um-hm.
Mike: Investing the time to help set the norms, set the routines, give kids a vision of what things look like, and the payoff is productive math talk.
Lori: Exactly. And that was definitely the payoff. They needed reminders on occasion, but for the most part, they really understood what was expected.
Mike: I think it's fascinating that you talked about your role and asked the kids to talk about that. I would love if you could say more about why you asked them to think about your role when it came to Work Places.
Lori: I wanted them to realize that I was there to help them. But at the same time, I was there to help their peers as well. So, if I was working with a small group, I wanted them to understand that they might need to go to another resource to help them answer a question. They needed to make sure that I was giving my attention to the, the small group or the individual that I was working with at that time. So, by talking about what was expected from me, my hope was that they would understand that there were times when they might have to wait a minute, or they might go to another resource to find an answer to their question, or to help them with the situation that they were in. And that seemed to be the case. I think I alleviated a lot of those interruptions just by talking about expectations.
Mike: So, I want to return to something that you said earlier, Lori, ’cause I think it's really important. I can imagine that there might be some folks who are listening who are wondering, “What exactly is the teacher doing while students are engaged in Work Places?”
Lori: Um-hm.
Mike: And I wanted to give you an opportunity to really help us understand how you thought about what your main focus was during that time. So, children are out, they're engaged with the Work Places. How do you think about what you want to do with that time?
Lori: OK. So, I often look at the needs of my students and, and think about “What have I seen during Problems & Investigations? What have I seen during Work Places previously? And where do I focus my time?” And then I kind of gravitate towards those students that I want to listen in on. So, I want to again, be like that fly on the wall and just listen to them, maybe ask a few questions, some clarifying questions about what they're doing, get an idea of what strategies or the thinking that they're going through as they're processing the problem. And then from there, I can start focusing on small groups, maybe adjust the Wo…
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Jun 22, 2023
Building a Broader Definition of Participation - Guest: Dr. Juanita Silva
Rounding Up
Season 1 | Episode 19 – Building a Broader Definition of Participation
Guest: Juanita Silva
Mike Wallus: Participation is an important part of learning to make sense of mathematics. But stop and ask yourself, “What counts as participation?” In this episode, we'll talk with Dr. Juanita Silva from Texas State University about an expanded definition of participation and what it might mean for how we engage with and value our students' thinking.
Mike: Welcome, Juanita. Thanks for joining us on the podcast.
Juanita Silva: Hi. Thank you for inviting me. I'm excited to talk about this topic.
Mike: I think I'd like to start by asking you to just talk about the meaning of participation. What is it and what forms can participation take in an elementary math classroom?
Juanita: Well, there's a mixture of nonverbal and verbal communication. And you can add in there gestures [as a] form of communication, not just in an interconnected space, but also thinking about students’ respect. And it's not just bidirectional, but there’s a lot of things that are kind of added in that space.
Mike: So, it strikes me that when I was a classroom teacher, when I look back, I probably overemphasized verbal communication when I was assessing my students' understanding of math concepts. And I have a feeling that I'm not alone in that. And I'm wondering if you could talk about the way that we've traditionally thought about participation and how that might have impacted student learning?
Juanita: Yes, this is a great question. In thinking about, “What does this look like, how to participate in the classroom?” Mostly teachers think about this as whole group discussions or in small group discussions. And I emphasize the word “their” discussions, where students can share verbally how they thought about the problem. So, for example, if a student is solving a fraction word problem, the teacher may ask, “OK, so how did you solve this problem? Can you share your strategy with the class? What does that look like?” And so, the student sometimes will say, “If I'm solving a fraction word problem about four parts or four chocolate bars, then I can cut those leftovers into four parts.” So that's usually what we think of, as in our teaching and practice in elementary schooling. We think of that as verbal communication and verbal participation, but there are others. ( laughs )
Mike: Let's talk about that. I think part of what you have pushed me to think about is that a student's verbal communication of their thinking, it really only offers a partial window into their actual thinking. What I'd like to do is just talk about what it might look like to consciously value participation that's nonverbal in an elementary classroom. Like, what are the norms and the routines that a teacher could use to value nonverbal communication, maybe in a one-to-one conversation in a small group or even in a whole group discussion?
Juanita: Yes. So, I can share a little bit for each one of those. For example, in a one-to-one environment, the teacher and student can more effectively actually communicate ideas if the teacher attends to that child's thinking in nonverbal ways as well. So, for instance, I've had a student before in the past where he would love to explain his thinking using unifix cubes and to share his thinking on a multiplication problem that was about three sets of cookies. And those sets were in groups of seven. So, there were seven cookies in each bag. And I asked him, “Well, how would you share? Could you explain your thinking to me?” And so, he showed me three sets of seven unifix cubes, and he pointed to each of the seven linking cubes and then wrote on his paper, the number sentence, “seven plus seven plus seven is 21.” And when I asked him if the seven represented the cookies, he simply nodded yes and pointed to his paper, saying and writing the words “21 total.”
Juanita: So, I didn't ask him to further explain anything else to me verbally because I had completely understood how he thought of the problem. And in this example, I'm showing that a student's gestures and a student's explanation on a piece of paper should be valued enough. And we don't necessarily need to engage in a verbal communication of mathematical ideas because this honors his ways of thinking. But at the same time, I could clearly understand how this child thought of the problem. So, I think that's one way to think about how we can privilege a nonverbal communication in a one-to-one setting.
Mike: That's really helpful. I think that part of the example that you shared that jumps out for me is attending to the ways that a child might be using manipulative tools as well, right?
Juanita: Correct.
Mike: So, it was kind of this interaction of the student's written work, their manipulative tools, the way that they gestured to indicate their thinking … that gave you a picture of how this child was thinking. And you didn't really need to go further than that. You had an understanding as an educator that would help you think about what you might do next with that child.
Juanita: Absolutely. And that is one of the tools that I find to be super useful, is to not just have students explain their thinking, but also just listen to their nonverbal cues. And so, paying attention to those and also valuing those is extremely important in our practice. I can share one of my favorites, which is a small group example. And this one is kind of foundational to think of the practice when we're teaching in our elementary math classrooms. It's not just that interactions between student and teacher, but the interactions between students and students can be very powerful. So, that's why this is one of my favorite examples. I had two students at one point in my practice. And this was Marco and José, and they were in fourth grade. They were having a hard time communicating verbally with one another, and José was trying to convince Marco of his strategy to split the leftovers of an equal-sharing problem into three parts instead of halves.
Juanita: But his verbal communication of these ideas were not clear to Marco. And José explains to Marco, “You have to cut it into halves.” And Marco would say, “Yes, that is what I did.” Like, frustrated, as if, like, “You have to cut this into halves.” And José would say, and Marco was like, “Yes, that's exactly what I did.” So, this exchange of verbal communication was not really helping both of them showcase how they were trying to communicate. So, then José started to insist, and he said, “No, look.” And then he showed Marco his strategy on his paper. And in his paper, he had split the bar into three parts. And then Marco looked at José and said, “Ah, OK.” Had José not shown this strategy on his paper, then Marco would have never really understood what he meant by “You have to cut it into halves.” And so, I share this example because it really showcases that sometimes what we're trying to say and communicate might come across differently verbally, but we mean something else when we showcase it nonverbally. So, in this instance, José was trying to explain that, but he couldn't figure out how to tell that to Marco. And so, in this instance, I feel like it really showcases the power of the nonverbal communication among students.
Mike: I think what's fascinating about that is, conceptually the strategy was right there. It was kind of like, “I'm going to equally partition into three parts.” The issue at hand was the language choice. I'm essentially referring to this equal partition as a half, this second equal partition as a half, and this third equal partition as a half. That's a question of helping figure out what is the language that we might use to describe those partitions. But if we step back and say, “Mathematically, does the child actually understand the idea of equal partitioning?” Yes. And then it seems…
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Jun 8, 2023
Why Progressions Matter - Guest: Graham Fletcher
Rounding Up
Season 1 | Episode 18 – Why Progressions Matter
Guest: Graham Fletcher
Mike Wallus: Many educators were first introduced to the content that they teach as a series of items on a checklist. What impact might that way of thinking have on a teacher's approach to instruction? And what if there were another way to understand the mathematics that our students are learning? In this podcast, we talk with Graham Fletcher about seeing mathematics as a progression and how this shift could have a profound impact on teaching and learning.
Mike: Welcome to the podcast, Graham. We're glad to have you with us.
Graham Fletcher: Yeah, really excited to just kind of play around, uh, in this space with you here talking about math and supporting teachers so that they can, in turn, support kids.
Mike: You bet. So, just as a starting point, we're talking about progressions, and we're talking about some of the work that you've done, building progression videos. I have, maybe, what is kind of a weird opening question: How would you define the term “progression” so that we're all starting with the same understanding?
Graham: So, when I think about progression, I think a lot of the times as teachers we can become, like, hyper focused on one grade level. And within that one grade level there can be a progression of where things are learned in a sequential order. It's probably not as linear as we'd like it to be, but I think that little micro progression, or sequence, of learning that we see in one grade level, we start thinking about what that might look like over a grade band, over like K–2 or even K–5. So, there’s things that happen within certain grade levels, and that's kind of where progressions happen. How do we move kids through this understanding of learning? And it's that progression of understanding that we tend to want to move kids through, where everything's kind of connected. And that's really where I see progressions.
Mike: So, I think you're kind of leading into my second question, which is—I love the work that you've put together on your website. I'm unabashedly going to say that this is a great place for teachers to go. But part of what strikes me is that there are a lot of things that you could have done to support elementary math educators and yet you chose to invest time to build this series of videos that unpack the ideas that underlie processes, like counting or addition and subtraction or fractions. Like, why that? Why was that a thing where you're like, “I should invest some time in putting this together.”
Graham: So, I guess we're all teachers at heart, and so I start thinking about how I'm in a place of privilege where I've had an opportunity to work with some really amazing educators that I've stood on their shoulders over the years. And I think about all the times that I've been able to huddle up in a classroom at the end of the day and just listen to those people who are brilliant and really understand those progressions and the smaller nuances of what it is to just understand student thinking and how to keep moving it forward. So, I started thinking about, “Well, what does this look like in one grade level?” But then, when I was starting to think about that whole idea, the big piece for me is: Not every teacher has a person that they can sit next to. And so, if I've had the opportunity to sit down and make sense of these things where, like, on a Friday night (laughs) maybe I'm sitting down with some math books, which most people don't choose to do, I enjoy doing that.
Graham: And so, if I've had the opportunity to do that, and I'm able to make these connections, I start thinking about those other teachers who, teachers that teach 75 subjects 54 days a week, right? And we want them to focus solely on math. So, maybe just sharing some of that knowledge to kind of lessen the burden of understanding that content. So, giving them like a 60,000-foot view of what those progressions could look like. And then them saying, “OK, well, wait a minute. Maybe I can do a deeper dive,” where we're giving them those [aha moments] that they might want or need to kind of do that deeper dive. And the big piece for it was, there's always talk about progressions. There's always talk about, “This is the content that you need to know,” content after content after content. But very seldom is it ever in a coherent, consumable manner. So, when I start thinking about teachers, we don't have that time to sit down and give hours and hours and hours to the work. So really, just what is a consumable amount of time to where teachers won't be overwhelmed? And I think that's why I tried to keep them at about 5 to 6 minutes; to where you can go kind of light that fire to go and continue building your own capacity. So, that's kind of where it was. My North Star: just building capacity and supporting teachers in their own growth. For sure.
Mike: You know, it's interesting, ’cause when I was a classroom teacher, the lion’s share of my time was kindergarten and first grade, with a little bit of time in second grade. So, I was thinking about that when I was watching these because I watched some of the ones for younger kids and I was like, “This makes a ton of sense to me.” But I really kind of perked up when I started watching the ones for kids in the intermediate grades. And I think for me it was kind of like, “Ah, these ideas that I was working on in K and 1, so often, I wasn't quite sure what seeds was I planting or how would those seeds grow in the long term—not just next year, but in the long term. I wonder if that's part of what you think about comes out of a teacher's experience with these.
Graham: Yeah, I definitely think so. I think finding that scalability in reasoning and relationships is key for students, and it's key for teachers as well. So, for instance, when we start thinking about, in kindergarten, where kids are sitting and they're practicing counting and they're counting by singular units; singular units of 1, where it's 1, 2, 3. Well, then when we start making that connection into third grade, where kids are counting by fractions instead of going ahead and saying, like, “One-fourth, two-fourth, three-fourths,” really focusing on that iteration of the unit, that rote counting where it's one one-fourth, two one-fourths, three one-fourths. And then, even that singular unit that we're talking about in kindergarten, which now is in fractions in third grade, well that begins to connect in sixth grade when we start talking about unit rate, when we start getting into ratios and proportions. So, that scalability of counting is massive. So, that's just one little example of taking something and seeing how it progresses throughout the grade level. And making those connections explicit becomes really powerful because I know, just in my own experiences, in talking with teachers as well, is when they start making those connections. Bingo, right? So, now when you're looking at students, it's like, “OK, they're able to count by unit fractions. Well, what now happens if we start grouping fractions together and units and we start counting by two-thirds?” So, now you start moving from counting strategies to additive strategies and then additive strategies to multiplicative, and seeing how it all kind of grows together. That scalability is what I'm really after a lot of the time, which falls in line with that idea of teaching through progressions.
Mike: Yeah, I think one of the things that's really hitting me about this, too, is that understanding children's mathematical thinking as a progression is really a different experience than thinking about math as a set of procedures or skills that kids need to leave second grade with. It feels really different. I wonder if you could talk about that.
Graham: Yeah, absolutely. So, working with Tracy Zager—good friend of mine—we've done a lot of work around fact fluenc…
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May 18, 2023
Asset-Based Assessment - Guest: Tisha Jones
Rounding Up
Season 1 | Episode 17 – Asset-Based Approach to Assessment
Guest: Tisha Jones
Mike Wallus: When you look at the results of your students work, what types of things are you attending to? Many of us were trained to look for the ways that students were not understanding concepts or ideas. But what if we flipped that practice on its head and focused on the things students did understand? Today on the podcast, we're talking with Tisha Jones, senior adviser for content development at The Math Learning Center, about building an asset-based approach to assessment.
Mike: Tisha, first of all, thanks for joining us. We're thrilled to have you with us.
Tisha Jones: I'm really excited to be here.
Mike: I have a sense that for a lot of people, the idea of asset-based assessment is something that we might need to unpack to offer, kind of, a basic set of operating principles or a definition. So, my first question is, how would you describe asset-based assessment? What would that mean for a practitioner?
Tisha: I think the first part of it is thinking just about assessment. Assessment is a huge part of every school that is in this country. So, there are formative assessments, which are ongoing assessments that teachers are doing while students are considered “in the process of learning”—although we know that students really are never not in the process of learning. And then there are also summative assessments, when we want to see if they have demonstrated proficiency or mastery of the concepts that they've been learning throughout that unit. But when we're thinking about assessments, oftentimes the idea of assessment is that we are looking for what students don't know. And asset-based assessment means that we're taking this idea and we're flipping it, and we're saying, “Let's start by looking at what students are showing us that they do know.” And we're trying to really focus on the things that our students are showing us that they're able to do.
Mike: So, that's a lot. And I think one of many of the things that's going on for me is that that's a pretty profound mind shift, I think, for a lot of folks in the field; not because they necessarily want to look at their students as a set of deficits, but because most of the training that a lot of us got actually was focused on “What are the deficits?”
Tisha: Most of the training when we're talking about kids casually, or with our colleagues or administrators, we're often worried about, “Well, our kids don't know this. Our kids are struggling here.” And that really becomes the way that we see our students, right? And our kids are so much more than that, right? And our kids are coming to us with knowledge, and we can forget that when we're only focused on what they don't know.
Mike: There's a great quote that you're making me think about. It's from the 14th century, and the person has said, essentially, “The language that we use becomes the world that we live in.” And I think that's a little bit of where you're going, is that deficit-focused language kind of lives in the DNA of a lot of either the training that we've had or the structures of schools. And so, flipping this is a mind shift, and I think it's really exciting that we're talking about this. I have two things on my mind. I think one is, let's talk about the assessments themselves first. So, if I want to start thinking about using my assessments in an asset-based way, if we just think about the assessments themselves, be they formative or summative, tell me about what you think an educator might do with the assessments that they're using, whether they're coming from a curriculum or whether they're some that they're designing on their own. How should I think about the assessment materials that I have, and are there ways that I should imagine shifting them?
Tisha: That's a great question. I think that when you're looking at your assessments, you may or may not need to change them. They might be fine the way that they are. But the way to know is when you see the opportunities kids have to give their answers, what is that going to tell you about what they understand? So, if you have, for example, a problem that is computation, if you have a problem that has just asked the kids for an answer, or if you have a problem that's multiple choice, what are you learning about their thinking, about their understanding from what they put on the paper? Now, I'm not saying don't ever use those questions. They have their purpose. But that is really what I am asking you to do, is to think about “What is their purpose? What is the intention behind the questions on the assessment?” So, are there ways for you to open up the assessment to give kids more ways of showing what they do understand as opposed to limiting them to saying, “You must show something in this way” or “You're either right or you're wrong”?
Mike: Yeah, that really hits home for me. And I think one of the operating principles that I'm hearing is, regardless of what assessment tools you're using, creating space for kids to show you how they're thinking is really a starting, foundational, kind of, centerpiece for asset-based assessment.
Tisha: Absolutely. And I want to also add that I'm talking a lot about paper and pencil because we think about assessments as paper and pencil. But assessment’s also not just paper and pencil. Assessment, especially formative assessment, it's your conversations that you have with kids in class. As far as I am concerned, there is no better way to know what a kid's thinking than to talk to them. Talk to your kids as much as you absolutely, possibly can. Ask them so many questions.
Mike: Well, you're bringing me to the second piece about the assessments themselves. One piece is, create space, regardless of whether it's a question in a conversation or whether it's a question in a paper-pencil assessment or what have you, for them to show their thinking. The other thing that it makes me think is, part of my work as an educator is to look at the questions and say, “What are the big ideas that I'm really looking for? And what is it that I'm hoping that I can understand about children's thinking with each of these questions that I'm asking?”
Tisha: Yes.
Mike: Beyond just right and wrong.
Tisha: Yes, this is hard work. But this, to me, is not extra work. When you think about a gap, sometimes that can feel very disheartening. It can feel like, “I can't close it. My kids don't know this. They're never going to get it.” It almost just drains the joy of teaching out. This is the job, and this is the part that I am hoping we can all get excited about. I am excited to know what my kids understand. I feel like that gives me a better entryway to being a better teacher for them. If we can start to shift how we think about assessing our students to looking for what they know, to me, that feels very different. It feels different for your kids, and it feels different for you. It's much more fun to walk into a classroom thinking about what my kids know than what they don't.
Mike: Yeah. And I think you're hinting at the next place that I wanted to go, which is, there's the assessments themselves and both how I use them and how I make space for kids to show their thinking. And then there's “How do I approach the things that kids are showing me in their assessments?” And I think that feels like another one of these mind-shift pieces where, what kept coming to mind for me is, if you and I and a colleague or two were sitting together at a table and we were teaching third grade and we had a set of student work in front of us, part of what I'm thinking about is what would a conversation sound like if we were really taking an asset-based perspective on looking at our students' work? What questions might we ask? What kind of a process might we use to, kind of, really focus on assets as opposed to focusing on deficits and gaps?
Tisha: So,…
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May 4, 2023
Math Talk in Kindergarten & Beyond - Guest: Dr. Hala Ghousseini
Rounding Up
Season 1 | Episode 16 – Math Talk in Kindergarten & Beyond
Guest: Dr. Hala Ghousseini
Mike Wallus: Kindergarten is a joyful, exciting, and challenging grade level to teach. It's also a time when educators can develop a set of productive norms and routines around discourse that can have long lasting effects on students. On today's podcast, we talk with Dr. Hala Ghousseini, a professor at the University of Wisconsin, about building a solid foundation from math talk in kindergarten and beyond.
Mike: Welcome, Hala. We're really excited to have you on the podcast today talking about math talk in kindergarten.
Hala Ghousseini: Thank you very much for having me. This is exciting. I love this topic, and the chance to really talk about this with you is great.
Mike: Well, I feel the same way. I spent eight of my 17 years teaching kindergarten, so I've been dreaming about a podcast like this for a long time.
Hala: ( laughs ) I can imagine the magic of kindergarten just because it's a time where people think that they know what to expect, but literally you don't know what to expect with children in kindergarten.
Mike: You started to hint at the first thing that I hope to talk about. I would love to talk about norms. This feels so important because the norms and the culture that we set in kindergarten, from my perspective, those might be some of the first messages students receive about what's valued in a mathematics classroom. And I'm wondering if you could talk just a bit about the norms that you think are important. I mean, perhaps what it looks like to support them in kindergarten.
Hala: Absolutely. And I just want to situate a little bit some of the things that I have been studying and thinking about. When I think of math in kindergarten, it very much exists within the learning altogether that happens in kindergarten; whether it's social-emotional skills, whether they're learning about other subject areas. So, when I think about the norms, I think often of them as embedded within the fabric of what's happening in kindergarten. In the research that we've done, we've seen it happening at two levels. One in relation to what we would call ‘norms related to what's conceptual,’ or what [people might] call more like the disciplinary aspects of norms. So, some of the things that we've seen is, first of all, centered on children's thinking. The idea that first as an individual in class, that I'm a contributor to everyone's understanding. So, the way that is typically continuously communicated by the teacher, in the sense that it's important to share our thinking. And it's important to share it, not just because I'm the teacher and I asked you to do it, but because it's going to contribute to everyone else's learning.
Hala: My learning as the teacher, others learning in the classroom. And we've seen examples from teachers where often, as they're asking students to get ready to go into their small groups, they would always say, ‘Remember, it's important to show our thinking and our work because we want to help someone else learn it.’ You want to help the class understand this idea better. And even with the use of representations, resources, those were all really in the service of helping someone make their thinking explicit so that someone else is going to understand it or use it or build on it. So, I'll give you another example. The idea of saying, ‘Remember, we want to listen now to Hala share her thinking because we want to think how we make sense of it, what Hala is helping us think about. So, those were the typical expressions or things that teachers would say in building these norms in the classroom.
Hala: The other norm, when it comes to the social aspects of the norm, was really this explicit work on the sense of the collective as an intellectual community. The idea that we are in this together. It's not about me and you as the teacher, but it's about the us. What do we make of it? How do we really flag certain things that may help the group process and think about something? And those were also done constantly across the times we've spent in these classrooms, in the way teachers would really point to something that may help us as a group later. ‘Hey, look at this, this might help us later in the way we're going to work on certain ideas together.’
Mike: Well, I do want to ask you about something else that really struck me when I was reading the article. So, you and your co-authors talked a great deal about orienting students to and then encouraging the use of resources to communicate their thinking. That really hit me as a person who used to teach these young kiddos. Can you talk a little bit about what this looks like?
Hala: Yes. This drew our attention, given where kindergartners are in their language development. They bring a lot of language from home that actually is going to be essential to build on in explaining the reasoning, talking about their thinking, reacting to someone else's thinking. So, we started thinking about the way students’ thinking, the way their language that they bring with them, becomes a resource that they could use. So, encouraging them that ‘Yes, that is one way you can explain your thinking,’ so that really they find that language that is going to give them an entry point into the collective as an intellectual community. The second thing in relation to resources, also availing in the classroom. We've noticed these teachers that—besides the fact that you have, like, a number line or a hundredth chart displayed on the board or even the physical tools that usually typically students play with—how those become things that the teacher points to and says, ‘Wow, you know what you're doing.’
Hala: This might help us think about this idea. So, let's remember that what struck us was that, when students were explaining their thinking, we rarely saw a student asking for permission to go and use something to come and support their thinking. We saw that they were really going to things and bringing them. So that was a norm in that class. That kind of intersects with the idea of normative ways of working. You can just go and reach it. You don't have to get that teacher's permission to do it. I think one more thing I'll say about resources. We've noticed the teacher, typically if a student used a particular resource that supported them in their thinking, when they're sharing, they make sure to actually highlight it, lift it up in what the student is saying so that others see that those resources could be contributions to supporting the reasoning in this class.
Mike: So, boy, there's a lot there. I think the first thing that really hits me is this idea that part of the culture that you want to establish, is that the resources are available and it's contingent on the teacher saying, ‘Yes, you can go get that right now.’
Hala: Absolutely. And it's a way of socializing the students to be aware of what's in their classroom that is actually part of what's supporting their learning. You know, there is a thing that I always work in when I'm working with teachers, this idea that, you know, children are sense makers. And we tend to think of children as sense makers beyond just mathematics. Of course they are, but also they're sense makers as learners in general. So, we treat them as sense makers in the way as teachers. We owe it to them to explain to them why, for example, we're asking them to do something. And we say, ‘So, I want you to show your work—not just to please me, because this contributes to the collective work in this way.’ And we reinforce this message continuously. Similarly, the idea of what's in our class, like, when we see, for example, base ten blocks. I have a few things in this corner. The idea that these are there to also support our learning. So, we treat them as sense makers in the sense, these are all shared tools for our classrooms. So, that's kind of how we think…
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Apr 20, 2023
Productive Ways to Build Fluency with Basic Facts - Guest: Dr. Jenny Bay Williams
Rounding Up
Season 1 | Episode 15 – Productive Ways to Build Fluency with Basic Facts
Guest: Dr. Jennifer Bay-Williams
Mike Wallus: 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 sense making.
Mike: 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 nine plus six, you can move one over and you have 10 plus five—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 raised 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 ten-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 10 or a 20 or a 30’ 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.
Jenny: 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 10 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 add a group strategy with multiplication. If you were going to do direct instruction, and you're doing six times eight, 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.’
Jenny: The explicit strategy instruction would have, for example—I like eight boxes of crowns because they oftentimes come in eight. So, but they'd have five boxes of crowns and then one more box of crowns. So, they could see you've got five boxes of crowns. 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 ten-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 are] a lot of steps to think through. So, if you're doing nine plus five, then in the pretend-to-10 strategy, you just pretend that nine is a 10. So now you've got 10 plus five and then you've got to compensate in the end. You’ve got to fix your answer because it's one too much. And so, you've got to come back one. 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.
Jenny: Otherwise, the strategy is not useful. The strategy is useful when they already know 10 plus five. So, you teach them this, you teach them that relationship, you know 10 and some more, and then they know that nine’s one less than 10. That relationship is hugely important, knowing nine is one less than 10. Um, and so then they know their answer has to be one less. Nine’s one less than 10. So, nine plus a number is one less than 10 plus the number. Huge idea. And there's been a lot of research done in kindergarten on students understanding things like seven’s one more than six, seven’s one less than eight. 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 one more, one less, um, two more, two 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…
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Apr 6, 2023
Enhancing Tasks for Multilingual Learners - Guest: Dr. Zandra de Araujo
Rounding Up
Season 1 | Episode 14 – Enhanced Tasks for Multilingual Learners
Guest: Dr. Zandra de Araujo
Mike Wallace: How can educators take concrete steps to enhance tasks for multilingual learners? That's the subject of today's podcast. Today we'll talk with Dr. Zandra de Araujo, the chief equity officer at the University of Florida's Lastinger Center for Learning, about three ways to enhance tasks for multilingual learners and how to implement them in an elementary mathematics setting. We'll also discuss practical strategies and resources for supporting multilingual learners regardless of their age or grade.
Mike: Hey, Zandra. Welcome to the podcast.
Zandra de Araujo: Thanks for having me. I'm excited to be here.
Mike: I'm super excited to be talking to you. So, I'd love to just start with a quote you and your co-authors wrote. You say, ‘Rather than focus on language before mathematics, research shows that multilingual learners both can and should develop mathematical knowledge and language proficiency simultaneously.’ Can you talk a little bit about that statement and share some of the research that informs it?
Zandra: Sure. So, basically, if you think about learning a new language, you need to use it to get better at it. And so, in the past, people were more likely to put language first and to hold off on academics until students learned English. And what we've learned since then from brilliant scholars like Judy Moskowitz and others, is that we should simultaneously grow math alongside language development. And there's a couple of reasons for that. One, it helps improve your math learning, your language learning at the same time, which is great. It doesn't put you below grade level for your math learning because you're waiting to catch up with English first. And we know that that proficiency in your first language also will lead to better proficiency in your second language in math and other areas. So, there's only benefits really.
Zandra: And also, if you think about kids who are native English speakers, they're also learning how to talk about mathematics in school and how to use math language. And so, you might as well do it with the whole class and practice discourse and use good multimodal representations and communication skills to enhance everybody's language learning and math learning because you learn through and with language. And so, you can't really put language—or mathematics—on hold completely for kids. It's just not the right thing to do.
Mike: I loved where you said you learn through and with language.
Zandra: Um-hm.
Mike: Could you just expand upon that? Because it really feels like there's a lot of wisdom in that statement.
Zandra: Yeah, I mean, the way that we learn is we listen, we participate, we talk, we discuss. We have to communicate ideas from one person to another. And it's this communication—and communication is not just in one language in one way. And language is more expansive than that. And we need to think about that. And the way you communicate what you've learned is through language. Or you show it visually. But usually as you're showing, you're gesturing and communicating in maybe non-verbal language communication. So, I think we forget that math is inherently language based as we communicate it in schools and as we typically experience it in schools.
Mike: Thank you. I want to shift a little bit and talk about the three types of enhancements that you and your co-authors are talking about. So, using and connecting multiple representations, thinking through language obstacles, and contextualizing concepts and problem-solving activities. And what I'd like to do is take time to discuss each one of these. So, to begin, can you talk a little bit about what you mean by using and connecting multiple representations?
Zandra: Sure. I tend to put things in my own frame of learning a second language. So, if you think about when you travel to a country that you don't speak the language in fluently, you probably do a lot of gesturing. You look for signs that don't have words in that language, necessarily, if you can't read it. You might draw something, you might do a lot of things. So, visuals and representations are very helpful when we're learning something new or trying to understand something that we already understand, we just can't communicate it. So, in mathematics, a lot of our representations are serving that purpose. They allow us to learn things in a more deep way.
Zandra: So, if you think about, I can show you something like the number five written out. I can show you five unifix cubes, I could show you five tally marks. Those are all different representations that very young children experience. And we're trying to communicate the same concept typically, of five; like the total set of five, the cardinality of five things, typically. And so, kids, when they experience all these different things in different ways, and we connect explicitly across them, it really helps them to understand something in a new or different way. But also, for students who are acquiring English, it allows them to connect the visual with their home language that they're thinking in their brain. And they probably have the words for it in their home language. They may just not understand just the spoken word. But when you see a representation, you have more ideas to anchor on.
Mike: Yeah. As you described that, you can see how critically important that would be for multilingual learners and how much that would both support them and allow them to make the connections.
Zandra: Um-hm. And it's not just for multilingual students. I can't imagine the number of times I've been in a classroom and a teacher might model something with base ten blocks and maybe draw on representation of base ten blocks on the board and then never take the extra step to explicitly link it to the numerals that it …
Mike: Um-hm.
Zandra: … they're representing or the bundles and things like that. But those connections are what we're hoping kids will make. And so, explicitly linking those things and talking across them. And ‘How do you see five here? And how do you see five here?’ is really important for all students. But it's especially beneficial if you're still acquiring the language of instruction.
Mike: Absolutely. So, let's shift gears and talk a little bit about language obstacles. So, as a monolingual English speaker, this is an enhancement that I'd really like to understand in more depth.
Zandra: ( laughs ) As a monolingual also, uh, English speaker that grew up in a Portuguese-speaking household and someone who is trained in mathematics teaching and learning and not in language teaching and learning specifically, this was very interesting to me, too. Essentially, it seems intuitive that you would take away language if that's an obstacle. And that is the main obstacle that students who are acquiring English in school are facing. It's not necessarily that they're below grade level in math. Sometimes they are. But many times they're not. They might be above grade level. But there are specific potential needs for support around English-language proficiency or acquisition. And so, when we think about language obstacles, it's those things that get in the way of learning the mathematics. And there's kind of two ways that you could address them: One is you remove them all, and then two is you scaffold up so that they can access it.
Zandra: I'm more in favor of that approach where we scaffold and try to help further their language alongside their mathematics. Because that goes through the very first thing we talked about, is that you're enhancing and developing English alongside mathematics. But there are some times where there's just unnecessary obstacles that are really getting in the way of understanding what you're trying to do in mathematics. And that's kind of what we provided in the article is the list of some of these…
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Mar 16, 2023
Keep Calm and Press for Reasoning - Guest: Nancy Anderson, EdD
Rounding Up
Season 1 | Episode 13 – Keep Calm
Guest: Nancy Anderson, EdD
Mike Wallace: We often ask students to share their strategies. But, what does it look like to uncover and highlight the reasoning that informs that strategy? Today on the podcast, we'll talk with Nancy Anderson, a classroom teacher and professional learning developer, about strategies to elicit the reasoning at the heart of the student's thinking.
Welcome to the podcast, Nancy. I am so excited to talk to you today.
Nancy Anderson: Thank you. Likewise, Mike.
Mike: I'd like to begin with a quote from your article, “Keep Calm and Press for Reasoning.” In it, you state: “Mathematical reasoning describes the process and tools that we use to determine which ideas are true and which are false.” And then you go on to say that “in the context of a class discussion, reasoning includes addressing the strategy’s most important ideas and highlighting how those ideas are related.” So, what I'm wondering is, can you talk a little bit about how eliciting a strategy and eliciting reasoning may or may not be different from one another?
Nancy: So, when we elicit a strategy, we're largely focused on what the student did to solve the problem. For example, what operations and equations they might have used, what were the steps, and even what tools they might have used. For example, might they have used concrete tools or a number line? Whereas eliciting reasoning focuses on the why behind what they did. Why did they choose a particular strategy or equation? What was it in the problem that signaled that particular equation or that particular operation made sense? And if the strategy included several steps, what told them to go from one step to the next? How did they know that? And then similarly for the tools, what is it in the problem that suggested to them a number line might be an effective strategy to use? And lastly, listening reasoning sort of focuses on putting all those different pieces together so that you talk about those different elements and the rationale behind them in such a way that the people listening are convinced that the strategy is sound.
Mike: That's actually really helpful. I found myself thinking about two scenarios that used to play out when I was teaching first grade. One was I had a group of children who were really engaging with the number line to help them think about difference unknown problems. And what it's making me think is, the focus of the conversation wasn't necessarily that they used the number line. And it's like, ‘Why did this particular jump that you're articulating via number line? What is it about the number line that helped you model this big idea or can help make this idea clearer for the other students in the class?’
Nancy: Exactly, yes. So, when I think about reasoning, I think about different pieces coming together to form a cohesive explanation that also serves as a bridge to using a particular strategy for one particular problem, [and] as a tool for solving something similar in the future.
Mike: So, I have a follow-up question. When teachers are pressing students for their reasoning, what counts as reasoning? What should teachers be listening for?
Nancy: Broadly, mathematical reasoning describes the processes and tools that we use to determine which ideas are true and which are false. Because mathematics is based upon logic and reasoning—not a matter of who says it or how loudly they say it or how convincingly they say it, but rather, what are the mathematical truths that undergird what they're saying? That's sort of a broad definition of mathematical reasoning, which I think certainly has its merits. But then I think about the work of teaching, particularly at the elementary level. I think it's helpful to get much more specific. So, when we think about elementary arithmetic, reasoning really focuses on connecting computational strategies to the operations and the principles that lie underneath. So, in the context of a class discussion, when we have a student explain their reasoning, we're really trying to highlight a particular strategy's most important ideas and how those ideas are related, but in such a way that others can listen and say, ‘Oh, I get it. If I were to try the problem again, I do believe that's going to lead to the correct answer.’ Or if it was this problem, which is similar, ‘I think I can see how it might make sense for me to use this approach here with these slight adjustments.’ So, do you want to take an example?
Mike: Yeah, I'd love to.
Nancy: So, for example, in a first-grade class, there might be a class discussion about different strategies for adding seven plus eight. And I think in a lot of classes at one point, the teacher would likely want to highlight the fact that you can find that sum using doubles plus one. So, in this particular instance, if a student were to talk about their reasoning, we'd want to encourage that student and certainly help that student talk about the following ideas: the connection between seven plus eight and seven plus seven, and the connection between their answers, namely because the second addend has changed from seven to eight, and noting the connections between the second addend and the answers, namely, if the second addend increases by one, so, too does the sum. And finally, we'd want to emphasize what it is we're doing here. Namely, we are using sums that we know to find sums we don't know.
Nancy: So, that's an effective example of what reasoning sounds like in the elementary grades. It's very specific. So even though reasoning is the thing that allows us to move from specific examples to generalizations in elementary mathematics, it's oftentimes by really focusing on what's going on with specific examples
Mike: Uh-hm.
Nancy: … that students can begin to make those leaps forward. Some of my thinking lately about what I do in the classroom comes from the book ‘Make It Stick,’ which talks a lot about learning processes and principles in general. And one of the points that the authors make in the book is that effective learners see important connections, for whatever reasons, sometimes more readily or more quickly than others. So, what I try to do with my teaching then is to say, ‘OK, well how can I help all learners see those relevant and important connections as well?’
Mike: Absolutely. So, it really does strike me that there are planning practices that educators could use that might make a press for reasoning more effective. I'm wondering if you could talk about how might an educator plan for pressing for reasoning?
Nancy: One thing that I think teachers can do is anticipate, in a very literal sense, what is it that they want students to say as a result of participating in the lesson? So, I think oftentimes we, as classroom teachers, focus on what we want students to learn, i.e., the lesson objective or the essential aim. But that can be a big jump from thinking about that to thinking about the words we literally want to hear come out of student's mouths. So, I think that that's one shift teachers can make to thinking not just about the lesson objective as you'd write on the board, but literally what you want students to say, such that when you walk around and you sort of listen in on small groups, those moments where you say like, ‘Oh yeah, they're on the right track.’ And then I think another key shift is thinking more towards specific examples rather than generalizations.
Nancy: So, as an example, suppose that in a third- or fourth- or fifth-grade classroom, students were talking about fraction comparison strategies, and the teacher had planned for a lesson where the objective was to determine if a fraction was more or less than a half by using the generalization about all fractions equal to a half. Namely, that the numerator is always half of the denominator. So, that certainly could be something that we might se…
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Mar 2, 2023
Open Tasks - Guest: Dr. Kim Markworth
Rounding Up
Season 1 | Episode 12 – Open Tasks
Guest: Dr. Kim Markworth
Mike Wallus: Lately, terms like ‘rich tasks,’ ‘multiple entry points,’ and ‘low floor,’ ‘high ceiling’ are being used so often in the world of mathematics education that many educators are confused about their meaning. Today we talk with Kim Markworth, director of content development at The Math Learning Center, about what these terms look like in practice and how they support student learning. Welcome to the podcast, Kim. It's great to have you.
Kim Markworth: Thank you, Mike. I'm really honored and excited to be here.
Mike: I would love to start this conversation by talking about what it means for a task to be open-ended and have a low floor and a high ceiling. So, is there a way that you think about these terms that might help educators clarify their meaning?
Kim: That's a great question, Mike. In truth, when we think about these terms, they're really all interconnected. And I don't know…
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Feb 16, 2023
Navigating a Successful Curriculum Adoption - Guest: Dana Nathanson
Rounding Up
Season 1 | Episode 11 – Successful Curriculum Adoption
Guest: Dana Nathanson
Mike Wallus: Adopting a new curriculum is not for the faint of heart. What makes this challenging? Well, beyond the materials themselves, a curriculum adoption may represent many things: changes to longstanding practices, beliefs, and classroom culture. On today's podcast, we'll talk with Dana Nathanson, the elementary math coordinator in Leander, Texas, about how leaders can effectively design, manage and sustain a successful curriculum adoption. Welcome to the podcast, Dana. I'm thrilled to have you and be able to talk with you a little bit about the work that goes into adopting and supporting the implementation of a new curriculum.
Dana Nathanson: I'm excited to be here. Thank you for the opportunity.
Mike: Absolutely. So, in your case, we're talking about the work that you did in Leander, Texas, when you supported the adoption of Bridges in Mathematics. I'd love to start by talking about…
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Feb 2, 2023
Building Asset Based Learning Environments - Guest: Dr. Jessica Hunt
Rounding Up
Season 1 | Episode 10: Asset-Based Learning Environments
Guest: Dr. Jessica Hunt
Mike Wallus: Take a moment to think about the students in your most recent class. What assets do each of them bring to your classroom and how might those assets provide a foundation for their learning? Today we're talking with Dr. Jessica Hunt about asset-based learning environments. We'll talk about how educators can build an asset-based learning environment in their classrooms, schools, and school districts. Welcome to the podcast, Jessica. Thanks for joining us.
Jessica Hunt: Thank you. I'm so excited to be here today.
Mike: Well, I would love to start our conversation asking you to help define some language that we're going to use throughout the course of the podcast.
Jessica: Sure.
Mike: I'm wondering if you can just describe the difference between an asset-based and a deficitfocused learning environment.
Jessica: I think historically what we see a lot of is deficit-based thinking.…
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Jan 19, 2023
Multilingual Learners - Guest: Jean Harvey, Shannon Lindstedt and Christa Beebe of TNTP (The New Teacher Project)
Rounding Up
Season 1 | Episode 9: Multilingual Learners
Guest: Jean Harvey, Shannon Lindstedt and Christa Beebe of TNTP (The New Teacher Project)
Mike: As a young educator, I was often unsure how to support the multilingual learners in my classroom. And my well-intended attempts didn't always have the impact that I hoped they would. Today we're returning to a topic we've discussed before on the podcast: support for multilingual learners in the mathematics classroom. We'll talk about some of the myths surrounding multilingual learners and dig into specific strategies educators can use to leverage their assets and support meaningful understanding of mathematics. Today we're joined by Shannon Lindstedt, Jean Harvey and Christa Beebe from TNTP (The New Teacher Project). We're going to talk with them about a set of tools and practices they've developed to support educators who serve multilingual learners.
Mike: Welcome, Shannon, Jean and Christa. Great to have you with us today.…
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Jan 5, 2023
Exploring a Framework for Equity in the Math Classroom - Guest: Dr. Pamela Seda and Dr. Kyndall Brown
Rounding Up
Season 1 | Episode 8 – Unpacking ICUCARE
Guests: Dr. Pamela Seda & Dr. Kyndall Brown
Mike Wallus: What does it mean to offer our students a culturally relevant experience in mathematics?
This is a question on the minds of many, particularly elementary mathematics educators. Today we're
talking with Pamela Seda and Kyndall Brown, authors of “Choosing to See: A Framework for Equity in the
Math Classroom.” We'll talk with our guests about what culturally relevant mathematics instruction
looks like and identify practical steps educators can take to start this important work in their classrooms.
Mike: So, hello, Pam and Kyndall. Welcome to the podcast. We're so glad to have you with us. I'm
wondering if both of you would be willing to take a turn and just talk a little bit about what brought you
to writing the book.
Pamela Seda: OK, well I'll start. This book really started with my dissertation research. And when I
started my Ph.D. program, I was very well…
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Dec 15, 2022
Cognitively Guided Instruction: Turning Big Ideas into Practice - Guest: Dr. Kendra Lomax
Rounding Up
Season 1 | Episode 7 – Cognitively Guided Instruction: Turning Big Ideas into Practice
Guest: Dr. Kendra Lomax
Mike Wallus: Have you ever had an experience during your teaching career that fundamentally changed how you thought about your students and the role that you play as an educator? For me, that shift occurred during a sweltering week in July of 2007, when I attended a course on cognitively guided instruction. Cognitively guided instruction, or CGI, is a body of research that has had a massive impact on elementary mathematics over the past 20 years. Today on the podcast, we're talking with Kendra Lomax, from the University of Washington, about CGI and the promise it holds for elementary educators and students. Well, Kendra, welcome to the podcast. It's so great to have you on.
Kendra Lomax: Well, thanks for having me.
Mike: Absolutely. I'm wondering if we can start today with a little bit of background; part history lesson, part primer to help listene…
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Dec 1, 2022
Cultivating Positive Math Identity - Guest: Nataki McClain and Annelly Rodas
Rounding Up
Season 1 | Episode 6 – Cultivating a Positive Math Identity
Guests: Nataki McClain and Annelly Rodas
Mike Wallus: Today I'd like to start our episode with a bit of a thought exercise. I'd like you to close your eyes and picture your childhood self, learning math in your elementary school. What are some of the memories and feelings that come to mind? And when you reflect on those memories, what do you think the unspoken messages you may have absorbed about what it means to be good at math were? And then maybe most importantly, how did those early experiences with mathematics shape your belief about yourself as a doer of math? Today on the podcast, we're talking about identity; specifically, math identity. What is it? And how can we as teachers shape our students' math identities. Let's get started.
Mike: Well, hey, everyone. Welcome to Rounding Up. I'm excited to have our friends Nataki and Annelly joining us today. And I think I'll just start by welcoming the…
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Nov 17, 2022
Learning Targets - Guest: Dr. Rachel Harrington
Rounding Up
Season 1 | Episode 5 – Learning Targets
Guest: Dr. Rachel Harrington
Mike Wallus: As a 17-year-veteran classroom teacher, I can't even begin to count the number of learning targets that I've written over the years. Whether it's writing ‘I can’ statements or developing success criteria, there's no denying that writing learning targets is an important part of teacher practice. That said, the thinking about what makes a strong learning target continues to evolve and the language that we select for those targets has implications for instructional practice. Today on the podcast, we're talking with Dr. Rachel Harrington from Western Oregon University about creating powerful and productive learning targets. Welcome to the podcast.
Rachel Harrington: Thank you for having me. I'm excited to be here.
Mike: Sure. So I'd love to just start our conversation by having you talk a little bit about how the ideas around learning targets have evolved, even just in the course of you…
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Oct 13, 2022
Multilingual Learners for Success - Guest: Dr. Erin Smith
Rounding Up
Season 1 | Episode 4 – Multilingual Learners for Success
Guest: Dr. Erin Smith
Mike Wallus: Multilingual learners represent approximately 10 percent of the U.S. K–12 student population. And they're the fastest growing subpopulation of students in the United States. That said, multilingual learners have been and continue to be underserved in mathematics. Today, we talk with Erin Smith, a mathematics education professor at the University of Southern Mississippi, about ways to support and position multilingual learners as competent doers of mathematics. Hey, Erin, thank you for joining us today on the podcast.
Erin Smith: Thank you so much for inviting me. I'm really happy to be here.
Mike: I was really fascinated by one of the concepts that you talked about your article. You referenced the idea of positioning, and I'm just fascinated by that because I think it has so much potential for how we support students’ math identities. Can you explain positioning…
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Sep 29, 2022
Recording Student Thinking During a Mathematics Discussion - Guest: Dr. Nicole Garcia
Rounding Up
Season 1 | Episode 3 – Recording Student Thinking During a Mathematics Discussion
Guest: Dr. Nicole Garcia
Mike Wallus: If you're anything like me, learning to record students’ mathematical thinking might best be described as on-the-job training, which meant trial and error, and a lot of practice. Our guest on today's podcast is Nicole Garcia, the co-author of an article, published in Mathematics Teacher, that explores the practice of recording student thinking, and offers insights and some principles for making them as productive as possible. Welcome to the podcast, Nicole.
Nicole Garcia: Thank you for having me.
Mike: So you and your co-authors start the article by acknowledging that representing and recording student thinking—when you're in the moment, in a public space, with students—it's challenging, even for veteran teachers. And I suspect that most teachers would agree and appreciate the recognition that this is a skill that takes time and it takes pract…
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Sep 15, 2022
Posing Purposeful Questions - Guest: Dr. DeAnn Huinker
Rounding Up
Season 1 | Episode 2 – Posing Purposeful Questions
Guest: Dr. DeAnn Huinker
Mike Wallus: Educational theorist Charles De Garmo once said, ‘To question well is to teach well. In the skillful use of the question, more than anything else, lies the fine art of teaching.’ Our guest today, DeAnn Huinker, is one of the co-authors of ‘Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K–5.’ We'll talk with DeAnn about the art and the science of questioning and the ways that teachers can maximize the impact of their questions on student learning. DeAnn, welcome to the podcast. It's great to have you.
DeAnn Huinker: I'm happy to be here, Mike. I'm looking forward to our conversation today.
Mike: So, I'd like to start by noting that NCTM (National Council of Teachers of Mathematics) has identified posing purposeful questions as a high-leverage practice in ‘Principles to Actions,’ and then again in 2017 with the publication of…
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Aug 26, 2022
Culturally Relevant Practices in the Elementary Math Classroom - Guest: Dr. Corey Drake
Rounding Up
Season 1 | Episode 1 – Culturally Relevant Practices in the Elementary Math Classroom
Guest: Dr. Corey Drake
Mike Wallus: There's a persistent myth in the world of education, that mathematics is abstract and its teaching is not influenced by cultural contexts. This, despite the fact that research and scholarship indicate when students see how math applies to a world that they recognize, they perform better. Today on the podcast, we'll talk with Dr. Corey Drake, senior director of academic programs at The Math Learning Center, about what it means to provide a culturally inclusive and relevant mathematics experience in the elementary classroom. This is a topic on everybody's mind, and we're excited to address it head on.
Mike: All right. Hello, everybody. Welcome to the podcast. We are excited today to have Dr. Corey Drake with us. And the topic of the day is culturally relevant practices in an elementary classroom. So, Corey, welcome. It's great to have you on the podc…
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Aug 26, 2022
Rounding Up - Opener
Welcome to Rounding Up, the professional learning podcast brought to you by The Math Learning Center.
Two things have always been true in education: Ongoing professional learning is essential, and teachers are extremely busy people. Rounding Up is a podcast designed to provide meaningful, bite-sized professional learning for busy educators and instructional leaders.
I'm Mike Wallus, vice president for educator support at The Math Learning Center and host of the show. In each episode, we'll explore topics important to teachers, instructional leaders, and anyone interested in elementary mathematics education. Topics such as posing purposeful questions, effectively recording student thinking, cultivating students' math identity, and designing asset-based instruction from multilingual learners. Don't miss out! Subscribe now wherever you get your podcasts. Each episode will also be published on the Bridges Educator Site.
We hope you'll give Rounding Up a try, and that the ideas we d…
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