Rounding Up

Season 2 | Episode 13 – Rough Draft Math

Guest: Dr. Amanda Jansen

Mike Wallus: What would happen if teachers consistently invited students to think of their ideas in math class as a rough draft? What impact might this have on students' participation, their learning experience, and their math identity? Those are the questions we'll explore today with Dr. Mandy Jansen, the author of “Rough Draft Math,” on this episode of Rounding Up. 

Mike: Well, welcome to the podcast, Mandy. We are excited to be talking with you. 

Mandy Jansen: Thanks, Mike. I'm happy to be here. 

Mike: So, I'd like to start by asking you where the ideas involved in “Rough Draft Math” originated. What drove you and your collaborators to explore these ideas in the first place? 

Mandy: So, I work in the state of Delaware. And there's an organization called the Delaware Math Coalition, and I was working in a teacher study group where we were all puzzling together—secondary math teachers—thinking about how we could create more productive classroom discussions. And so, by productive, one of the ways we thought about that was creating classrooms where students felt safe to take intellectual risks, to share their thinking when they weren't sure, just to elicit more student participation in the discussions. One way we went about that was, we were reading chapters from a book called “Exploring Talk in School” that was dedicated to the work of Doug Barnes. And one of the ideas in that book was, we could think about fostering classroom talk in a way that was more exploratory. Exploratory talk, where you learn through interaction. Students often experience classroom discussions as an opportunity to perform. "I want to show you what I know.” And that can kind of feel more like a final draft. And the teachers thought, “Well, we want students to share their thinking in ways that they're more open to continue to grow their thinking.” So, in contrast to final draft talk, maybe we want to call this rough draft talk because the idea of exploratory talk felt like, maybe kind of vague, maybe hard for students to understand. And so, the term “rough draft talk” emerged from the teachers trying to think of a way to frame this for students. 

Mike: You're making me think about the different ways that people perceive a rough draft. So, for example, I can imagine that someone might think about a rough draft as something that needs to be corrected. But based on what you just said, I don't think that's how you and your collaborators thought about it, nor do I think that probably is the way that you framed it for kids. So how did you invite kids to think about a rough draft as you were introducing this idea? 

Mandy: Yeah, so we thought that the term “rough draft” would be useful for students if they have ever thought about rough drafts in maybe language arts. And so, we thought, “Oh, let's introduce this to kids by asking, ‘Well, what do you know about rough drafts already? Let's think about what a rough draft is.’” And then we could ask them, “Why do you think this might be useful for math?” So, students will brainstorm, “Oh yeah, rough draft, that's like my first version” or “That's something I get the chance to correct and fix.” But also, sometimes kids would say, “Oh, rough drafts … like the bad version. It's the one that needs to be fixed.” And we wanted students to think about rough drafts more like, just your initial thinking, your first ideas; thinking that we think of as in progress that can be adjusted and improved. And we want to share that idea with students because sometimes people have the perception that math is, like, you're either right or you're wrong, as opposed to something that there's gradients of different levels of understanding associated with mathematical thinking. And we want math to be more than correct answers, but about what makes sense to you and why this makes sense. So, we wanted to shift that thinking from rough drafts being the bad version that you have to fix to be more like it's OK just to share your in-progress ideas, your initial thinking. And then you're going to have a chance to keep improving those ideas. 

Mike: I'm really curious, when you shared that with kids, how did they react? Maybe at first, and then over time?

Mandy: So, one thing that teachers have shared that's helpful is that during a class discussion where you might put out an idea for students to think about, and it's kind of silent, you get crickets. If teachers would say, “Well, remember it's OK to just share your rough drafts.” It's kind of like letting the pressure out. And they don't feel like, “Oh wait, I can't share unless I totally know I'm correct. Oh, I can just share my rough drafts?” And then the ideas sort of start popping out onto the floor like popcorn, and it really kind of opens up and frees people up. “I can just share whatever's on my mind.” So that's one thing that starts happening right away, and it's kind of magical that you could just say a few words and students would be like, “Oh, right, it's fine. I can just share whatever I'm thinking about.” 

Mike: So, when we were preparing for this interview, you said something that has really stuck with me and that I've found myself thinking about ever since. And I'm going to paraphrase a little bit, but I think what you had said at that point in time was that a rough draft is something that you revise. And that leads into a second set of practices that we could take up for the benefit of our students. Can you talk a little bit about the ideas for revising rough drafts in a math classroom? 

Mandy: Yes. I think when we think about rough drafts in math, it's important to interact with people thinking by first, assuming those initial ideas are going to have some merit, some strength. There's going to be value in those initial ideas. And then once those ideas are elicited, we have that initial thinking out on the floor. And so, then we want to think about, “How can we not only honor the strengths in those ideas, but we want to keep refining and improving?” So inviting revision or structuring revision opportunities is one way that we then can respond to students’ thinking when they share their drafts. So, we want to workshop those drafts. We want to work to revise them. Maybe it's peer-to-peer workshops. Maybe it's whole-class situation where you may get out maybe an anonymous solution. Or a solution that you strategically selected. And then work to workshop that idea first on their strengths, what's making sense, what's working about this draft, and then how can we extend it? How can we correct it, sure. But grow it, improve it.

Mandy: And promoting this idea that everyone's thinking can be revised. It's not just about your work needs to be corrected, and your work is fine. But if we're always trying to grow in our mathematical thinking, you could even drop the idea of correct and incorrect. But everyone can keep revising. You can develop a new strategy. You can think about connections between representations or connections between strategies. You can develop a new visual representation to represent what makes sense to you. And so, just really promoting this idea that our thinking can always keep growing. That's sort of how we feel when we teach something, right? Maybe we have a task that we've taught multiple times in a row, and every year that we teach it we may be surprised by a new strategy. We know how to solve the problem—but we don't have to necessarily just think about revising our work but revising our thinking about the ideas underlying that problem. So really promoting that sense of wonder, that sense of curiosity, and this idea that we can keep growing our thinking all the time. 

Mike: Yeah, there's a few things that popped out when you were talking that I want to explore just a little bit. I think when we were initially planning this conversation, what intrigued me was the idea that this is a way to help loosen up that fear that kids sometimes feel when it does feel like there's a right or a wrong answer, and this is a performance. And so, I think I was attracted to the idea of a rough draft as a vehicle to build student participation. I wonder if you could talk a little bit about the impact on their mathematical thinking, not only the way that you've seen participation grow, but also the impact on the depth of kids' mathematical thinking as well. 

Mandy: Yes, and also I think there's impact on students' identities and sense of self, too. So, if we first start with the mathematical thinking. If we're trying to work on revising—and one of the lenses we bring to revising, some people talk about lenses of revising as accuracy and precision. I think, “Sure.” But I also think about connectedness and building a larger network or web of how ideas relate to one another. So, I think it can change our view of what it means to know and do math, but also extending that thinking over time and seeing relationships. Like relationships between all the different aspects of rational number, right? Fractions, decimals, percents, and how these are all part of one larger set of ideas. So, I think that you can look at revision in a number of different grain sizes. 

Mandy: You can revise your thinking about a specific problem. You can revise your thinking about a specific concept. You can revise your thinking across a network of concepts. So, there's lots of different dimensions that you could go down with revising. But then this idea that we can see all these relationships with math … then students start to wonder about what other relationships exist that they hadn't thought of and seen before. And I think it can also change the idea of, “What does it mean to be smart in math?” Because I think math is often treated as this right or wrong idea, and the smart people are the ones that get the right idea correct, quickly. But we could reframe smartness to be somebody who is willing to take risk and put their initial thinking out there. Or someone who's really good at seeing connections between people's thinking. Or someone who persists in continuing to try to revise. And just knowing math and being smart in math is so much more than this speed idea, and it can give lots of different ways to show people's competencies and to honor different strengths that students have. 

Mike: Yeah, there are a few words that you said that keep resonating for me. One is this idea of connections. And the other word that I think popped into my head was “insights.” The idea that what's powerful is that these relationships, connections, patterns, that those are things that can be become clearer or that one could build insights around. And then, I'm really interested in this idea of shifting kids' understanding of what mathematics is away from answer-getting and speed into, “Do I really understand this interconnected bundle of relationships about how numbers work or how patterns play out?” It's really interesting to think about all of the ramifications of a process like rough draft work and how that could have an impact on multiple levels. 

Mandy: I also think that it changes what the classroom space is in the first place. So, if the classroom space is now always looking for new connections, people are going to be spending more time thinking about, “Well, what do these symbols even mean?” As opposed to pushing the symbols around to get the answer that the book is looking for. 

Mike: Amen.

Mandy: And I think it's more fun. There are all kinds of possible ways to understand things. And then I also think it can improve the social dimension of the classroom, too. So, if there's lots of possible connections to notice or lots of different ways to relationships, then I can try to learn about someone else's thinking. And then I learn more about them. And they might try to learn about my thinking and learn more about me. And then we feel, like, this greater connection to one another by trying to see the world through their eyes. And so, if the classroom environment is a space where we're trying to constantly see through other people's eyes, but also let them try to see through our eyes, we're this community of people that is just constantly in awe of one another. Like, “Oh, I never thought to see things that way.” And so, people feel more appreciated and valued. 

Mike: So, I'm wondering if we could spend a little bit of time trying to bring these ideas to life for folks who are listening. You already started to unpack what it might look like to initially introduce this idea, and you've led me to see the ways that a teacher might introduce or remind kids about the fact that we're thinking about this in terms of a rough draft. But I'm wondering if you can talk a little bit about, how have you seen educators bring these ideas to life? How have you seen them introduce rough draft thinking or sustain rough draft thinking? Are there any examples that you think might highlight some of the practices teachers could take up? 

Mandy: Yeah, definitely. So, I think along the lines of, “How do we create that culture where drafting and revising is welcome in addition to asking students about rough drafts and why they might make sense of math?” Another approach that people have found valuable is talking with students about … instead of rules in the classroom, more like their rights. What are your rights as a learner in this space? And drawing from the work of an elementary teacher in Tucson, Arizona, Olga Torres, thinking about students having rights in the classroom, it's a democratic space. You have these rights to be confused, the right to say what makes sense to you, and represent your thinking in ways that make sense to you right now. If you honor these rights and name these rights, it really just changes students' roles in that space. And drafting and revising is just a part of that. 

Mandy: So different culture-building experiences. And so, with the rights of a learner brainstorming new rights that students want to have, reflecting on how they saw those rights in action today, and setting goals for yourself about what rights you want to claim in that space. So then, in addition to culture building and sustaining that culture, it has to do—right, like Math Learning Center thinks about this all the time—like, rich tasks that students would work on. Where students have the opportunity to express their reasoning and maybe multiple strategies because that richness gives us so much to think about. 

And drafts would a part of that. But also, there's something to revise if you're working on your reasoning or multiple strategies or multiple representations. So, the tasks that you work on make a difference in that space. And then of course, in that space, often we're inviting peer collaboration. 

Mandy: So, those are kinds of things that a lot of teachers are trying to do already with productive practices. But I think the piece with rough draft math then, is “How are you going to integrate revising into that space?” So eliciting students' reasoning and strategies—but honoring that as a draft. But then, maybe if you're having a classroom discussion anyway, with the five practices where you're selecting and sequencing student strategies to build up to larger connections, at the end of that conversation, you can add in this moment where, “OK, we've had this discussion. Now write down individually or turn and talk. How did your thinking get revised after this discussion? What's a new idea you didn't have before? Or what is a strategy you want to try to remember?” So, adding in that revision moment after the class discussion you may have already wanted to have, helps students get more out of the discussion, helps them remember and honor how their thinking grew and changed, and giving them that opportunity to reflect on those conversations that maybe you're trying to already have anyway, gives you a little more value added to that discussion. 

Mandy: It doesn't take that much time, but making sure you take a moment to journal about it or talk to a peer about it, to kind of integrate that more into your thought process. And we see revising happening with routines that teachers often use, like, math language routines such as stronger and clearer each time where you have the opportunity to share your draft with someone and try to understand their draft, and then make that draft stronger or clearer. Or people have talked about routines, like, there's this one called “My Favorite No,” where you get out of student strategy and talk about what's working and then why maybe a mistake is a productive thing to think about, try to make sense out of. But teachers have changed that to be “My Favorite Rough Draft.” So, then you're workshopping reasoning or a strategy, something like that. And so, I think sometimes teachers are doing things already that are in the spirit of this drafting, revising idea. But having the lens of rough drafts and revising can add a degree of intentionality to what you already value. And then making that explicit to students helps them engage in the process and hopefully get more out of it. 

Mike: It strikes me that that piece that you were talking about where you're already likely doing things like sequencing student work to help tell a story, to help expose a connection. The power of that add-on where you ask the question, “How has your thinking shifted? How have you revised your thinking?” And doing the turn and talk or the reflection. It's kind of like a marking event, right? You're marking that one, it's normal, that your ideas are likely going to be refined or revised. And two, it sets a point in time for kids to say, “Oh yes, they have changed.” And you're helping them capture that moment and notice the changes that have already occurred even if they happened in their head. 

Mandy: I think it can help you internalize those changes. I think it can also, like you said, kind of normalize and honor the fact that the thinking is continually growing and changing. I think we can also celebrate, “Oh my gosh, I hadn't thought about that before, and I want to kind of celebrate that moment.” And I think in terms of the social dimension of the classroom, you can honor and get excited about, “If I hadn't had the opportunity to hear from my friend in the room, I wouldn't have learned this.” And so, it helps us see how much we need one another, and they need us. We wouldn't understand as much as we're understanding if we weren't all together in this space on this day and this time working on this task. And so, I love experiences that help us both develop our mathematical understandings and also bond us to one another interpersonally. 

Mike: So, one of the joys for me of doing this podcast is getting to talk about big ideas that I think can really impact students' learning experiences. One of the limitations is, we usually spend about 20 minutes or so talking about it, and we could talk about this for a long time, Mandy. I'm wondering, if I'm a person who's listening, and I'm really interested in continuing to learn about rough draft math, is there a particular resource or a set of resources that you might recommend for someone who wants to keep learning?

Mandy: Thank you for asking. So, like you said, we can think about this for a long time, and I've been thinking about it for seven or eight years already, and I still keep growing in my thinking. I have a book called “Rough Draft Math: Revising to Learn” that came out in March 2020, which is not the best time for a book to come out, but that's when it came out. And it's been really enjoyable to connect with people about the ideas. And what I'm trying to do in that book is show that rough draft math is a set of ideas that people have applied in a lot of different ways. And I think of myself kind of as a curator, curating all the brilliant ideas that teachers have had if they think about rough drafts and revising a math class. And the book collects a set of those ideas together. 

Mandy: But a lot of times, I don't know if you're like me, I end up buying a bunch of books and not necessarily reading them all. So, there are shorter pieces. There's an article in Mathematics Teaching in the Middle School that I co-wrote with three of the teachers in the Delaware Teacher Study Group, and that is at the end of the 2016 volume, and it's called “Rough-Draft Talk.” And that's only 1,800 words. That's a short read that you could read with a PLC or with a friend. And there's an even shorter piece in the NCTM Journal, MTLT, in the “Ear to the Ground” section. And I have a professional website that has a collection of free articles because I know those NCTM articles are behind a paywall. And so, I can share that. Maybe there's show notes where we can put a link and there's some pieces there. 

Mike: Yes, absolutely. Well, I think that's probably a good place to stop. Thank you again for joining us, Mandy. It really has been a pleasure talking with you. 

Mandy: Thank you so much, Mike.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up

Season 2 | Episode 12 – Counting

Guest: Dr. Kim Hartweg

Mike Wallus: Counting is a process that involves a complex and interconnected set of concepts and skills. This means that for most children, the path to counting proficiency is not a linear process. Today we're talking with Dr. Kim Hartweg from Western Illinois University about the big ideas and skills that are a part of counting, and the ways educators can support their students on this important part of their math journey.

 Mike: Well, hey, Kim, welcome to the podcast. We're excited to be talking with you about counting. 

 Kim Hartweg: Ah, thanks for having me. I'm excited, too. 

Mike: So, I'm fascinated by all of the things that we're learning about how young kids count, or at least the way that they attend to quantities. 

Kim: Yeah, it's exciting what all is taking place, with the research and everything going on with early childhood education, especially in regards to number and number sense. And I think back to an article I read about a 6-month-old baby who's in a crib and there's three pictures in this crib. One of them has two dots on it, another one has one dot, and then a third one has three dots. And a drum sounds, and it goes boom, boom, boom. And the 6-month-old baby turns their head and eyes and they look at the picture with three dots on it. And I just think that's exciting that even at that age they're recognizing that three dots [go] with three drum beats. So, it's just exciting. 

Mike: So, you're actually taking us to a place that I was hoping we could go to, which is, there are some ideas and some concepts that we associate with counting. And I'm wondering if we could start the podcast by naming and unpacking a few of the really important ones. 

Kim: OK, sure. I think of the fundamental counting principles, three different areas. And for me, the first one is that counting sequence, or just learning the language and that we count 1, 2, 3, 4, 5. However, in the English language, it's much more difficult [than] in other languages when we get beyond 10 because we have numbers like 11, 12, 13 that we never hear again. Like, you hear 21, 31, 41, but you don't hear 11. Again, it's the only time it's ever mentioned. So, I think it's harder for students to get that counting sequence for those who speak English. 

Mike: I appreciate you saying that because I remember reading at one point that in certain Asian languages, the number 11, the translation is essentially 10 and 1, as opposed to for English speakers where it really is 11, which doesn't really follow the cadence of the number sequence that kids are learning: 1, 2, 3, 4, and so on. 

Kim: Exactly. Yes. 

Mike: It picks up again at 21, but this interim space where the teen numbers show up and we're first talking about a 10 and however many more, it's not a great thing about the English language that suddenly we decided to call those things that don't have that same cadence. 

 Kim: Yeah, after you get past 20, yes. And if you think of kids when they hear the number 16, a lot of times they'll say, “A 1 and a 6 or a 6 and a 1?” Because they hear 16, so you hear the 6 first. But like you said, in other languages, it's 10 six, 10 seven, 10 eight. So, it kind of fits more naturally with the way we talk and the language. 

 Mike: So, there's the language of the counting sequence. Let's talk about a couple of the other things.

 Kim: OK. One-to-one correspondence is a key idea, and I think of this when I was first starting to teach undergraduate students about early math education. I had kids at the same age, so at a restaurant or wherever we happened to be, I'd get out the sugar packets and I would have them count. And at first when they're maybe 2 years old or so, and they're just learning the language, they may count those sugar packets as 1, 2, 3. There may be two packets. There may be five packets. But everything is 1, 2, 3, whether there's again, five packets or two packets. So, once they get that idea that each time they say a number word that it counts for an actual object and they can match them up, that's that idea of one-to-one correspondence to where they say a number and they either point or move the object so you can tell they're matching those up. 

 Mike: OK, let's talk about cardinality because this is one that I think when I first started teaching kindergarten, I took for granted how big of a leap this one is. 

 Kim: Yeah, that's interesting. So, once they can count out and you have five sugar packets and they count 1, 2, 3, 4, 5, and you ask how many are there, they should be able to say five. That's cardinality of number. If they have to count again, 1, 2, 3, 4, 5, then they don't have cardinality of number, where whatever number they count last is how many is in that set. 

 Mike: Which is kind of amazing actually. We're asking kids to decide that “I've figured out this idea that when I say a number name, I'm talking about an individual part of the count until I say the last one, and then I'm actually talking about the entire set.” That's a pretty big leap for kids to start to make sense of. 

 Kim: It is, and it's fun to watch because hear some of them say, “One, 2, 3, 4, 5. Five, there's five.”
( laughs ) So, they kind of get that idea. But yeah, that cardinality of number is a key principle and leads into the conservation of numbers.

 Mike: Let's talk about conservation of number. What I'm loving about this conversation is the way that you're using these concrete examples from your own children, from sugar packets, to help us make sense of something that we might be seeing, but we might not have a name for. 

 Kim: Yeah, so the conservation of number, this is my favorite task when I have young kids around. I want to see if they can serve number or not. So, I might first do the sugar pack thing or whatever and see if they can tell you how many there are. But the real fun is, do they conserve that number? So, I think back to a friend of mine who brought her daughter over one time, and I had these toy matchbox cards on my table, and her name was Katie. And I said, “Katie, how many cars are there?” And she counted “One, 2, 3, 4, 5 … there's five toy cars.” And I moved them around and I said, “Now how many cars are there?” And she counted “One, 2, 3, 4, 5 … there's five toy cars.” So, she has cardinality of number. However, I kept moving those cars into different positions, never adding or taking any away. 

 Kim: That's all that were there the whole time. And after about seven or eight times, I said, “Now how many cars are there?” And her mom finally jumped in and said, “Katie, you've counted those already. There's five cars.” ( laughs ) And I said, “No, no, no. This is just whether she conserves number or not, it's a developmental-type thing.” But you know they conserve number when you ask them, “Well, now how many cars are there?” And they look at you and like, “Well, why would you ask that again? There's five.”
( chuckles ) So, then they can conserve number. It's real fun to do that with elementary students who are getting their number sense going and even before they enter school. However, there will be some that may not get that conservation of number until they're 5 or 6 years old. 

 Mike: Let's talk about something you named earlier. I've heard people pronounce this as (soobitizing) or (subitizing), but in any case, it's really an important idea for people, especially if you're teaching young children to make sense of this. Can you talk about what that means? 

 Kim: Yeah, so subitizing, I think that's interesting. We work so hard getting kids to count and learn the language and have one-to-one correspondence, and then be able to eventually conserve number. But then we want them to just be able to recognize a set of numbers without counting. And that's when they're really starting to develop some number sense. I think of dice. And if you roll a single [die], we want students to just know that when there's an arrangement of four dice, they know it's four without having to count 1, 2, 3, 4. So the subitizing idea, a lot of dice games, maybe some ten-frame cards, dot cards, lots of things like that can help students develop a little bit more of that subitizing, or recognizing a set of items without having to count those. 

 Mike: So, when I look at a set of three dots, I can just say that's three, as opposed to an earlier point where a child might actually say, “One, 2, 3 … that's three.”

 Kim: Exactly. So, that would be subitizing—just instantly knowing what's there without having to count. 

 Mike: So, I wonder if we could unpack two other counting behaviors that sometimes pop up with kids when they're combining or separating quantities. And what I'm thinking about is the difference between the child who counts everything and the child who either counts on from a number or counts back from a number. And I'm wondering if you can talk about what these two behaviors can tell us about how kids are thinking about the numbers that they're operating on. 

 Kim: Yeah, it's so interesting when you have activities like a cup … and maybe you have eight counters and you put three under the cup and you say, “How many are here? Three.” And then you cover those up and you ask, “Well, how many are altogether?” There are some kids who don't have any trouble with counting on 4, 5, 6, 7, 8, but there's other kids who have to lift up the cup and start again at 1. So, they don't have that idea that there's three items under that cup whether you can see them or not. So, it's difficult for them to be able to count on, and we shouldn't as teachers force that on them until they're ready to do it. So, it's a hard concept for kids to get, but especially if they're not developmentally ready for it. 

 Mike: I think that's a really nice caution because I think you could accidentally potentially get kids to mimic a practice that you're trying to show them, but without understanding there's some real danger that you're just causing confusion. 

 Kim: And we want to give kids the idea that counting collections and things, it's a fun thing to do. And I know there may be teachers that have seashells or rocks or different types of collections they might count, and we want students to count those and then discuss how they counted them, arrange them. And I'm thinking of this little girl that I saw on a video where she was counting eight bears, and she arranged them first by color, then counted how many there were. And the teacher then went on to use that and make a problem-solving task for her, such as, “Well, how many green bears do you have?” And she would count them. “Well, what if you gave me those green bears? Do you know what you would have left?” And she said, “Well, I don't know. Let's try it.” And I love that because I think that's the kind of idea we want students to have. They're counting, and “I don't know, let's try it.” They're excited about it. They're not afraid to take chances, and we don't want them to think that “Oh, this is difficult to do.” It's just, “Hey, let's try it. Give it a try here.” 

 Mike: Well, I've heard people talking a lot about this idea of counting collections lately. It seems like we are almost rediscovering the value of a routine like that. I'm wondering if you could talk about the value that can come out of an experience of counting collections and help bring that idea to life for people. 

 Kim: The idea here is that we want students to get good at counting. And the research is showing that students who maybe don't show one-to-one correspondence when they count out, maybe eight counters, might show one-to-one correspondence when they count out 31 pennies, which seems like it shouldn't happen. But there's research out there that over 70 percent of them did better counting 31 pennies than they did with eight counters. So, I think what you count makes a big difference for kids—and to not hold them back, to not think that “OK, we've got to get one-to-one correspondence before we count this collection of 50 items.” I don't think that's the case. And the research is even showing that these ideas that we've talked about all develop concurrently. It's not a linear process. But this counting collections is kind of a big deal with that. And having students count, again, collections that they're interested in, writing number sentences about their collections, comparing what they counted with another partner, and then turning it into problem-solving questions where they're actually doing what happens if you lost five of yours. Or what happens if you combined your collection with somebody else? And turning it into where they're actually doing addition and subtraction, but not actually the formal process of that. 

 Mike: The other thing that you made me think about is, I would imagine you could also have kids finish counting a collection and then you could ask them to represent it either on paper or in some other way. 

 Kim: Exactly. And writing out those number sentences or even creating their own word problems so that they can ask a friend or a partner, it makes it fun. And it relates to what they've done. And let's face it, once you've taken that time to count those collections, you may as well get as much use out of it as you can. ( chuckles )

 Mike: Kim, you're making me think of something that I don't know that I had words for when I was teaching kindergarten, which is, when I look back now, I was looking to see that kids could do a particular thing like one-to-one correspondence or that they had cardinality before I would give them access to a task like counting collections. And I think what you're making me think is that those things shouldn't be a gatekeeper; that they actually develop by doing those things. Am I making sense to you? 

 Kim: Yes. I always thought you had to have the language first. You had to be able to do one-to-one correspondence before you could get cardinality of number, and you needed cardinality of number before you could do conservation of number. But what the research is showing is, it develops concurrently with students; that it's not something that is a linear process by any means. So, when we have these activities, it's OK if they don't have one-to-one correspondence, and you're doing problem-solving tasks with counters. We need to be planning these activities so they're getting all of this, and they will develop it as it fits in the schema of what they're working on and thinking of in their minds. 

 Mike: So, I want to bring up a set of manipulatives that are actually attached to our bodies, particularly when it comes to counting. I'm thinking about fingers. And part of what's on my mind is, again, to go back to my practice, there was a point in time where I was really hung up on whether kids should make use of their fingers when they're counting or when they're operating on numbers. And I'm wondering if you could just offer some guidance around that. 

 Kim: Yes. I think again, it goes to that idea that we have these 10 fingers that are great manipulatives, that we shouldn't stop students from doing that. And I know there was a time when teachers would say, “Don't use your fingers, don't count on your fingers.” And I get the idea that we want students to start to subitize eventually and make combinations and not have to count on their fingers, but to stop them from doing it when they need that would be very detrimental to them. And I actually have a story. When I was supervising student teachers, one teacher was telling a student don't count with their fingers. And I saw them nodding their head, and I went over and I said, “What are you doing?” He said, “Well, I can't count my fingers, so I'm using my tongue, and I'm counting my teeth.” ( laughs ) So, coming up with a problem that way, still using a manipulative, but it wasn't their fingers. 

 Mike: That's pretty creative. 

 Kim: ( laughs ) Yeah. 

 Mike: Well, part of what strikes me about it, too, is our entire number system is based on 10s and ones, and we've got a set of them right in front of us, right? We're trying to get kids to make sense of shifting from units of one to units of 10 or maybe units of five. So, these tools that are attached to our bodies are actually pretty helpful because they're really the basis for our number system in a lot of ways. 

 Kim: Yes, exactly. And being able to come up with even using your fingers to answer questions … I'm thinking, we want students to subitize. So, even something to where there's a dot card that a teacher flashes for 3 seconds, and it's in the formation of maybe a five on a [die]. And you could have students hold up how many there are. And you could do that five or 10 times, with dot flashes. Or you could hold up one more than what you see on the [die]. So, they only see it for 5 seconds and the number's five, but they hold up six. So just uses of fingers to kind of make those connections can be very helpful. 

 Mike: So, before we go, you mentioned that you work with pre-service teachers, folks who are getting ready to go into the field and work with elementary children in the area of mathematics. I'm wondering if there are any particular resources that really help your students and perhaps teachers who are already in the field just make sense of counting and number to really understand some of the ideas that we've been talking about today. Do you have anything in particular that you would recommend to teachers? 

 Kim: Yeah, I'll just mention a few that we use a lot of. We do the two-color counters a lot where one side is yellow and one is red. But we do a lot of dot cards, where again, there are arrangements of dots on a card that you could just flash to a student kind of like I've already explained. There's lots of resources on the National Council of Teachers of Mathematics website. That has ten-frame activities. And if you haven't used rekenreks before, I think those are pretty amazing as well—along with hundreds charts. And just being able to have students create some of their own manipulatives and their own numbers makes a huge difference for kids. 

 Mike: I think that's a great place to close the conversation. Thank you so much for joining us, Kim. It's really been a pleasure chatting with you. 

Kim: Hey, thanks so much. It's been fun, Mike. Thanks for asking me.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2024 The Math Learning Center | www.mathlearningcenter.org

What comes to mind when you think of Pythagoras? His triangle, his theorem and a foundational principle of mathematics? Absolutely, but he was also the enigmatic founder of Pythagoreanism, a mysterious secret society that strove to create a Utopia on earth. 

Today Keith delves into this ancient social experiment, and talks about other attempts at paradise in the millennia since, and comes to a conclusion with some advise for Elon Musk.

Rounding Up

Season 2 | Episode 11 – Translanguaging

Guest: Tatyana Kleyn

Mike Wallus: Over the past two years, we've done several episodes on supporting multilingual learners in math classrooms. Today we're going back to this topic to talk about “translanguaging,” an asset-focused approach that invites students to bring their full language repertoire into the classroom. We'll talk with Tatyana Kleyn about what translanguaging looks like and how all teachers can integrate this practice into their classrooms. 

Mike: Well, welcome to the podcast, Tatyana. We're excited to be talking with you today. 

Tatyana Kleyn: Thank you. This is very exciting. 

Mike: So, your background with the topic of multilingual learners and translanguaging, it's not only academic. It's also personal. I'm wondering if you might share a bit of your own background as a starting point for this conversation. 

Tatyana: Yes, absolutely. I think for many of us in education, we don't randomly end up teaching in the areas that we're teaching in or doing the work that we're doing. So, I always like to share my story so people know why I'm doing this work and where I'm coming from. So, my personal story, I work a lot at the intersection of language migration and education, and those are all three aspects that have been critical in bringing me here. So, I was actually born in what was the Soviet Union many, many years ago, and my family immigrated to the United States as political refugees, and I was just 5½ years old. So, I actually never went to school in the Soviet Union. Russian was my home language, and I quickly started speaking English, but my literacy was not quick at all, and it was quite painful because I never learned to read in my home language. I never had that foundation. 

Tatyana: So, when I was learning to read in English, it wasn't meaning making, it was just making sounds. It was kind of painful. I once heard somebody say, “For some people, reading is like this escape and this pure joy, and for other people it's like cleaning the toilet. You get in and you get out.” And I was like, “That's me. I'm the toilet cleaner.” ( laughs ) So, that was how reading was for me. I always left my home language at the door when I came into school, and I wanted it that way because I, as a young child, got this strong message that English was the language that mattered in this country. So, for example, instead of going by Tatyana, I went by Tanya. So, I always kind of kept this secret that I spoke this other language. I had this other culture, and it wasn't until sixth grade where my sixth-grade teacher, Ms. Chang, invited my mom to speak about our immigration history. 

Tatyana: And I don't know why, but I thought that was so embarrassing. I think in middle school, it's not really cool to have your parents around. So, I was like, “Oh my God, this is going to be horrible.” But then I realized my peers were really interested—and in a good way—and I was like, “Wait, this is a good thing?” So, I started thinking, “OK, we should be proud of who we are and let just people be who they are.” And when you let people be who they are, they thrive in math, in science, in social studies, instead of trying so hard to be someone they're not, and then focusing on that instead of everything else that they should be focusing on as students. 

Mike: So, there's a lot there. And I think I want to dig into what you talked about over the course of the interview. I want to zero in a little bit on translanguaging though, because for me, at least until quite recently, this idea of translanguaging was really a new concept, a new idea for me, and I'm going to guess that that's the case for a lot of the people who are listening to this as well. So, just to begin, would you talk briefly about what translanguaging is and your sense of the impact that it can have on learners? 

Tatyana: Sure. Well, I'm so glad to be talking about translanguaging in this space specifically, because often when we talk about translanguaging, it's in bilingual education or English as a second language or is a new language, and it's important in those settings, right? But it's important in all settings. So, I think you're not the only one, especially if we're talking about math educators or general elementary educators, it's like, “Oh, translanguaging, I haven't heard of that,” right? So, it is not something brand new, but it is a concept that Ofelia García and some of her colleagues really brought forth to the field in the early 2000s … around 2009. And what it does is instead of saying English should be the center of everything, and everyone who doesn't just speak English is peripheral. It's saying, “Instead of putting English at the center, let's put our students' home language practices at the center. And what would that look like?” So, that wouldn't mean everything has to be in English. It wouldn't mean the teacher's language practices are front and center, and the students have to adapt to that. But it's about centering the students and then the teacher adapting to the languages and the language practices that the students bring. Teachers are there to have students use all the language at their resource—whatever language it is, whatever variety it is. And all those resources will help them learn. The more you can use, when we're talking about math, well, if we're teaching a concept and there are manipulatives there that will help students use them, why should we hide them? Why not bring them in and say, “OK, use this.” And once you have that concept, we can now scaffold and take things away little by little until you have it on your own. And the same thing with sometimes learning English. 

Tatyana: We should allow students to learn English as a new language using their home language resources. But one thing I will say is we should never take away their home language practices from the classroom. Even when they're fully bilingual, fully biliterate, it's still about, “How can we use these resources? How can they use that in their classroom?” Because we know in the world, speaking English is not enough. We're becoming more globalized, so let's have our students grow their language practices. And then students are allowed and proud of the language practices they bring. They teach their language practices to their peers, to their teachers. So, it's really hard to say it all in a couple of minutes, but I think the essence of translanguaging is centering students' language practices and then using that as a resource for them to learn and to grow, to learn languages and to learn content as well. 

Mike: How do you think that shifts the experience for a child? 

Tatyana: Well, if I think about my own experiences, you don't have to leave who you are at the door. We are not saying, “Home language is here, school language is there, and neither shall the two meet.” We're saying, “Language, and in the sense that it's a verb.” And when you can be your whole self, it allows you to have a stronger sense of who you are in order to really grow and learn and be proud of who you are. And I think that's a big part of it. I think when kids are bashful about who they are, thinking who they are isn't good enough, that has ripple effects in so many ways for them. So, I think we have to bring a lens of critical consciousness into these kind of spaces and make sure that our immigrant-origin students, their language practices, are centered through a translanguaging lens. 

Mike: It strikes me that it matters a lot how we as educators—internally, in the way that we think and externally, in the things that we do and the things that we say—how we position the child's home language, whether we think of it as an asset that is something to draw upon or a deficit or a barrier, that the way that we're thinking about it makes a really big difference in the child's experience. 

Tatyana: Yes, absolutely. Ofelia García, Kate Seltzer and Susana Johnson talk about a translanguaging stance. So, translanguaging is not just a practice or a pedagogy like, “Oh, let me switch this up, or let me say this in this language.” Yes, that's helpful, but it's how you approach who students are and what they bring. So, if you don't come from a stance of valuing multilingualism, it's not really going to cut it, right? It's something, but it's really about the stance. So, something that's really important is to change the culture of classrooms. So, just because you tell somebody like, “Oh, you can say this in your home language, or you can read this book side by side in Spanish and in English if it'll help you understand it.” Some students may not want to because they will think their peers will look down on them for doing it, or they'll think it means they're not smart enough. So, it's really about centering multilingualism in your classroom and celebrating it. And then as that stance changes the culture of the classroom, I can see students just saying, “Ah, no, no, no, I'm good in English.” Even though they may not fully feel comfortable in English yet, but because of the perception of what it means to be bilinguals. 

Mike: I'm thinking even about the example that you shared earlier where you said that an educator might say, “You can read this in Spanish side by side with English if you need to or if you want to.” But even that language of you can implies that, potentially, this is a remedy for a deficit as opposed to the ability to read in multiple languages as a huge asset. And it makes me think even our language choices sometimes will be a tell to kids about how we think about them as a learner and how we think about their language. 

Tatyana: That's so true, and how do we reframe that? “Let's read this in two languages. Who wants to try a new language?” Making this something exciting as opposed to framing it in a deficit way. So that's something that's so important that you picked up on. Yeah. 

Mike: Well, I think we're probably at the point in the conversation where there’s a lot of folks who are monolingual who might be listening and they're thinking to themselves, “This stance that we're talking about is something that I want to step into.” And now they're wondering what might it actually look like to put this into practice? Can we talk about what it would look like, particularly for someone who might be monolingual to both step into the stance and then also step into the practice a bit? 

Tatyana: Yes. I think the stance is really doing some internal reflection, questioning about what do I believe about multilingualism? What do I believe about people who come here, to come to the United States? In New York City, about half of our multilingual learners are U.S. born. So, it's not just immigrant students, but their parents, or they're often children of immigrants. So, really looking closely and saying, “How am I including respecting, valuing the languages of students regardless of where they come from?” And then, I think for the practice, it's about letting go of some control. As teachers, we are kind of control freaks. I can just speak for myself. ( laughs ) I like to know everything that's going on. 

Mike: I will add myself to that list, Tatyana. 

Tatyana: It's a long list. It's a long list. ( laughs ) But I think first of all, as educators, we have a sense when a kid is on task, and you can tell when a kid is not on task. You may not know exactly what they're saying. So, I think it's letting go of that control and letting the students, for example, when you are giving directions … I think one of the most dangerous things we do is we give directions in English when we have multilingual students in our classrooms, and we assume they understood it. If you don't understand the directions, the next 40 minutes will be a waste of time because you will have no idea what's happening. So, what does that mean? It means perhaps putting the directions into Google Translate and having it translate the different languages of your students. Will it be perfect? No. But will it be better than just being in English? A million times yes, right? 

Tatyana: Sometimes it's about putting students in same-language groups. If there are enough—two or three or four students that speak the same home language—and having them discuss something in their home language or multilingually before actually starting to do the work to make sure they're all on the same page. Sometimes it can mean if asking students if they do come from other countries, sometimes I'm thinking of math, math is done differently in different countries. So, we teach one approach, but what is another approach? Let's share that. Instead of having kids think like, “Oh, I came here, now this is the bad way. Or when I go home and I ask my family to help me, they're telling me all wrong.” No, again, these are the strengths of the families, and let's put them side by side and see how they go together. 

Tatyana: And I think what it's ultimately about is thinking about your classroom, not as a monolingual classroom, but as a multilingual classroom. And really taking stock of who are your students? Where are they and their families coming from, and what languages do they speak? And really centering that. Sometimes you may have students that may not tell you because they may feel like it's shameful to share that we speak a language that maybe other people haven't heard of. I'm thinking of indigenous languages from Honduras, like Garífuna, Miskito, right? Of course, Spanish, everyone knows that. But really excavating the languages of the students, the home language practices, and then thinking about giving them opportunities to translate if they need to translate. I'm not saying everything should be translated. I think word problems, having problems side by side, is really important. Because sometimes what students know is they know the math terms in English, but the other terms, they may not know those yet. 

Tatyana: And I'll give you one really powerful example. This is a million years ago, but it stays with me from my dissertation. It was in a Haitian Creole bilingual classroom. They were taking a standardized test, and the word problem was where it was like three gumballs, two gumballs, this color, what are the probability of a blue gumball coming out of this gumball machine? And this student just got stuck on gumball machine because in Haiti people sell gum, not machines, and it was irrelevant to the whole problem. So, language matters, but culture matters, too, right? So, giving students the opportunity to see things side by side and thinking about, “Are there any things here that might trip them up that I could explain to them?” So, I think it's starting small. It's taking risks. It's letting go of control and centering the students. 

Mike: So, from one recovering control freak to another, there are a couple of things that I'm thinking about. One is expanding a little bit on this idea of having two kids who might speak to one another in their home language, even if you are a monolingual speaker and you speak English and you don't necessarily have access to the language that they're using. Can you talk a little bit about that practice and how you see it and any guidance that you might offer around that? 

Tatyana: Yeah, I mean, it may not work the first time or the second time because kids may feel a little bit shy to do that. So maybe it's, “I want to try out something new in our class. I really am trying to make this a multilingual class. Who speaks another language here? Let's try … I am going to put you in a group and you're going to talk about this, and let's come back. And how did you feel? How was it for you? Let me tell you how I felt about it.” And it may be trying over a couple times because kids have learned that in most school settings, English is a language you should be using. And to the extent that some have been told not to speak any other language, I think it's just about setting it up and, “Oh, you two spoke, which language? Wow, can you teach us how to say this math term in this language?” 

Tatyana: “Oh, wow, isn't this interesting? This is a cognate, which means it sounds the same as the English word. And let's see if this language and this language, if the word means the same thing,” getting everyone involved in centering this multilingualism. And language is fun. We can play with language, we can put language side by side. So, then if you're labeling or if you have a math word wall, why not put key terms in all the languages that the students speak in the class and then they could teach each other those languages? So, I think you have to start little. You have to expect some resistance. But over time, if you keep pushing away at this, I think it will be good for not only your multilingual students, but all your students to say like, “Oh, wait a minute, there's all these languages in the world, but they're not just in the world. They're right here by my friend to the left and my friend to the right” and open up that space. 

Mike: So, I want to ask another question. What I'm thinking about is participation. And we've done an episode in the past around not privileging verbal communication as the only way that kids can communicate their ideas. We were speaking to someone who, their focus really was elementary years mathematics, but specifically, with multilingual learners. And the point that they were making was, kids gestures, the way that they use their hands, the way that they move manipulatives, their drawings, all of those things are sources of communication that we don't have to only say, “Kids understand things if they can articulate it in a particular way.” That there are other things that they do that are legitimate forms of participation. The thing that was in my head was, it seems really reasonable to say that if you have kids who could share an explanation or a strategy that they've come up with or a solution to a problem in their home language in front of the group, that would be perfectly legitimate. Having them actually explain their thinking in their home language is accomplishing the goal that we're after, which is can you justify your mathematical thinking? I guess I just wanted to check in and say, “Does that actually seem like a reasonable logic to follow that that's actually a productive practice for a teacher, but also a productive practice for a kid to engage in?” 

Tatyana: That makes a lot of sense. So, I would say for every lesson you, you may have a math objective, you may have a language objective, and you may have both. If your objective is to get kids to understand a concept in math or to explain something in math, who cares what language they do it in? It's about learning math. And if you're only allowing them to do it in a language that they are still developing in, they will always be about English and not about math. So, how do you take that away? You allow them to use all their linguistic resources. And we can have students explain something in their home language. There are now many apps where we could just record that, and it will translate it into English. If you are not a speaker of the language that the student speaks, you can have a peer then summarize what they said in English as well. So, there's different ways to do it. So yes, I think it's about thinking about the objectives or the objective of the lesson. And if you're really focusing on math, the language is really irrelevant. It's about explaining or showing what they know in math, and they can do that in any language. Or even without spoken language, but in written language artistically with symbols, et cetera. 

Mike: Well, and what you made me think, too, is for that peer, it's actually a great opportunity for them to engage with the reasoning of someone else and try to make meaning of it. So, there's a double bonus in it for that practice. 

Tatyana: Exactly. I think sometimes students don't really like listening to each other. They think they only need to listen to the teacher. So, I think this really has them listen to each other. And then sometimes summarizing or synthesizing is a really hard skill, and then doing it in another language is a whole other level. So, we're really pushing kids in those ways as well. So, there's many advantages to this approach. 

Mike: Yeah, absolutely. We have talked a lot about the importance of having kids engage with the thinking of other children as opposed to having the teacher be positioned as the only source of mathematical knowledge. So, the more that we talk about it, the more that I can see there's a lot of value culturally for a mathematics classroom in terms of showing that kids thinking matters, but also supporting that language development as well. 

Tatyana: Yes, and doing it is hard. As I said, none of this is easy, but it's so important. And I think when you start creating a multilingual classroom, it just has a different feel to it. And I think students can grow so much in their math, understanding it and in so many other ways. 

Mike: Absolutely. Well, before we close the interview, I invite you to share resources that you would recommend for an educator who's listening who wants to step into the stance of translanguaging, the practice of translanguaging, anything that you would offer that could help people continue learning. 

Tatyana: I have one hub of all things translanguaging, so this will make it easy for all the listeners. So, it is the CUNY New York State Initiative on Emergent Bilinguals. And let me just give you the website. It's C-U-N-Y [hyphen] N-Y-S-I-E-B.org. And I'll say that again. C-U-N-Y, N-Y-S-I-E-B.org, cuny-nysieb.org. That's the CUNY New York State Initiative on Emergent Bilinguals. And because it's such a mouthful, we just say “CUNY NYSIEB,” as you could tell by my own, trying to get it straight. You can find translanguaging resources such as guides. You can find webinars, you can find research, you can find books. Literally everything you would want around translanguaging is there in one website. Of course, there's more out there in the world. But I think that's a great starting point. There's so many great resources just to start with there. And then just start small. Small changes sometimes have big impacts on student learning and students' perceptions of how teachers view them and their families. 

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

Tatyana: Yes, it's been wonderful. Thank you so much. And we will just all try to let go a little bit of our control little by little.

Both: ( laugh) 

Tatyana: Because at the end of the day, we really don't control very much at all. ( laughs ) 

Mike: Agreed. ( chuckles ) Thank you. 

Tatyana: Thank you.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up

Season 2 | Episode 10 – Place Value

Guest: Dr. Eric Sisofo

Mike Wallus: If you ask an educator to share some of the most important ideas in elementary mathematics, I'm willing to bet that most would include place value on that list. But what does it mean to understand place value really? And what types of language practices and tools support students as they build their understanding? Today we're digging deep into the topic of place value with Dr. Eric Sisofo from the University of Delaware. 

Mike: Welcome to the podcast, Eric. We're glad to have you with us. 

Eric Sisofo: Thanks for having me, Mike. Really excited to be here with you today. 

Mike: I'm pretty excited to talk about place value. One of the things that's interesting is part of your work is preparing pre-service students to become classroom elementary teachers. And one of the things that I was thinking about is what do you want educators preparing to teach to understand about place value as they're getting ready to enter the field? 

Eric: Yeah, that's a really great question. In our math content courses at the University of Delaware, we focus on three big ideas about place value with our novice teachers. The first big idea is that place value is based on the idea of grouping a total amount of stuff or bundling a total amount of stuff into different size units. So, as you know, we use groups of ones, tens, hundreds, thousands and so on, not just ones in our base 10 system to count or measure a total amount of stuff. And we write a numeral using the digit 0 through 9 to represent the amount of stuff that we measured. So interestingly, our novice teachers come to us with a really good understanding of this idea for whole numbers, but it's not as obvious to them for decimal quantities. So, we spend a lot of time with our novice teachers helping them think conceptually about the different groupings, or bundlings, that they're using to measure a decimal amount of stuff. In particular, getting them used to using units of size: one-tenth, one-hundredth, one-thousandth, and so on. So, that's one big idea that really shines through whether you're dealing with whole numbers or decimal numbers, is that place value is all about grouping, or bundling, a total amount of stuff with very specific, different-size units. 

Eric: The second big idea we'd help our novice teachers make sense of at UD is that there's a relationship between different place value units. In particular, we want our novice teachers to realize that there's this 10 times relationship between place value units. And this relationship holds true for whole numbers and decimal numbers. So, 10 of one type of grouping will make one of the next larger-sized grouping in our decimal system. And that relationship holds true for all place value units in our place value system. So, there might be some kindergarten and first-grade teachers listening who try to help their students realize that 10 ones are needed to make one 10. And some second- and third-grade teachers who try to help their students see that 10 tens are needed to make 100. And 10 hundreds are needed to make 1,000, and so on. In fourth and fifth grade, we kind of extend that idea to decimal amounts. So, helping our students realize that 10 of these one-tenths will create a one. Or 10 of the one-hundredths are needed to make one-tenth, and so on and so on for smaller and smaller place value units. So, that's the second big idea.

Eric: And the third big idea that we explicitly discuss with our pre-service teachers is that there's a big difference between the face value of a digit and the place value of a digit. So, as you know, there are only 10 digits in our base 10 place value system. And we can reuse those digits in different places, and they take on a different value. So, for example, for the number 444, the same digit, 4, shows up three different times in the numeral. So, the face value is four. It's the same each digit in the numeral, but each four represents a different place value or a different grouping or an amount of stuff. So, for 444, the 4 in the hundreds place means that you have four groupings of size 100, the four in the tens place means you have four groupings of size 10, and the four in the ones place means you have four groupings of size one. 

Eric: So, this happens with decimal numbers, too. With our novice teachers, we spend a lot of time trying to get them to name those units and not just say, for example, 3.4 miles when they're talking about a numeral. We wouldn't want them to say 3.4. We instead want them to say three and four-tenths, or three ones and four-tenths miles. So, saying the numeral 3.4 focuses mostly just on the face value of those digits and removes some of the mathematics that's embedded in the numeral. So, instead of saying the numerals three ones and four-tenths or three and four-tenths really requires you to think about the face value and the place value of each digit. So those are the three big ideas that we discuss often with our novice teachers at the University of Delaware, and we hope that this helps them develop their conceptual understanding of those ideas so that they're better prepared to help their future students make sense of those same ideas.

Mike: You said a lot there, Eric. I'm really struck by the point two where you talk about the relationship between units, and I think what's hitting me is that I don't know that when I was a child learning mathematics—but even when I was an adult getting started teaching mathematics—that I really thought about relationships. I think about things like add a zero, or even the language of point-something. And how in some ways some of the procedures or the tricks that we've used have actually obscured the relationship as opposed to shining a light on it. Does that make sense? 

Eric: I think the same was true when I was growing up. That math was often taught to be a bunch of procedures or memorized kinds of things that my teacher taught me that I didn't really understand the meaning behind what I was doing. And so, mathematics became more of just doing what I was told and memorizing things and not really understanding the reasoning why I was doing it. Talking about relationships between things I think helps kids develop number sense. And so, when you talk about how 10 tenths are required to make 1 one, and knowing that that's how many of those one-tenths are needed to make 1 one, and that same pattern happens for every unit connected to the next larger unit, seeing that in decimal numbers helps kids develop number sense about place value. And then when they start to need to operate on those numerals or on those numbers, if they need to add two decimal numbers together and they get more than 10 tenths when they add down the columns or something like that in a procedure—if you're doing it vertically. If they have more of a conceptual understanding of the relationship, maybe they'll say, “Oh, I have more than 10 tenths, so 10 of those tenths will allow me to get 1 one, and I'll leave the others in the tens place,” or something like that. So, it helps you to make sense of the regrouping that's going on and develop number sense so that when you operate and solve problems with these numbers, you actually understand the reasoning behind what you're doing as opposed to just memorizing a bunch of rules or steps.

Mike: Yeah. I will also say, just as an aside, I taught kindergarten and first grade for a long time and just that idea of 10 ones and 1 ten, simultaneously, is such a big deal. And I think that idea of being able to say this unit is comprised of these equal-sized units, how challenging that can be for educators to help build that understanding. But how rich and how worthwhile the payoff is when kids do understand that level of equivalence between different sets of units. Eric: Absolutely, and it starts at a young age with children. And getting them to visualize those connections and that equivalence that a 10, 1 ten, can be broken up into these 10 ones or 10 ones can create 1 ten, and seeing that visually multiple times in lots of different situations really does pay off because that pattern will continue to show up throughout the grades. When you're going into second, third grade, like I said before, you’ve got to realize that 10 of these things we call tens, then we'll make a new unit called 100. Or 10 of these 100s will then make a unit that is called a thousand. And a thousand is equivalent to 10 hundreds. So, these ideas are really critical pieces of students understanding about place value when they go ahead and try to add or subtract with these using different strategies or the standard algorithm, they're able to break numbers up, or decompose, numbers into pieces that make sense to them. And their understanding of the mathematical relationships or ideas can just continue to grow and flourish. 

Mike: I'm going to stay on this for one more question, Eric, and then I think you're already headed to the place where I want to go next. What you're making me think about is this work with kids not as, “How do I get an answer today?” But “What role is my helping kids understand these place value relationships going to play in their long-term success?” 

Eric: Yeah, that's a great point. And learning mathematical ideas, it just doesn't happen in one lesson or in one week. When you have a complex idea like place value that … it spans over multiple years. And what kindergarten and first-grade teachers are teaching them with respect to the relationship, or the equivalence, between 10 ones and 1 ten is setting the foundation, setting the stage for the students to start to make sense of a similar idea that happens in second grade. And then another similar idea that happens in third grade where they continue to think about this 10 times relationship between units, but just with larger and larger groupings. And then when you get to fourth, fifth, sixth, seventh grade, you're talking about smaller units, units smaller than 1, and seeing that if we're using a decimal place value system, that there's still these relationships that occur. And that 10 times relationship holds true. And so, if we're going to help students make sense of those ideas in fourth and fifth grade with decimal units, we need to start laying that groundwork and helping them make sense of those relationships in the earlier grades as well. 

Mike: That's a great segue because I suspect there are probably educators who are listening who are curious about the types of learning activities that they could put into place that would help build that deeper understanding of place value. And I'm curious, when you think about learning activities that you think really do help build that understanding, what are some of the things that come to mind for you? 

Eric: Well, I'll talk about some specific activities in response to this, and thankfully there are some really high-quality instructional materials and math curricula out there that suggest some specific activities for teachers to use to help students make sense of place value. I personally think there are lots of cool instructional routines nowadays that teachers can use to help students make sense of place value ideas, too. Actually, some of the math curricula embed these instructional routines within their lesson plans. But what I love about the instructional routines is that they're fairly easy to implement. They usually don't take that much time, and as long as you do them fairly consistently with your students, they can have real benefits for the children's thinking over time. So, one of the instructional routines that could really help students develop place value ideas in the younger grades is something called “counting collections.” 

Eric: And with counting collections, students are asked to just count a collection of objects. It could be beans or paper clips or straws or unifix cubes, whatever you have available in your classroom. And when counting, students are encouraged to make different bundles that help them keep track of the total more efficiently than if they were just counting by ones. So, let's say we asked our first- or second-grade class to count a collection of 36 unifix cubes or something like that. And when counting, students can put every group of 10 cubes into a cup or make stacks of 10 cubes by connecting them together to represent every grouping of 10. And so, if they continue to make stacks of 10 unifix cubes as they count the total of 36, they'll get three stacks of 10 cubes or three cups of 10 cubes and six singletons. And then teachers can have students represent their count in a place value table where the columns are labeled with tens and ones. So, they would put a 3 in the tens column and a 6 in the ones column to show why the numeral 36 represents the total. So, giving students multiple opportunities to make the connection between counting an amount of stuff and using groupings of tens and ones, writing that numeral that corresponds to that quantity in a place value table, let's say, and using words like 3 tens and 6 ones will hopefully help students over time to make sense of that idea.

Mike: You're bringing me back to that language you used at the beginning, Eric, where you talked about face value versus place value. What strikes me is that counting collections task, where kids are literally counting physical objects, grouping them into, in the case you used tens, you actually have a physical representation that they've created themself that helps them think about, “OK, here's the face value. Where do you see this particular chunk of that and what place value does it hold?” That's a lovely, super simple, as you said, but really powerful way to kind of take all those big ideas—like 10 times as many, grouping, place value versus face value—and really touch all of those big ideas for kids in a short amount of time.

 Eric: Absolutely. What's nice is that this instructional routine, counting collections, can be used with older students, too. So, when you're discussing decimal quantities let's say, you just have to make it very clear what represents one. So, suppose we were in a fourth- or fifth-grade class, and we still wanted students to count 36 unifix cubes, but we make it very clear that every cup of 10 cubes, or every stack of 10 cubes, represents, let's say, 1 pound. Then every stack of 10 cubes represents 1 pound. So, every cube would represent just one-tenth of a pound. Then as the students count the 36 unifix cubes, they would still get three stacks of 10 cubes, but this time each stack represents one. And they would get six singleton cubes where each singleton cube represents one-tenth of a pound. So, if you have students represent this quantity in a place value table labeled ones and tenths, they still get 3 in the ones place this time and 6 in the tenths place. So over time, students will learn that the face value of a digit tells you how many of a particular-size grouping you need, and the place value tells you the size of the grouping needed to make the total quantity.

Mike: That totally makes sense.

Eric: I guess another instructional routine that I really like is called “choral counting.” And with coral counting, teachers ask students to count together as a class starting from a particular number and jumping either forward or backward by a particular amount. So, for example, suppose we ask students to start at 5 and count by tens together. The teacher would record their counting on the board in several rows. And so, as the students count together, saying “5 15, 25, 35,” and so on, the teacher's writing these numerals across the board. He or she puts 10 numbers in a row. That means that when the students get to 105, the teacher starts a new row beginning at 105 and records all the way to 195, and then the third row would start at 205 and go all the way to 295. And after a few rows are recorded on the board, teachers could ask students to look for any patterns that they see in the numerals on the board and to see if those patterns can help them predict what number might come in the next row.

Eric: So, students might notice that 10 is being added across from one number to the next going across, or 100 is being added down the columns. Or 10 tens are needed to make a hundred. And having students notice those patterns and discuss how they see those patterns and then share their reasoning for how they can use that pattern to predict what's going to happen further down in the rows could be really helpful for them, too. Again, this can be used with decimal numbers and even fractional numbers. So, this is something that I think can also be really helpful, and it's done in a fun and engaging way. It seems like a puzzle. And I know patterns are a big part of mathematics and coral counting is just a neat way to incorporate those ideas.

Eric: Yeah, I've seen people do things like counting by unit fractions, too, and in this case counting by tenths, right? One-tenths, two-tenths, three-tenths, and so on. And then there's a point where the teacher might start a new column and you could make a strategic choice to say, “I'm going to start a new column when we get to ten-tenths.” Or you could do it at five-tenths. But regardless, one of the things that's lovely is choral counting can really help kids see structure in a way that counting out loud, if it doesn't have the, kind of, written component of building it along rows and columns, it's harder to discern that. You might hear it in the language, but choral accounting really helps kids see that structure in a way that, from my experience at least, is really powerful for them.

Eric: And like you said, the teacher, strategically, chooses when to make the new row happen to help students, kind of, see particular patterns or groupings. And like you said, you could do it with fractions, too. So even unit fractions: zero, one-seventh, two-sevenths, three-sevenths, four-sevenths all the way to six-sevenths. And then you might start a new row at seven-sevenths, which is the same as 1. And so, kind of realize that, “Oh, I get a new 1 when I regroup 7 of these sevenths together.” And so, with decimal numbers, I need 10 of the one-tenths to get to 1. And so, if you help kids, kind of, realize that these numerals that we write down correspond with units and smaller amounts of stuff, and you need a certain amount of those units to make the next-sized unit or something like that, like I said, it can go a long way even into fractional or decimal kinds of quantities.

Mike: I think you're taking this conversation in a place I was hoping it would go, Eric, because to be autobiographical, one thing that I think is an advance in the field from the time when I was learning mathematics as a child is, rather than having just a procedure with no visual or manipulative support, we have made progress using a set of manipulative tools. And at the same time, there's definitely nuance to how manipulatives might support kids' understanding of place value and also ways where, if we're not careful, it might actually just replace the algorithm that we had with a different algorithm that just happens to be shaped like cubes. What I wanted to unpack with you is what's the best-case use for manipulatives? What can manipulatives do to help kids think about place value? And is there any place where you would imagine asking teachers to approach with caution?

Eric: Well, yeah. To start off, I'll just begin by saying that I really believe manipulatives can play a critical role in developing an understanding of a lot of mathematical ideas, including place value. And there's been a lot of research about how concrete materials can help students visualize amounts of stuff and visualize relationships among different amounts of stuff. And in particular, research has suggested that the CRA progression, have you heard of CRA before?

 Mike: Let me check. Concrete, Representational and Abstract. Am I right?

Eric: That's right. So, because “C,” the concrete representation, is first in this progression, this means that we should first give students opportunities to represent an amount of stuff with concrete manipulatives before having them draw pictures or write the amount with a numeral. To help kindergarten and first-grade students begin to develop understandings of our base 10 place value system, I think it's super important to maybe use unifix cubes to make stacks of 10 cubes. We could use bundles of 10 straws wrapped up with a rubber band and singleton straws. We could use cups of 10 beans and singleton beans … basically use any concrete manipulative that allows us to easily group stuff into tens and ones and give students multiple opportunities to understand that grouping of tens and ones are important to count by. And I think at the same time, making connections between the concrete representation, the “C” in CRA, and the abstract representation, the “A,” which is the symbol or the numeral we write down, is so important.

Eric: So, using place value tables, like I was saying before, and writing the symbols in the place value table that corresponds with the grouping that children used with the actual stuff that they counted will help them over time make sense that we use these groupings of tens and ones to count or measure stuff. And then in second grade, you can start using base 10 blocks to do the same type of thing, but for maybe groupings of hundreds, thousands, and beyond. And then in fourth and fifth grade, base 10 blocks are really good for tenths and hundredths and ones, and so on like that. But for each of these, making connections between the concrete stuff and the abstract symbols that we use to represent that stuff. So, one of the main values that concrete manipulatives bring to the table, I think, is that they allow students to represent some fairly abstract mathematical ideas with actual stuff that you can see and manipulate with your hands.

Eric: And it allows students to get visual images in their heads of what the numerals and the symbols mean. And so, it brings meaning to the mathematics. Additionally, I think concrete manipulatives can be used to help students really make sense of the meaning of the four operations, too, by performing actions on the concrete stuff. So, for example, if we're modeling the meaning of addition, we can use concrete manipulatives to represent the two or more numerals as amounts of stuff and show the addition by actually combining all the stuff together and then figuring out, “Well, how much is this stuff altogether?” And then if we're going to represent this with a base 10 numeral, we got to break all the stuff into groupings that base 10 numerals use. So, ones, tens, hundreds if needed, tenths, hundredths, thousandths. And one thing that you said that maybe we need to be cautious about is we don't want those manipulatives to always be a crutch for students, I don't think. So, we need to help students make the transition between those concrete manipulatives and abstract symbols by making connections, looking at similarities, looking at differences.

Eric: I guess another concern that educators should be aware of is that you want to be strategic, again, which manipulatives you think would match the students’ development in terms of their mathematical thinking? So, for example, I probably wouldn't use base 10 blocks in kindergarten or first grade, to be honest. When students are just learning about tens and ones, because the long in a base 10 block is already put together for them. The 10-unit cubes are already formed into a long. So, some of the cognitive work is already done for them in the base 10 blocks, and so you're kind of removing some of the thinking. And so that's why I would choose unifix cubes over base 10 blocks, or I would choose straws to, kind of, represent this relationship between ones and tens in those early grades before I start using base 10 blocks. So, those are two things that I think we have to be thoughtful about when we're using manipulatives.

Mike: My wife and I have this conversation very often, and it's fascinating to me. I think about what happens in my head when a multi-edition problem gets posed. So, say it was 13 plus 46, right? In my head, I start to decompose those numbers into place value chunks, and in some cases I'll round them to compensate. Or in some cases I'll almost visualize a number line, and I'll add those chunks to get to landmarks. And she'll say to me, “I see the standard algorithm with those two things lined up.” And I just think to myself, “How big of a gift we're actually giving kids, giving them these tools that can then transfer.” Eventually they become these representations that happen in their heads and how much more they have in their toolbox when it comes to thinking about operating than many of us did who grew up learning just a set of algorithms.

Eric: Yeah, and like you said, decomposing numerals or numbers into place value parts is huge because the standard algorithm does the same thing. When you're doing the standard addition algorithm in vertical form, you're still adding things up, and you're breaking the two numbers up by place value. It's just that you're doing it in a very specific way. You're starting with the smallest unit first, and you add those up, and if you get more than 10 of that particular unit, then you put a little 1 at the top to represent, “Oh, I get one of the next size unit because 10 of one unit makes one of the next size.” And so, it's interesting how the standard algorithm kind of flows from some of these more informal strategies that you were talking about—decomposing or compensating or rounding these numbers and other strategies that you were talking about—really, I think help students understand, and manipulatives, too, help students understand that you can break these numbers up into pieces where you can figure out how close this amount of stuff is to another amount of stuff and round it up or round it down and then compensate based off of that. And that helps prepare students to make sense of those standard algorithms when we go ahead and teach those.

Mike: And I think you put your finger on the thing. I suspect that some people would be listening to this and they might think, “Boy, Mike really doesn't like the standard algorithm.” What I would say is, “The concern I have is that oftentimes the way that we've introduced the algorithm obscures the place value ideas that we really want kids to have so that they're actually making sense of it.” So, I think we need to give kids options as opposed to giving them one way to do it, and perhaps doing it in a way that obscures the mathematics.

Eric: And I'm not against the standard algorithm at all. We teach the standard algorithms at the University of Delaware to our novice teachers and try to help them make sense of those standard algorithms in ways that talk about those big ideas that we've been discussing throughout the podcast. And talking about the place values of the units, talking about how when you get 10 of a particular unit, it makes one of the next-size unit. And thinking about how the standard algorithm can be taught in a more conceptual way as opposed to a procedural, memorized kind of set of steps. And I think that's how it sounds like you were taught the standard algorithm, and I know I was taught that, too. But giving them the foundation with making sense of the mathematical relationships between place value units in the early grades and continuing that throughout, will help students make sense of those standard algorithms much more efficiently and soundly.

Mike: Yeah, absolutely. One of the pieces that you started to talk about earlier is how do you help bring meaning to both place value and, ultimately, things like standard algorithms. I'm thinking about the role of language, meaning the language that we use when we talk in our classrooms, when we talk about numbers and quantities. And I'm wondering if you have any thoughts about the ways that educators can use language to support students understanding of place value?

Eric: Oh, yeah. That's a huge part of our teaching. How we as teachers talk about mathematics and how we ask our students to communicate their thinking, I think is a critical piece of their learning. As I was saying earlier, instead of saying 3.4, but expecting students to say three and four-tenths, can help them make sense of the meaning of each digit and the total value of the numeral as opposed to just saying 3.4. Another area of mathematics where we tend to focus on the face value of digits, like I was saying before, rather than the place value, is when we teach the standard algorithms. So, it kind of connects again. I believe it's really important that students and teachers alike should think about and use the place value words of the digits when they communicate their reasoning. So, if we're adding 36 plus 48 using the standard addition algorithm and vertical format, we start at the right and say, “Well, 6 plus 8 equals 14, put the 4 carry the 1 … but what does that little 1 represent, is what we want to talk about or have our students make sense of. And it's actually the 10 ones that we regrouped into 1 ten.

Eric: So, we need to say that that equivalence happened or that regrouping or that exchange happened, and talk about how that little 1 that's carried over is actually the 1 ten that we got and not just call it a 1 that we carry over. So, continuing with the standard algorithm for 36 plus 48, going over to the tens column, we usually often just say, “Three plus 4 plus the 1 gives us 8,” and we put down the 8 and get the answer of 84. But what does the 3 and the 4 and the 1 really represent? “Oh, they're all tens.” So, we might say that we're combining 3 tens, or 30, with 4 tens, or 40. And the other 10 that we got from the regrouping to get 8 tens, or 80, as opposed to just calling it 8.

Eric: So, talking about the digits in this way and using the place value meaning, and talking about the regrouping, all of this is really bringing meaning to what's actually happening mathematically. That's a big part of it. I guess to add onto that, when I was talking about the standard algorithm, I didn't use the words “add” or “plus,” I was saying “put together,” “combine,” to talk about the actual action of what we're doing with those two amounts of stuff. Even that language is, I think, really important. That kind of emphasizes the action that we're taking when we're using the plus symbol to put two things together. And also, I didn't say “carry.” Instead, I said, we want to “regroup” or “exchange” these 10 ones for 1 ten. So, I'm a big believer in using language that tries to precisely describe the mathematical ideas accurately because I just have seen over and over again how this language can benefit students' understanding of the ideas, too.

Mike: I think what strikes me, too, is that the kinds of suggestions you're talking about in terms of describing the units, the quantities, the actions, these are things that I hope folks feel like they could turn around and use tomorrow and have an immediate impact on their kids.

Eric: I hope so, too. That would be fantastic.

Mike: Well, before we close the interview, I wanted to ask you, for many teachers thinking about things like place value or any big idea that they're teaching, often is kind of on the job learning and you're learning along with your kids, at least initially. So, I wanted to step back and ask if you had any recommendations for an educator who's listening to the podcast. If there are articles, books, things, online, particular resources that you think would help an educator build that understanding or think about how to build that understanding with their students?

Eric: Yeah. One is to listen to podcasts about mathematics teaching and learning like this one. There's a little plug for you, Mike.

Both: (laugh)

Eric: I guess …

Mike: I'll take it.

Eric: Yeah! Another way that comes to mind is if your school uses a math curriculum that aims to help students make sense of ideas, often the curriculum materials have some mathematical background pages that teachers can read to really deepen their understanding of the mathematics. There's some really good math curricula out there now that can be really educative for teachers. I think teachers also can learn from each other. I believe teachers should collaborate with each other, talk about teaching specific lessons with each other, and through their discussions, teachers can learn from one another about the mathematics that they teach and different ways that they can try to help their students make sense of some of those ideas. Another thing that I would suggest is to become a member of an organization like NCTM, the National Council of Teachers of Mathematics. I know NCTM has some awesome resources for practitioners to help teachers continue to learn about mathematical ideas and different ways to teach particular ideas to kids. And you can attend a regional or national conference with some of these organizations.

Eric: I know I've been to several of them, and I always learn some really great ideas about teaching place value or fractions or early algebraic thinking. Whatever it is, there's so many neat ideas that you can learn from others. I've been teaching math for so many years. What's cool is that I'm still learning about math and how to teach math in effective ways, and I keep learning every day, which is really one of the fun things about teaching as a profession. You just keep learning. So, I guess one thing I would suggest is to keep plugging away. Stay positive as you work through any struggles you might experience, and just know that we all wrestle with parts of teaching mathematics especially. So, stay curious and keep working to make sense of those concepts that you want your students to make sense of so that they can be problem-solvers and thinkers and sensemakers.

Mike: I think it's a great place to leave it. Eric, thank you so much for joining us. It's really been a pleasure talking to you.

Eric: Thanks, Mike. It's been a pleasure.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up

Season 2 | Episode 9 – Instructional & Assessment Practices

Guest: Dr. Kim Morrow-Leong

Mike Wallus: What are the habits of mind that educators can adopt to be more responsive to our students' thinking? And how can we turn these habits of mind into practical steps that we can take on a regular basis? Dr. Kim Morrow-Leong has some thoughts on this topic. Today, Kim joins the podcast, and we'll talk with her about three mental shifts that can profoundly impact educators instructional and assessment practices. 

Mike: Kim Morrow-Leong, welcome to the podcast. We're excited to have you.

Kim Morrow-Leong: Thank you, Mike. It's nice to be here.

Mike: I'm really excited to talk about the shifts educators can make to foster responsive interpretations of student thinking. This is an idea that for me has been near and dear for a long time, and it's fun to be able to have this conversation with you because I think there are some things we're going to get into that are shifts in how people think. But they're also practical. You introduced the shift that you proposed with a series of questions that you suggested that teachers might ask themselves or ask their colleagues, and the first question that you posed was, “What is right?” And I'm wondering what do you mean when you suggest that teachers might ask themselves or their colleagues this question when they're interpreting student thinking?

Kim: So, I'm going to rephrase your question a little bit and change the emphasis to say, “What is right?” And the reason I want to change the emphasis of that is because we often talk about what is wrong, and so rather than talking about what is wrong, let's talk about what's right. When we look at student work, it's a picture. It's a snapshot of where they are at that particular moment. And the greater honesty that we can bring to that situation to understand what their thinking is, the better off we're going to be. So, there's a lot of talk lately about asset-based instruction, asset-based assessment, and I think it's a great initiative and it really gets us thinking about how we can think about what students are good at and what they bring to the table or what they bring to the classroom culture. But we don't often talk a lot about how we do that, how we break the mold. Because many of our metaphors and our language about learning are linear, and they indicate that students are moving from somewhere to achieve a goal somewhere down the path, somewhere down the line.

Kim: How do you switch that around? Well, rather than looking at what they're missing and what part of the path they haven't achieved yet, we can look at where they are at the moment because that reflects everything they've learned up to that moment. So, one of the ways we can do this is to unpack our standards a little more carefully, and I think a lot of people are very good at looking at what the skills are and what our students need to be able to do by the end of the year. But a lot of what's behind a standard are concepts. What are some big ideas that must be in place for students to be successful with the skills? So, I'm going to give a very specific example. This one happens to be about a fourth-grade question that we've asked before in a district I used to work at. The task is to sketch as many rectangles as you can that are 48 square units.

Kim: There's some skills behind this, but understanding what the concepts are is going to give us a little more insight into student thinking. So, one of the skills is to understand that there are many ways to make 48: to take two factors and multiply them together and only two factors, and to make a product of 48 or to get the area. But a concept behind that is that 48 is the product of two numbers. It's what happens when you multiply one dimension by the other dimension. It's not the measure of one of the dimensions. That's a huge conceptual idea for students to sort out what area is and what perimeter is, and we want to look for evidence of what they understand about the differences between what the answer to an area problem is and what the answer to, for example, a perimeter problem is. Another concept is that area indicates that a space is covered by squares.

Kim: The other big concept here is that this particular question is going to have more than one answer. You're going to have 48 as a product, but you could have six times eight and four times 12 and many others. So that's a lot of things going into this one, admittedly very rich, task for students to take in. One of the things I've been thinking a lot about lately is this idea of a listening stance. So, a listening stance describes what you're listening for. It describes how you're listening. Are you listening for the right answer? Are you listening to understand students' thinking? Are you listening to respond or are you listening to hear more—and asking for more information from your student or really from any listener? So, one of the ways we could think about that, and perhaps this sounds familiar to you, is you could have what we call an evaluative listening stance.

Kim: An evaluative listening stance is listening for the right answer. As you listen to what students say, you're listening for the student who gives you the answer that you're looking for. So, here's an example of something you might see. Perhaps a student covers their space and has dimensions for the rectangle of seven times six, and they tell you that this is a space that has an area of 48 square units. There's something right about that. They are really close. Because you can look at their paper and you can see squares on their paper and they're arranged in an array and you can see the dimensions on this side and the dimensions on that side, and you can see that there's almost 48 square units. I know we all can see what's wrong about that answer, but that's not what we're thinking about right now. We're thinking about what's right. And what's right is they covered that space with an area that is something by six. This is a great place to start with this student to figure out where they got that answer. If you're listening evaluatively, that's a wrong answer and there's nowhere else to go. So, when we look at what is right in student work, we're looking for the starting point. We're looking for what they know so that we can begin there and make a plan to move forward with them. You can't change where students are unless you meet them where they are and help them move forward.

Mike: So, the second question that you posed was, “Can you cite evidence for what you're saying?” So again, talk us through what you're asking, when you ask teachers to pose this question to themselves or to their colleagues.

Kim: Think about ways that you might be listening to a student's answer and very quickly say, “Oh, they got it,” and you move on. And you grab the next student's paper or the next student comes up to your desk and you take their work and you say, “Tell me what you're thinking.” And they tell you something. You say, “That's good,” and you move to the next one. Sometimes you can take the time to linger and listen and ask for more and ask for more and ask for more information. Teachers are very good at gathering information, at a glance. We can look at a stack of papers and in 30 seconds get a good snapshot of what's happening in that classroom. But in that efficiency we lose some details. We lose information about specifics, about what students understand, that we can only get by digging in and asking more questions.

Kim: Someone once told me that every time a student gives an answer, you should follow it with, “How do you know?” And somebody raised their hand and said, “Well, what if it's the right answer?” And the presenter said, “Oh, you still ask it. As a matter of fact, that's the best one to ask. When you ask, ‘How do you know?’ you don't know what you're going to hear, you have no idea what's going to happen.” And sometimes those are the most delightful surprises, is to hear some fantastical creative way to solve a problem that you never would've thought about. Unless you ask, you won't hear these wonderful things. Sometimes you find out that a correct answer has some flawed reasoning behind it. Maybe it's reasoning that only works for that particular problem, but it won't work for something else in the future. You definitely want to know that information so that you can help that student rethink their reasoning so that the next time it always works.

Kim: Sometimes you find out the wrong answers are accidents. They're just a wrong computation. Everything was perfect up until the last moment and they said three times two is five, and then they have a wrong answer. If you don't ask more either in writing or verbally, you have incorrect information about that student's progress, their understandings, their conceptual development and even their skills. That kind of thing happens to everyone because we're human. By asking for more information, you're really getting at what is important in terms of student errors and what is not important, what is just easily fixable. I worked with a group of teachers once to create some open-ended tasks that require extended answers, and we sat down one time to create rubrics. And we did this with student work, so we laid them all out and someone held up a paper and said, “This is it!”

Kim: “This student gets it.” And so, we all took a copy of this work and we looked at it. And we were trying to figure out what exactly does this answer communicate that makes sense to us? That seems to be an exemplar. And so, what we did was we focused on exactly what the students said. We focused on the evidence in front of us. This one was placing decimal numbers on a number line. We noted that the representation was accurate, that the position of the point on the number line was correct. We noticed that the label on the point matched the numbers in the problem, so that made sense. But then all of a sudden somebody said, “Well, wait a minute. There's an answer here, but I don't know how this answer got here.” Something happened, and there's no evidence on the page that this student added this or subtracted this, but magically the right answer was there. And it really drove home for this group—and for me, it really stuck with me—the idea that you can see a correct answer but not know the thinking behind it.

Kim: And so, we learned from that point on to always focus on the evidence in front of us and to make declarative statements about what we saw, what we observed, and to hold off on making inferences. We saved our inferences for the end. After I had this experience with the rubric grading and with this group of teachers and coaches, I read something about over attribution and under attribution. And it really resonated with me. Over attribution is when you make the claim that a student understands something when there really isn't enough evidence to make that statement. It doesn't mean that's true or not true, it means that you don't have enough information in front of you. You don't have enough evidence to make that statement. You over attribute what it is they understand based on what's in front of you. Similarly, you get under attribution. You have a student who brings to you a drawing or a sketch or a representation of some sort that you don't understand because you've never seen anybody solve a problem this way before.

Kim: You might come to the assumption that this student doesn't understand the math task at hand. That could be under attribution. It could be that you have never seen this before and you have not yet made sense of it. And so, focusing on evidence really gets us to stop short of making broad, general claims about what students understand, making broad inferences about what we see. It asks us to cite evidence to be grounded in what the student actually put on the paper. For some students, this is challenging because they mechanically have difficulties putting things on paper. But we call a student up to our desk and say, “Can you tell me more about what you've done here? I'm not following your logic.” And that's really the solution is to ask more questions. I know, you can't do this all the time. But you can do it once in a while, and you can check yourself if you are assigning too much credit for understanding to a student without evidence. And you can also check yourself and say, “Hmm, am I not asking enough questions of this student? Is there something here that I don't understand that I need to ask more about?”

Mike: This is really an interesting point because what I'm finding myself thinking about is my own practice. What I feel like you're offering is this caution, which says, “You may have a set of cumulative experiences with children that have led you to a set of beliefs about their understanding or how they come to understanding. But if we're not careful—and even sometimes even if we are careful—we can bring that in a way that's actually less helpful, less productive,” right? It's important to look at things and actually say, “What's the evidence?” Rather than, “What's the body of my memory of this child's previous work?” It's not to say that that might not have value, but at this particular point in time, “What's the evidence that I see in front of me?”

Kim: That's a good point, and it reminds me of a practice that we used to have when we got together and assessed these open-ended tasks. The first thing we would do is we put them all in the middle of the table and we would not look at our own students' work. That's a good strategy if you work with a team of people, to use these extended assessments or extended tasks to understand student thinking, is to share the load. You put them all out there. And the other thing we would do is we would take the papers, turn them over and put a Post-it note on the back. And we would take our own notes on what we saw, the evidence that we saw. We put them on a Post-it note, turn them over and then stick the Post-it note to the back of the work. There are benefits to looking at work fresh without any preconceived notions that you bring to this work. There are other times when you want all that background knowledge. My suggestion is that you try it differently, that you look at students' work for students you don't know and that you not share what you're seeing with your colleagues immediately, is that you hold your opinions on a Post-it to yourself, and then you can share it afterwards. You can bring the whole conversation to the whole table and look at the data in front of you and discuss it as a team afterwards. But to take your initial look as an individual with an unknown student.

Mike: Hmm. I'm going to jump to the third shift that you suggest, which is less of a question and more of a challenge. You talk about the idea of moving from anticipating to targeting a learning trajectory, and I'm wondering if you could talk about what that means and why you think it's important.

Kim: Earlier we talked about how important it is to understand and unpack our standards that we're teaching so that we know what to look for. And I think the thing that's often missed, particularly in standards in the older grades, is that there are a lot of developmental steps between, for example, a third-grade standard and a fourth-grade standard. There are skills and concepts that need to grow and develop, but we don't talk about those as much as perhaps we should. Each one of those conceptual ideas we talked about with the area problem we discussed may come at different times. It may not come during the unit where you are teaching area versus perimeter versus multiplication. That student may not come to all of those conceptual understandings or acquire all of the skills they need at the same time, even though we are diligently teaching it at the same time.

Kim: So, it helps to look at third grade to understand, what are these pieces that make up this particular skill? What are the pieces that make up the standard that you're trying to unpack and to understand? So, the third shift in our thinking is to let go of the standard as our goal, but to break apart the standard into manageable pieces that are trackable because really our standards mean by the end of the year. They don't mean by December, they mean by the end of the year. So that gives you the opportunity to make choices. What are you going to do with the information you gather? You've asked what is right about student work. You've gathered evidence about what they understand. What are you going to do with that information? That perhaps is the hardest part. There's something out there called a learning trajectory that you've mentioned.

Kim: A learning trajectory comes out of people who really dig in and understand student thinking on a fine-grain level, how students will learn … developmentally, what are some ideas they will develop before they develop other ideas? That's the nature of a learning trajectory. And sometimes those are reflected in our standards. The way that kindergartners are asked to rote count before they're asked to really understand one-to-one correspondence. We only expect one-to-one correspondence up to 20 in kindergarten, but we expect counting up to a hundred because we acknowledge that that doesn't come at the same time. So, a learning trajectory to some degree is built into your standards. But as we talked about earlier, there are pieces and parts that aren't outlined in your standards. One of the things we know about students and their interactions with grids and arrays is that a student might be able to recognize an array that is six by eight, but they may not yet be able to draw it.

Kim: The spatial structuring that's required to create a certain number of lines going vertically and a certain number of lines going horizontally may not be in place. At the same time, they are reading a arrays and understanding what they mean. So, the skill of structuring the space around you takes time. The task where we ask them to draw these arrays is asking something that some kids may not yet be able to do, to draw these grids out. If we know that we can give them practice making arrays, we can give them tools to make arrays, we can give them blocks to make arrays, and we can scaffold this and help them move forward. What we don't want to assume is that a student who cannot yet make a six by eight array can't do any of it because that's not true. There's parts they can do. So, our job as teachers is to look at what they do, look carefully at the evidence of what they do, and then make a plan. Use all of that skill and experience that's on our teams. Even if you're a new teacher, all those people on your teams know a lot more than they're letting on, and then you can make a plan to move forward and help that student make these smaller steps so they can reach the standard.

Mike: When we talked earlier, one of the things that you really shifted for me was some of the language that I found myself using. So, I know I have been in the habit of using the word “misconception” when we're talking about student work. And the part of the conversation that we had that really has never left me is this idea of, what do we actually mean when we say “misconception”? Because I found that the more I reflected on it, I used that language to describe a whole array of things that kids were doing, and not all of them were what I think a misconception actually is. Can you just talk about this language of misconception and how we use it and perhaps what we might use instead to be a little bit more precise?

Kim: I have stopped using the word misconception myself. Students understand what they understand. It's our job to figure out what they do understand. And if it's not at that mature level we need it to be for them to understand the concept, what disequilibrium do I need to introduce to them? I'm borrowing from Piaget there. You have to introduce some sort of challenge so that they have the opportunity to restructure what it is they understand. They need to take their current conception, change it with new learning to become a new conception. That's our teaching opportunity right there. That's where I have to start.

Mike: Before we close, I have to say one of the big takeaways from this conversation is the extent to which the language that I use, and I mean literally what I say to myself internally or what I say to my colleagues when we're interpreting student work or student thinking, that that language has major implications for my instruction and that the language that surrounds my assessing, my interpreting and my planning habits really matters.

Kim: It does. You are what you practice. You are what you put forth into the world. And to see a truly student-centered point of view requires a degree of empathy that we have to learn.

Mike: So, before you go, Kim, I'm wondering, can you share two or three resources that have really shaped your thinking on the interpretation of student learning?

Kim: Yes, I could. And one of them is the book, “Children's Mathematics.” There's a lot of information in this book, and if you've ever engaged with the work of cognitively guided instruction, you're familiar with the work in this book. There's plenty of content knowledge, there's plenty of pedagogical content knowledge in this book. But the message that I think is the most important is that everything they learn, they learn by listening. They listen to what students were saying. And the second piece is called “Warning Signs!” And this one is one of my favorites. And in this book, they give three warning signs that you as a teacher are taking over students' learning. And one example that comes to mind for me is you take the pencil from the student. It's such a simple thing that we would just take it into quickly get something out, but to them, they expressed that that's a warning sign that you're about to take over their thinking. So, I highly recommend that one. And there's another one that I always recommend. It was published in Mathematics Teaching in the Middle School. It's called “Never Say Anything a Kid Can Say!” That's a classic. I highly recommend it if you've never read it.

Mike: Kim Morrow-Leong, thank you for joining us. It's really been a pleasure.

Kim: Mike, thank you for having me. This has been delightful.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

© 2024 The Math Learning Center | www.mathlearningcenter.org

Rounding Up

Season 2 | Episode 7 – Making Sense of Fractions

Guest: Dr. Susan Empson

Mike Wallus: For quite a few adults, fractions were a stumbling block in their education that caused many to lose their footing and begin to doubt their ability to make sense of math. But this doesn't have to be the case for our students. Today on the podcast, we're talking with Dr. Susan Empson about big ideas and fractions and how we can make them more meaningful for our students. Welcome to the podcast. Susan. Thanks for joining us.

Susan Empson: Oh, it's so great to be here. Thank you for having me.

Mike: So, your book was a real turning point for me as an educator, and one of the things that it did for me at least, it exposed how little that I actually understood about the meaning of fractions. And I say this because I don't think that I'm alone in saying that my own elementary school experience was mostly procedural. So rather than attempting to move kids quickly to procedures, what types of experiences can help children build a more meaningful understanding of fractions?

Susan: Great question. Before I get started, I just want to acknowledge my collaborators because I've had many people that I've worked with. There's Linda Levi, co-author of the book, and then my current research partner, Vicki Jacobs. And of course, we wouldn't know anything without many classroom teachers we've worked with in the current and past graduate students. In terms of the types of experiences that can help children build more meaningful experiences of fractions, the main thing we would say is to offer opportunities that allow children to use what they already understand about fractions to solve and discuss story problems. Children's understandings are often informal and early on, for example, may consist mainly partitioning things in half. What I mean by informal is that understandings emerge in situations out of school. So, for example, many children have siblings and have experienced situations where they have had to share, let's say three cookies or slices of pizza between two children. In these kinds of situations, children appreciate the need for equal shares, and they also develop strategies for creating them. So, as children solve and discuss story problems in school, their understandings grow. The important point is that story problems can provide a bridge between children's existing understandings and new understandings of fractions by allowing children to draw on these informal experiences. Generally, we recommend lots of experiences with story problems before moving on to symbolic work to give children plenty of opportunity to develop meaningful fractions. And we also recommend using story problems throughout fraction instruction. Teachers can use different types of story problems and adjust the numbers in those problems to address a range of fraction content. There are also ideas that we think are foundational to understanding fractions, and they're all ideas that can be elicited and developed as children engage in solving and discussing story problems. 

Susan: So, one idea is that the size of a piece is determined by its relationship to the whole. What I mean is that it's not necessarily the number of pieces into which a whole is partitioned that determines the size of a piece. Instead, it's how many times the piece fits into the whole. So, in their problem-solving, children create these amounts and eventually name them and symbolize them as unit fractions. That's any fraction with 1 in the numerator.

Mike: You know, one of the things that stands out for me in that initial description that you offered, is this idea of kids don't just make meaning of fractions at school, that their informal lived experiences are really an asset that we can draw on to help make sense of what a fraction is or how to think about it.

Susan: That's a wonderful way to say it. And absolutely, the more teachers get to know the children in their classrooms and the kinds of experiences those children might have outside of school, the more of that can be incorporated into experiences like solving story problems in school.

Mike: Well, let's dig into this a little bit. Let's talk a little bit about the kinds of story problems or the structure that actually provides an entry point and can build understanding of fractions for students. Can you talk a bit about that, Susan?

Susan: Yes. So, I'll describe a couple types of story problems that we have found especially useful to elicit and develop children's fraction understandings. So first, equal sharing story problems are a powerful type of story problem that can be used at the beginning of and even throughout instruction. These problems involve sharing multiple things among multiple sharers. So, for example, four friends equally sharing 10 oranges. How much orange would each friend get? Problems like this one allow children to create fractional amounts by drawing things, partitioning those things, and then attaching fraction names and symbols. So, let's [talk] a little bit about how a child might solve the oranges problem. A child might begin by drawing four friends and then distributing whole oranges one by one until each friend has two whole oranges. Now, there are two oranges left and not enough to give each friend another whole orange. So, they have to think about how to partition the remaining oranges.

Susan: They might partition each orange in half and give one more piece to each friend, or they might partition each of the remaining oranges into fourths and give two pieces to each friend. Finally, they have to think about how to describe how much each friend gets in terms of the wholes and the pieces. They might simply draw the amount, they might shade it in, or they might attach number names to it. I also want to point out that a problem about four friends equally sharing 10 oranges can be solved by children with no formal understanding of fraction names and symbols because there are no fractions in the story problem. The fractions emerge in children's strategies and are represented by the pieces in the answer. The important thing here is that children are engaged in creating pieces and considering how the pieces are related to the wholes or other pieces. The names and symbols can be attached gradually.

Mike: So, the question that I wanted to ask is how to deal with this idea of how you name those fractional amounts, because the process that you described to me, what's powerful about it is that I can directly model the situation. I can make sense of partitioning. I think one of the things that I've always wondered about is, do you have a recommendation for how to navigate that naming process? I've got one of something, but it's not really one whole orange. So how do I name that?

Susan: That's a great question. Children often know some of the informal names for fractions, and they might understand halves or even fourths. Initially, they may call everything a half or everything a piece or just count everything as one. And so, what teachers can do is have conversations with children about the pieces they've created and how the pieces relate to the whole. A question that we've found to be very helpful is, how many of those pieces fit into the whole?

Mike: Got it.

Susan: Not a question about how many pieces are there in the whole, but how many of the one piece fit into the whole. Because it then focuses children on thinking about the relationship between the piece and the whole rather than simply counting pieces.

Mike: Let's talk about the other problem type that was kind of front and center in your thinking.

Susan: Yes. So, another type of story problem that can be used early in fraction instruction involves what we think of as special multiplication and division story problems that have a whole number of groups and a unit fraction amount in each group. So, what do I mean by that? For example, let's say there are six friends and they each will get one-third of a sub sandwich for lunch. So, there's a whole number of groups—that's the six friends—and there's a unit fraction amount in each group that's the one-third of a sandwich that they each get. And then the question is how many sandwiches will be needed for the friends? So, a problem like this one essentially engages children in reasoning about six groups of one-third. And again, as with the equal sharing problem about oranges, they can solve it by drawing out things. They might draw each one-third of a sandwich, and then they have to consider how to combine those to make whole sandwiches. An important idea that children work on with this problem then is that three groups of one-third of a sandwich can be combined to make one whole sandwich. There are other interesting types of story problems, but teachers have found these two types, in particular, effective in developing children's understandings of some of the big ideas and fractions.

Mike: I wonder if you have educators who hear you talk about the second type of problem and are a little bit surprised because they perceive it to be multiplication.

Susan: Yes, it is surprising. And the key is not that you teach all of multiplying and dividing fractions before adding and subtracting fractions, but that you use these problem types with special number combinations. So, a whole number of groups, for example, the six groups unit fractions in each group—because those are the earliest fractions children understand. And I think maybe one way to think about it is that fractions come out of multiplying and dividing, kind of in the way that whole numbers come out of adding and counting. And the key is to provide situations story problems that have number combinations in them that children are able to work with.

Mike: That totally makes sense. Can you say more about the importance of attending to the number combinations?

Susan: Yes. Well, I think that the number combinations that you might choose would be the ones that are able to connect with the fraction understandings that children already have. So, for example, if you're working with kindergartners, they might have a sense of what one half is. So, you might choose equal sharing problems that are about sharing things among two children. So, for example, three cookies among two children. You could even, once children are able to name the halves, they create in a problem like that, you can even pose problems that are about five children who each get half of a sandwich, how many sandwiches is that? But those are all numbers that are chosen to allow children to use what they understand about fractions. And then as their understandings grow and their repertoire of fractions also grows, you can increase the difficulty of the numbers. So, at the other end, let's think about fifth grade and posing equal sharing problems. If we take that problem about four friends sharing 10 oranges, we could change the number just a little bit to make it a lot harder to, four friends sharing 10 and a half oranges, and then fifth-graders would be solving a problem that's about finding a fraction of a fraction, sharing the half orange among the four children.

Mike: Let me take what you've shared and ask a follow-up question that came to me as you were talking. It strikes me that the design, the number choices that we use in problems matter, but so does the space that the teacher provides for students to develop strategies and also the way that the teacher engages with students around their strategy. Could you talk a little bit about that, Susan?

Susan: Yes. We think it's important for children to have space to solve problems, fraction story problems, in ways that make sense to them and also space to share their thinking. So, just as teachers might do with whole number problem-solving in terms of teacher questioning in these spaces, the important thing is for the teacher to be aware of and to appreciate the details of children's thinking. The idea is not to fix children's thinking with questioning, but to understand it or explore it. So, one space that we have found to be rich for this kind of questioning is circulating. So, that's the time when as children solve problems, the teacher circulates and has conversations with individual children about their strategies. So, follow-up questions that focus on the details of children's strategies help children to both articulate their strategies and to reflect on them and help teachers to understand what children's strategies are. We've also found that obvious questions are sometimes underappreciated. So, for example, questions about what this child understands about what's happening in a story problem, what the child has done so far in a partial strategy, even questions about marks on a child's paper; shapes or tallies that you as a teacher may not be quite sure about, asking what they mean to the child. “What are those? Why did you make those? How did they connect with the problem?” So, in some it benefits children to have the time to articulate the details of what they've done, and it benefits the teacher because they learn about children's understandings.

Mike: You're making me think about something that I don't know that I had words for before, which is I wonder if, as a field, we have made some progress about giving kids the space that you're talking about with whole number operations, especially with addition and subtraction. And you're also making me wonder if we still have a ways to go about not trying to simply funnel kids to, even if it's not algorithms, answer-getting strategies with rational numbers. I'm wondering if that strikes a chord for you or if that feels off base.

Susan: It feels totally on base to me. I think that it is as beneficial, perhaps even more beneficial for children to engage in solving story problems and teachers to have these conversations with them about their strategies. I actually think that fractions provide certain challenges that whole numbers may not, and the kinds of questioning that I'm talking about really depend on the details of what children have done. And so, teachers need to be comfortable with and familiar with children's strategies and how they think about fractions as they solve these problems. And then that understanding, that familiarity, lays the groundwork for teachers to have these conversations. The questions that I'm talking about can't really be planned in advance. Teachers need to be responsive to what the child is doing and saying in the moment. And so that also just adds to the challenge.

Mike: I'm wondering if you think that there are ways that educators can draw on the work that students have done composing and decomposing whole numbers to support their understanding of fractions?

Susan: Yes. We see lots of parallels just as children's understandings of whole numbers develop.
They're able to use these understandings to solve multi-digit operations problems by composing and decomposing numbers. So, for example, to take an easy addition, to add 37 plus eight, a child might say, “I don't know what that is, but I do know how to get from 37 to 40 with three.” So, they take three from the eight, add it to the 37 get to 40, and then once at 40 they might say, “I know that 40 plus five more is 45.” So, in other words, they decompose the eight in a way that helps them use what they understand about decade numbers. Operations with fractions work similarly, but children often do not think about the similarities because they don't understand fractions or numbers to, versus two numbers one on top of the other.

Susan: If children understand that fractions can be composed and decomposed just as whole numbers can be composed and decomposed, then they can use these understandings to add, subtract, multiply, and divide fractions. For example, to add one and four-fifths plus three-fifths, a child might say, “I know how to get up to two from one in four-fifths. I need one more fifth, and then I have two more fifths still to add from the three-fifths. So, it's two and two-fifths.” So, in other words, just as they decompose the eight into three and five to add eight to 37, they decompose the three-fifths into one-fifth and two-fifths to add it to one and four-fifths.

Mike: I could imagine a problem like one and a half plus five-eighths. I could say, “Well, I know I need to get a half up. Five-eighths is really four-eighths and one-eighths, and four-eighths is a half.”

Susan: Yep.

Mike: “So, I'm actually going from one and a half plus four-eighths. OK. That gets me to two, and then I've got one more eighth left. So, it's two and an eighth.”

Susan: Nice. Yeah, that's exactly the kind of reasoning this approach can encourage.

Mike: Well, I have a final question for you, Susan. “Extending Children's Mathematics” came out in 2011, and I'm wondering what you've learned since the book came out. So, are there ideas that you feel like have really been affirmed or refined, and what are some of the questions about the ways that students make meaning of fractions that you're exploring right now?

Susan: Well, I think, for one, I have a continued appreciation for the power of equal sharing problems. You can use them to elicit children's informal understandings of fractions early in instruction. You can use them to address a range of fraction understandings, and they can be adapted for a variety of fraction content. So, for example, building meaning for fractions, operating with fractions, concepts of equivalence. Vicki and I are currently writing up results from a big research project focused on teachers' responsiveness to children's fraction thinking during instruction. And right now, we're in the process of analyzing data on third-, fourth-, and fifth-grade children’s strategies for equal sharing problems. We specifically focused on over 1,500 drawing-based strategies used by children in a written assessment at the end of the school year. We've been surprised both by the variety of details in these strategies—so, for example, how children represent items, how they decide to distribute pieces to people—and also by the percentages of children using these drawing-based strategies. For each of grades three, four, and five, over 50 percent of children use the drawing-based strategy. There are also, of course, other kinds of strategies that don't depend on drawings that children use, but by far the majority of children were using these strategies.

Mike: That's interesting because I think it implies that we perhaps need to recognize that children actually benefit from time using those strategies as a starting point for making sense of the problems that they're solving.

Susan: I think it speaks to the length of time and the number of experiences that children need to really build meaning for fractions that they can then use in more symbolic work. I'll mention two other things that we've learned for which we actually have articles in the NCTM publication MTLT, which is “Mathematics Teacher: Learning and Teaching in PK–I2.” So first, we've renewed appreciation for the importance of unit fractions and story problems to elicit and develop big ideas. Another idea is that unit fractions are building blocks of other fractions. So, for example, if children solve the oranges problem by partitioning both of the extra oranges into fourths, then they have to combine the pieces in their answer. One-fourth from each of two oranges makes two-fourths of an orange. Another idea is that one whole can be seen as the same amount as a grouping of same-sized unit fractions. So, those unit fractions can all come from the same hole or different wholes, for example, to solve the problem about six friends who will each get one-third of a sub sandwich. A child has to group the one-third sandwiches to make whole sandwiches. Understanding that the same sandwich can be seen in these two ways, both as three one-third sandwiches or as one whole sandwich, provides a foundation for flexibility and reasoning. For those in the audience who are familiar with CGI, this idea is just like the IDM base ten, that 1 ten is the same amount as ten 1s, or what we describe in shorthand as 10 as a unit. And we also have an article in MTLT. It's about the use of follow-up equations to capture and focus on fraction ideas in children's thinking for their story problems. So basically, teachers listen carefully as children solve problems and explain their thinking to identify ideas that can be represented with the equations.

Susan: So, for example, a child solving the sub-sandwiches problem might draw a sandwich partitioned into thirds and say they know that one sandwich can serve three friends because there are three one-thirds in the sandwich. That idea for the child might be drawn, it might be verbally stated. A follow-up equation to capture this idea might be something like one equals one-third plus one-third plus blank, with the question for the child, “Could you finish this equation or make it a true equation?” So, follow-up equation[s] often make ideas about unit fractions explicit and put them into symbolic form for children. And then at the same time, the fractions in the equations are meaningful to children because they are linked to their own meaning-making for a story problem. And so, while follow-up equations are not exactly a question, they are something that teachers can engage children with in the moment as a way to kind of put some symbols onto what they are saying, help children to reflect on what they're saying or what they've drawn, in ways that point towards the use of symbols.

Mike: That really makes sense.

Susan: So, they could be encouraged to shade in the piece and count the total number of pieces into which an orange is cut. However, we have found that a better question is, how many of this size piece fit into the whole? Because it focuses children on the relationship between the piece and the whole, and not on only counting pieces.

Mike: Oh, that was wonderful. Thank you so much for joining us, Susan. It's really been a pleasure talking with you.

Susan: Thank you. It's been my pleasure. I've really enjoyed this conversation.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

Rounding Up

Season 2 | Episode 6 – Multiplicative Thinking

Guest: Dr. Anderson Norton

Mike Wallus: One of the most important shifts in students' thinking during their elementary years is also one of the least talked about. I'm talking about the shift from additive to multiplicative thinking. If you're not sure what I'm talking about, I suspect you're not alone. Today we talk with Dr. Anderson Norton about this important but underappreciated shift. 

Mike: Welcome to the podcast, Andy. I'm excited to talk with you about additive and multiplicative thinking.

Andy Norton: Oh, thank you. Thanks for inviting me. I love talking about that.

Mike: So, I want to start with a basic question. When we're talking about additive and multiplicative thinking, are we just talking about strategies or operations that students would carry out to find a sum or a product of a problem? Or are we talking about something larger?

Andy: Yeah, definitely something larger, and it doesn't come down to strategies. Students can solve multiplication tasks, what to us look like multiplication tasks, using additive reasoning. And they often do, I think, they get through a lot of elementary school using, for example, repeated edition. If I gave a task like what is four times five? Then they might just say that's five and five and five and five, which is fine. They're solving a multiplication problem, but their method for solving it is repeated addition, so it's basically additive reasoning. But it starts to catch up to them in later grades where that kind of additive reasoning requires them to do more and more sophisticated or complicated strategies that maybe their teachers can teach them, but it starts to add up, especially when they get to fractions or algebra.

Mike: So, let's dig into this a little bit deeper. How would you describe the difference between additive and multiplicative thinking? And I'm wondering if there's an example of the differences in how a student might approach a task or a problem that could maybe highlight that distinction.

Andy: The main distinction is with additive reasoning, you're working within one level of unit. So, for example, if I want to know, and going back to that four times five example, really what I'm doing is I'm working with ones. So, I say I have five ones and five ones and five ones and five ones, and that's 20 ones. But in a multiplication problem, you're really transforming across units. If I want to understand four times five as a multiplication problem, what I'm saying is, “If I measure a quantity with a unit of five, the measure is four,” just to make it a little more concrete. Suppose my unit of measure is like a stick that's 5 feet long, and then I say, “OK, I measured this length, and it was four of these sticks. So, it's four of these 5-foot sticks. But I want to know what it is in just feet.” So, I've changed my unit. I'm saying, “I measured this thing in one unit, this stick length, but I want to understand its measure in a different unit, a unit of ones.” So, you're transforming between this one kind of unit into another kind of unit, and it's a five-to-one transformation. So, I'm not just doing five plus five plus five plus five, I'm saying every one of that stick length contains 5 feet, five of these 1-foot measures. And so, it's a transformation from one unit into another, one unit for measuring into a different unit for measuring.

Mike: I mean, that's a really big shift, and I'm glad that you were able to describe that with a practical example, that someone could listen to this and visualize. I think understanding that for me clarifies the importance of not thinking about this in terms of just procedural steps that kids would take to either add or multiply; that really there's a transformation in how kids are thinking about what's happening rather than just the steps that they're following.

Andy: Yeah, that's right. And a lot of times as teachers or even as researchers studying children, we're frustrated like the kids are when they're solving tasks, when they're struggling. And so we try to give them those procedures. We might give them a visual model, we might give them an array model for multiplication, which can solve a lot of problems. You just sort of think about things going vertically and things going horizontally, and then you're looking at an area or a number of intersections. So, that makes it possible for them to solve these individual tasks. And there's a lot of pressure on teachers to cover curriculum. So, we feel like we have to support them by giving them these strategies. But in the end, it just becomes more and more of these complicated strategies without really necessitating the need for something we might call a “productive struggle”; that is, where students can actually start to go through developmental changes by allowing them to struggle so that they actually develop these kinds of multiplicative structures instead of just giving them a bunch of strategies for dealing with that one task at a time.

Mike: I'm wondering if you might share some examples of what multiplicative thinking might look like or sound like in different scenarios. For example, with whole numbers, with fractions or decimals … 

Andy: Uh-hm.

Mike: … and perhaps even in a context like measurement. What might an educator who was listening or observing students' work, what might they see that would indicate to them that multiplicative reasoning or multiplicative thinking was something that was happening for the student?

Andy: So, it really is that sort of transformation of units. Like imagine, I know something is nine-fifths, and nine-fifths doesn't make a whole lot of sense unless I can think about it as nine units of one-fifth. We have to think about it as a measure like it's nine of one-fifth. And then I have to somehow compare that to, OK, it's nine of this one unit, this one-fifth unit, but what is it of a whole unit? A unit of one? So, having an estimate for how big nine-fifths is, yes, it's nine units of one-fifth. But at the same time, I want to know how big that is relative to a one. So, there's this multiplicative nature kind of built into tasks like that, and it's one explanation for why students struggle so much with improper fractions.

Mike: So, I'm going to put my teacher hat on for a second because what you've got me thinking is, what are the types of tasks or experiences or even questions that an educator could put in front of students that would nudge them to make this shift without potentially pushing them to a place where they're not quite ready to go yet? 

Andy: Hmm.

Mike: Could you talk a little bit about what types of tasks or experiences or questions might help provide a little bit of that nudge?

Andy: Yeah, that's a really good question, because it goes back to this idea that students are already solving the kinds of tasks that should involve multiplicative reasoning, but they might be using additive strategies to do it. Those strategies get more and more complicated, and we as teachers facilitate students just, sort of, doing something more procedural instead of really struggling with the issue. And what the issue should be is opportunities to work with multiple levels of units and then to reflect on their activity and working with them. So, for example, one task I like to give students is, I'll cut out a piece of construction paper and I'll hand it to the student, and I'll have hidden what I'm going to label a whole, and I'll have hidden what I'm going to label to be the unit fraction that might be appropriate for measuring this thing I gave them. So, I'll give them this piece of construction paper and I'll say, “Hey, this is five-sevenths of my whole.” Now what I've given them as a rectangular strip of paper without any partitions in it, I've hidden the whole from which I created this five-sevenths. I've hidden one-seventh, and I've put them away, maybe inside of envelopes. So, it becomes like a game. Can you guess what I have in this envelope? I just gave you five-sevenths. Can you guess, what is this five of? What is the unit that this is five of and what is the whole this five-sevenths fraction is? So, it's getting them thinking about two different levels of units at once. They've been given this one measurement, but they don't know the unit in which it's measured, and they don't even have visually present for them what the whole unit would be.

Andy: So, what they might do, is they might engage in partitioning activity. Sometimes they might partition what I give them into seven equal parts instead of five because I told them five sevenths and five sevenths to them, that means partition it a seventh. Well, that could lead to problems, and if they see that their unit is smaller than the one I have hidden, they might have to reason through what went wrong, “Why might have you have gotten a different answer than I did?” So, it's those kinds of activities—of partitioning or iterating a unit, measuring out with a unit, and then reflecting on that activity—that give them a basis for starting to coordinate these units at higher and higher levels and, therefore, in line with Amy Hackenberg’s framing, develop multiplicative concepts.

Mike: I think that example is really helpful. I was picturing it in my head, and I could see the opportunities that that affords for, kind of, pressing on some of those big ideas. One of the things that you made me think about is the idea of manipulatives, or even if we broaden it out a little bit, visual models. Because the question I was going to ask is, “What role might a visual model or a manipulative play in supporting a shift from additive to multiplicative thinking?” I'm curious about how you would respond to that initially. And then I think I have a follow-up question for you as well.

Andy: OK. I can think of two important roles for visual models—or at least two for manipulatives—and at least one works with visual models as well. But before answering that, the bigger answer is, no one manipulative is going to be the silver bullet. It's how we use them. We can use manipulatives in ways where students are just following our procedures. We can use visual models where students are just doing what we tell them to do and reading off the answer on paper. That really isn't qualitatively any different than when we just teach them an algorithm. They don't know what they're doing. They get the answer, they read it off the paper. You could consider that to be a visual model, what they're doing on their paper or even a manipulative, they're just following a procedure. What manipulatives should afford is opportunities for students to manipulate. They should be able to carry out their mental actions. So, maybe when they're trying to partition something and then iterate it, or they're thinking about different units. That's too much for them to keep in mind in their visual imagination. So, a visual model or a manipulative gives them a way to carry those actions out to see how they work with each other, to notice the effects of those actions. 

Andy: So, if the manipulative is used truly as a manipulative, then it's an opportunity for them to carry out their mental actions to coordinate them with a physical material and to see what happens. And visual models could be similar, gives them a way to sort of carry out their mental actions, maybe a little more abstractly because they're just using representations rather than the actual manipulative, but maybe gives them a way to keep track of what would happen if I partitioned this into three parts and then took one of those parts and partitioned into five. How would that compare to the whole? So, it's their actions that have to be afforded by the manipulative or the visual model. And to decide what is an appropriate manipulative or an appropriate task, we need to think about, “OK, what can they already do without it?” And I'm trying to push them to do the next thing where it helps them coordinate at a level they can't just do in their imagination, and then to reflect on that activity by looking at what they wrote or looking at what they did. So, it's always that: Carrying out actions in slightly more powerful ways than they could do in their mind. That's sort of the sense in which mathematics builds on itself. After they've reflected on what they've done and they've seen the results, now maybe that's something that they can take as an object, as something that's just there for them in imagination so they can do the next thing, adding complexity.

Mike: OK. So, I take it back. I don't think I have a follow-up question because you answered it in that one. What I was kind of going to dig into is the thing that you said, which is, there's a larger question about the role that a manipulative plays, and I think that your description of a manipulative should be there to manipulate … 

Andy: Uh-hm.

Mike: … to help kids carry out the mental action and make meaning of that. I think that piece to me is one that I really needed clarified, just to think about my own teaching and the role the manipulatives are going to play when I'm using them to support student thinking.

Andy: And I'll just add one thing, not to use too many fractions examples, but that is where most of my empirical research has been, was working with elementary and middle-school children with fractions. But I have to make these decisions based on the child. So, sometimes I'll use these cuisenaire rods, the old fraction rods, the colored fraction rods. Sometimes I'll use those with students because then it sort of simplifies the idea. They don't have to wonder whether a piece fits in exactly a certain number of times. The rods are made to fit exactly. And maybe I'm not as concerned about them cutting a construction paper into equal parts or whatever. So, the rods are already formed. But other times I feel like they might be relying too much on the rods, where they start to see the brown rod as a four. They're not even really comparing the red rod, which fits into it twice. They're just, “Oh, the red is a two, the brown is a four. I know it's in there twice because two and two is four.” So, you start to think about them whole numbers. And so sometimes I'll use the rods because I want them to manipulate them in certain ways, and then other times I'll switch to the construction paper to sort of productively frustrate this idea that they're just going to work with whole numbers. I actually want them to create parts and to see the measurements and actually measure things out. So, it all depends on what kind of mental action I want them to carry out that would determine what manipulative as well. Because manipulatives have certain affordances and certain constraints. So, sometimes cuisenaire rods have the affordances I want, and other times they have constraints that I want to go beyond with, say, construction paper.

Mike: Absolutely. So, there's kind of a running theme that started to develop on the podcast. And one of the themes that comes to mind is this idea that it's important for us to think about what's happening with our students thinking as a progression rather than a checklist. What strikes me about this conversation is this shift from additive to multiplicative thinking has really major implications for our students beyond simple calculation. And I'm wondering if you could just afford us a view of, why does this shift in thinking matter for our students both in elementary school, and then also when they move beyond elementary school into middle and high school? Could you just talk about the ramifications of that shift and why it matters so much that we're not just building a set of procedures, we're building growth in the way that kids are thinking?

Andy: Yeah. So, one big idea that comes up starting in middle school—but becomes more and more important as they move into algebra and calculus, any kind of engineering problem—is a rate of change. So, a rate of change is describing a relationship between units. It's like, take a simple example of speed. It's taking units of distance and units of time and transforming them into a third level of unit that is speed. So, it's that intensive relationship that's defining a new unit. When I talk about units coordination, I'm not usually talking about physical units like distance, time and speed. I'm just talking about different numerical units that students might have to coordinate. But to get really practical when we talk about the sciences, units coordinations have to happen all the time. So, students are able to be successful with their additive reasoning up to a point, and I would argue that point is probably around where they first see improper fractions. ( chuckles ) They're able to work with them up to a point, and then after that, things [are] going to be less and less sensible if they're just relying on these additive sort of strategies that each have a separate rule for a different task instead of being able to think more generally in terms of multiplicative relationships.

Mike: Well, I will say from a former K–12 math curriculum director, thank you for making a very persuasive case for why it's important to help kids build multiplicative thinking. You certainly hit on some of the things that can be pitfalls for kids who are still thinking in an additive way when they start to move into upper elementary, middle school and beyond. Before we go, Andy, I suspect that this idea of shifting from additive to multiplicative thinking, that it's probably a new idea for our listeners. And you've hinted a bit about some of the folks who have been powerful in the field in terms of articulating some of these ideas. I'm wondering if there are any particular resources that you'd recommend for someone who wants to keep learning about this topic?

Andy: Yeah. So, there are a bunch of us developing ideas and trying to even create resources that teachers can pick up and use. Selfishly, I'll mention one called “Developing Fractions Knowledge,” used by the U.S. Math Recovery Council in their professional development programs for teacher-leaders across the country. That book is probably, at least as far as fractions, that book is maybe the most comprehensive. But then beyond that, there are some research articles that people can access, even going in Google Scholar and looking up units, coordination and multiplicative reasoning, maybe put in Steffe's name for good measure, S-T-E-F-F-E. You'll find a lot of papers there. Some of them have been written in teacher journals as well, like journals published by the National Council of Teachers of Mathematics, like Teaching Children Mathematics materials that are specifically designed for teachers.

Mike: Andy, thank you so much for joining us. It's really been a pleasure talking with you.

Andy: OK. Yeah, thank you. This was fun.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

Rounding Up

Season 2 | Episode 5 – Horizontal Enrichment

Guest: Tisha Jones

Mike Wallus: At their best, programs with titles such as “gifted and talented” seek to provide enrichment to a subset of learners. That said, these initiatives sometimes have unintended consequences, sending messages about which students are, or are not, capable doers of mathematics. What if there was a way educators could offer problems that extend grade-level learning to each and every student? Today we'll explore the concept of horizontal enrichment with Tisha Jones, MLC's senior manager of assessment. 

Mike: Well, thanks for joining us, Tisha. I am excited to explore this idea of horizontal enrichment.

Tisha Jones: I am excited to be here and talk about it.

Mike: So, we're using the term “horizontal enrichment,” and I think we should define the term and talk about, what do we mean when we say that?

Tisha: When we're talking about horizontal enrichment, we are looking at how do we enrich the curriculum, but on grade level. So, not trying to accelerate into the next grade level. But how do we help them go deeper with the content that is at their developmental level currently?

Mike: That's really interesting because when I was teaching, I would've said enrichment and acceleration are exactly the same thing, which, I think, leads me to the next question, which is: What are the features of a task that might be designed with horizontal enrichment in mind?

Tisha: So, I like to think about horizontal enrichment as an opportunity to engage the practice standards. So, how do we help kids do more of the things that we think being a [mathematician] actually is? So, how can we get them more invested in problem-solving? How can we get them using tools? How can we get them thinking creatively in math and not just procedurally. And, of course, we try to do that on a daily basis in math, but when we're enriching, we want to give them tasks that raise the ceiling of their thinking, where they can approach things in lots of different ways and push their thinking in ways that maybe they haven't, where they can apply the concepts that they're using to solve interesting and novel problems.

Mike: I think that's really helpful because you're really clarifying for me, one way that we could “enrich” kids would be to teach them procedures that they might learn in a grade or several grades that are of beyond where they're at right now. But what you're suggesting is that enrichment really looks like problem-solving and novelty and creativity. And we can do that with grade-level ideas. Am I making sense of that correctly?

Tisha: Absolutely, and I get excited because I also think that it's fun working a problem where the path is not clear-cut to get to the answer and try some things out and see what happens and look at how can I learn from what I did to make new decisions to try to get to where I'm going? To me, that's bringing in the joy of doing math.

Mike: So, this is interesting. I think that maybe the best way to unpack these ideas might be to look at a specific task. So, I'm wondering, is there a specific task that you could help us take a look at more closely?

Tisha: Absolutely. So, we're going to take a look at a task from third grade, and it comes out of Concept Quests, which is a supplemental resource that's published by Math Learning Center, and this task is called “The Lasagna Task.” So, I'm just going to read it and then we can talk about what is it asking kids to do. So, it says, “You need to assume that you like lasagna and would like as much lasagna as possible. For each of the ‘Would you rather…?’ scenarios below, justify your reasoning with equations, pictures, or both.” So, that's the setup for the kids. And then there's three “Would you rather…?” scenarios. So, the first is, “Would you rather: a.) share three lasagnas between two families or share four lasagnas between three families? b.) Would you rather share four lasagnas between six families or share three lasagnas between four families?” And the last one is, “c.) Would you rather share five lasagnas between three families or share six lasagnas between four families?”

Mike: Ahh, this is so great. There's so much to unpack here to step back and try to analyze this. What are some things that you would want us to notice about the way this task is set up for kids?

Tisha: So, there's a few things. The first thing is, I love that there's this progression of questions, of scenarios. I think what's also really important is, when you're looking at this on the page, there's no front-loading here. No, “Well, let me tell you about how to do this.” This is just, “I'm going to give you this problem, and I'm going to ask you to just take a stab at it, give it a shot.” So, what we want kids to do is start to learn, how do you approach a problem? What is your first step? What things do you do to make sense of what it's asking? Do you draw a picture? Do you start with numbers? Do you try to find important information? How do you even get started on a problem? And that's so important, right? That's a huge part of the process of problem-solving. And when we front-load for kids, we take away their opportunities to work on those skills.

Mike: So, there's a couple things that jump out for me when I've been reading the text of what you were reading aloud to the group. One bit is this language at the end where it says, “For each of the ‘Would you rather…?’ scenarios below, justify your reasoning with equations, pictures, or both.” And that language just pops out for me. I'm wondering if you could talk a little bit about the choice of that language in the way that this is set up for kids.

Tisha: Ahh, I love that language. So, I think this is amazing for kids because as a teacher, we've all had kids that come up to us and they hand us their paper and they say, “Is this right?” And when we ask them to justify their response, I think we're putting the responsibility back on them to be able to come up to me and say, “I think this is right because of this.” So now, who is owning what they did? The kids are owning what they did, right? And they're owning it because they've gone through this process of trying to prove it not just to somebody else but to themselves. If you're justifying it, you should be able to go back through and say, “Well, because I did this and this is this and because I did this next step and this is how this worked out, this is why I know my answer is correct.” And I love that kids can own their own answers and their own work to be able to determine whether it makes sense or not.

Mike: I'm going to read a part of this again because I just think it's worth lingering on and spending a little bit of time thinking about how this question structure impacts kids or has the potential to impact kids. So, I'm going to read it again for the audience: “Would you rather: a.) share three lasagnas between two families or share four lasagnas between three families?” So, listeners, just pause for a second and think about the mathematics in that question, and then also think about what mathematics might come out of it. What is it about the structure of that question that creates space for kids to solve problems, encounter novelty, and make decisions? Well, Tisha, since we can't hear their answer, I would love it if you could share a little bit of your thinking. What is it about the design that you think creates those conditions for kids?

Tisha: So, while there is an implied operation, it's not necessarily an obvious operation, right? I think that it is something that easily lends itself to drawing a picture, which, I think, when students start modeling the scenario, they now have … that opens up all kinds of creativity, right? They're going to model in the way that they're seeing it in their head. They're not focused on trying to divide this number by that number. They may not even, at first, realize that they're working with fractions. But by the end of it, because it's something that they can model, there's still a lot of room for them to be able to find success on this task, which I think is really important.

Mike: It seems like there's also opportunities for teachers to engage with kids because there's a fair number of assumptions that live inside of this question structure, right? Like three lasagnas for two families, four lasagnas for three families, but we haven't talked about how large those families are, how many people are in each family.

Tisha: How much lasagna there is ( chuckles ).

Mike: Yeah! Right?

Tisha: Absolutely. So, I think it's also fair to say that maybe a kid would decide that the four lasagnas between three families, those are going to need to be bigger pans of lasagna. So, how are they bringing in their world experience with feeding people and having to make these decisions? There's nothing in here that says that the lasagnas have to be the same size or that the families have to be the same size. So, as they're justifying the way that they would go as a teacher, I'm looking for: Is their justification, a sound justification?

Mike: Well, the thing that I started to think about, too, is, if you did introduce the variable that, “Oh, this family has three members and this family has, say, 12. Well, how many lasagnas would you need in order to give an equal share to the family with 12 versus the family with three?” There's a lot of ways as a teacher that I can continue to adapt and play with the ideas and really press kids to examine their own assumptions and their own logic.

Tisha: Absolutely, yeah. So, I think that's a really great point, too, is that, there's a lot of room to even extend these problems further. Would your answer change if you knew that one family was a family of six people, so you can even push their thinking even further than what's just on the paper.

Mike: I keep going back to this notion of justification. And we've talked about the structure of the problems as a way to differentiate for kids, to really press them on justification. But the other side of the coin is, as an educator, [it] really gives me a chance to understand my students' thinking and then continue to make moves or offer tasks that either shine a light on the blind spots that they have or extend some of the ideas in interesting and productive ways.

Tisha: Yes, I would agree with that.

Mike: So, I want to play with a couple more questions, Tisha. One of the ones that we touched on right at the beginning was this idea that a task can be characterized as enriching and challenging, and yet it can still be at a student's grade level. And I think that really stands out for me, and I suspect it probably might be a challenging idea for educators to get their heads around, especially if you've been a teacher, and for the majority of your career, acceleration and enrichment have meant the same thing. Can you unpack this just a little bit for the audience, this idea of enrichment?

Tisha: So, I like to think about enrichment as, how do we help our students think more deeply? There's so much room within a school year for a particular concept, for example. Like, let's say with fractions. There's a lot of room for students to think about things in ways they haven't thought about or ways that maybe we don't ask them to think about things in the curriculum; that, if we don't give them the opportunity, they're not going to, right? With enrichment, it's like we're giving them more opportunities to apply what they're learning about concepts. The other thing that I think is really important about enrichment is that it isn't just for the kids that may be characterize as being your high-level students. Because enrichment is still important. Problem-solving is still important for all kids. No matter where they are computationally, we want to make sure that all kids are getting opportunities to be problem-solvers, to apply their thinking in ways that work for them and not just the ways that we're asking them to through our curriculum. Acceleration, I think, often applies when kids are just well beyond grade level—but enrichment is really for every single kid.

Mike: Yeah, I think you answered, at least partly, the question that I was going to pose next, which was a question about access. Because at least with Concepts Quests, which is the MLC supplemental resource, we would describe this as a tool that should be made available to all students, not a particularly small subset of students. And I'm wondering if you can talk a little bit more about the case for that.

Tisha: So, if we go back to our lasagna problem, once our kids have had opportunities to read it and make sense out of it, at that point, I truly believe that there is an entry point in these problems for any kid. These are not dependent on computation. So, a student can draw pictures. I believe that all of my students that I've had throughout my years of teaching were capable of drawing a picture to model a problem. Then, I really believe that a good problem can have an entry point for every student.

Mike: The other thing that you're really making me think about is, how much we've equated the idea of enrichment, acceleration. We've fused those ideas, and we've really associated it with procedure and calculation versus problem-solving and thinking creatively.

Tisha: I think that happens a lot. I think that's a lot of how people think about math. You know, it's who can do it fast, who can get there? But what I think our goal is, is to create students who are not just able to be calculators, but who are able to apply their understandings of multiplication, addition, subtraction, division. They can apply them to novel problems.

Mike: Yeah, and the real world isn't designed with a set of “Free set, here's what you should do, repeat directions.”

Tisha: ( laughs ) I would love some of those. Where can I find them?

Mike and Tisha: ( laugh)

Mike: This has been fascinating, and I think we could and probably should do more work on Rounding Up talking about these versions of enrichment that are available for all kids. And I have a suspicion that this conversation is going to cause a lot of folks to reassess, reevaluate, and reflect on how they've understood the idea of enrichment. I'm wondering if we can help those folks out. If I'm an educator who's really interested in exploring the idea of horizontal enrichment in more detail, where might I get started? Or, perhaps, where are there some resources out there that might contain the types of problems that you introduced us to today?

Tisha: Well, of course, I have to say Concept Quests. We've put a lot of work into creating some really great tasks. But some other places where you can find tasks that are engaging and help kids to think more deeply are “Open Middle” and “NRICH” and “YouCubed” are just a few resources that I can think of off the top of my head.

Mike: Ahh, those are great ones. Tisha, thank you so much for joining us. It's really been a pleasure to have this conversation.

Tisha: This has been so fun.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

Rounding Up

Season 2 | Episode 4 – Joy in the Elementary Math Classroom

Guest: Amy Parks, Ph.D.

Mike Wallus: Teaching is a complex and challenging job. It's also one where educators experience moments of deep joy and satisfaction. What might it look like to build a culture of joy in an elementary mathematics classroom? Michigan State professor Amy Parks has some ideas. Today on the podcast, we explore ways educators can construct joyful experiences for their youngest mathematics learners. 

Mike: Well, welcome to the podcast, Amy. I'm so excited to be talking with you about joy in the elementary mathematics classroom.

Amy Parks: I'm so happy to be here.

Mike: So, your article in MTLT was titled, “Creating Joy in PK–Grade 2 Mathematics Classrooms.” And early on you draw a distinction between math classrooms where students are experiencing joy and those that are fun. And you quote Desmond Tutu and the Dalai Lama, who say, “Being joyful is not just about having more fun, we're talking about a more empathic, more empowered, more spiritual state of mind that's totally engaged with the world.” That really is powerful. So, I'm wondering if you could tell me about the difference between classrooms that foster joy and those that are just more fun.

Amy: Yeah, I was very struck by that quote when I read it the first time in “The Book of Joy.” And I think one of the reasons that book is powerful for me is that the two people writing it didn't have these super easy lives, right? Particularly the Archbishop Desmond Tutu was imprisoned in the country that was openly hostile to him, and yet he was still really committed to approaching his work and the world with joy. And so, I often think if he could do that, then surely the rest of us can get up and do that. And it also tied into something I often see in elementary classrooms, which is this focus on activities that are fun, like sugary cereal, right? They're immediately attractive, but they don't stick with us and maybe they're not really good for us. I often think the prototypical example is, like, analyses of packets of M&Ms. When I think about the intellectual energy that has gone into counting and sorting and defining colors of M&Ms, it makes me a little sad, given all the big questions that are out there that even really young kids can engage with. And so, yes, I want children to be playful and to laugh and to engage with materials they enjoy. But also, I think there is this quieter kind of joy that comes from making mathematical connections and understanding the world in new ways and grasping the thinking and ideas of others. And so, when I'm pointing toward joy, that's part of what I'm trying to point toward.

Mike: So, I want to dig into this a little bit more because one of your first recommendations for sparking joy is this idea that we need to make some room for play. And my guess is that that raises many questions for elementary educators, like “What do you mean by play?” and “What role does the teacher play in play?” Can you talk a little bit about this recommendation, Amy?

Amy: Yeah. So, when I have more time than that very short article to talk about, one of the things that I like to bring out to teachers is that we can think of play in sort of three broad buckets. So, one is “free play,” and this is an area where the teacher may not have a lot of roles except to sort of define health and safety limits. So certainly, recess is a place of free play. But there are places at recess where children are encountering mathematical ideas, right? There are walking in straight lines and they're balancing on things and they're seeing whether they all have the same amount of materials and toys. So, those are all mathematical contexts that we can, as teachers later bring in and highlight in places where they can engage. But they're not places where teachers are setting learning goals and reinforcing things. And particularly in the lower grades, we might see also free play opportunities in the classroom.

Amy: You know, many kindergarten classrooms have opportunities for free play during the school day. So, while kids are playing in the kitchen for example, or doing puzzles, they may be again encountering mathematical ideas and teachers certainly can capitalize on that. But they're not directing or shaping the play. And then there are these two other categories where the teacher's role is maybe more present. So, one I would call “guided play.” And this is a case where the teacher and the children are really handing responsibility back and forth. So, the teacher might set up a relatively open-ended task like pattern block puzzles or a commercial game that gets at counting or something like that. And so, the teacher has an intended mathematical goal. She has set some limits to keep children focused on that in some way. But the task is in the hands of the kids. They're playing together, they're negotiating roles, they have that more central responsibility. And the learning goals may be a little bit broader and more open because of that. Because since you're not centrally involved, you can't be so specific.

Amy: And then the last kind of play I talk with teachers about are “playful lessons.” Children might not have as much choice in the activity that they do. They might not be able to stop and start it or move in certain ways, but teachers are intentionally bringing aspects of play into the mathematics lesson. And that could be by using engaging materials. It could be by creating places for creativity. It could be by creating spaces for social collaboration. It could be just by inviting children to use their bodies in ways that are comfortable to them instead of being really constrained. But the mathematical task might be much more specific and “Build this cube and identify the vertices on it.” So, the task is constrained, but because they're using materials, because they can do it in different ways, there's this playful aspect to it. So, I like to encourage teachers to sort of think those three buckets of play and where kids are getting access to them during the day.

Mike: Yeah, I think that's really helpful. Because I did teach kindergarten for a long time, and so I think my definition of play was really the first one that you were talking about, which is free play. But hearing you talk about the other two definitions actually helps open space up for me. I feel like with that broader definition, it helps me consider the choices that I've got in front of me.

Amy: Yeah, and if you talk [to]—or read even—mathematicians, they will often talk about playing with ideas. So, there is a part of play that is inherently mathematical, the part that is about experimenting and investigating and trying things out and recognizing that you might be wrong and getting this engagement from others. So, I think sometimes even mathematics lessons that look relatively traditional can also have this playful spirit if we bring that to it.

Mike: I would love to talk to you a little bit about the way that choice can be a key component in sparking joy. So, what are some of the options that teachers have at their disposal to offer choice to learners in their classrooms?

Amy: Yeah, I think that this is something that's often overlooked. And I think that for kids in school right now, they often have so few choices. Their experiences are often so constrained by adults. And simply by allowing children to choose when they can, we can make experiences more joyful for them. So, one easy thing is who or whether children will work with other people. So yes, there are all kinds of benefits to group tasks and social interactions, but also lots of children are introverts. And being in a small room for six hours a day with 25 other people can be exhausting. And so, simply giving the children the choice to say, “I'm going to do this one on my own,” can be a huge relief to some children. Other children, like, need to talk—just like other adults—talk to others to know what they're thinking. And so, they need these groups.

Amy: And then I think also teachers can get really involved in choosing the magic right group, but often there is no magic right group as we know because we're constantly rejuggling these groups because they didn't work in the magic way we thought. And so just letting kids pick their groups, because then they have responsibility for that interaction. And it's not that they never have difficult social interactions, but they've chosen to be with this person and they have to work through it. So that's one. The other thing is letting children choose physically where they work. Some children lie on the floor while they work, or some children stand up at their seat. Allowing some choice in freedom of movement doesn't mean allowing total chaos. And I think even pretty young children can be taught that they can move within limits in the classroom. And I think if children get to stop expending so much energy trying to control their bodies in the ways adults find helpful, they can engage more fully in the academics of the day.

Amy: And then, like, choices of materials. So, we can make different things available to kids as they engage with mathematics, choices of problems. They may choose to do some and not others. Lower grades like using centers. If we have multiple centers that all get at the same mathematical idea, maybe it doesn't actually matter whether all kids get to all of them, right? As long as they're engaging with making units of 10, however they're doing that, can work for us. So, I think in general, the more often we can give children choices about anything, the better off all of us are.

Mike: I think that last bit is really interesting. I just want to pause for a second on it. Because what you've got me thinking is, if I have options available and they're all really addressing some of the same mathematical goals or a range of goals that I have in my class, this idea that I can release control and invite kids to make choices, that seems like a really practical first step that a teacher could take to think about, “What are the options? What are the goals that they meet?” And then, “To what degree can I offer those as choices?”

Amy: Yeah, and in a really basic way, right? Sometimes we might have a game that works with kids on making tens, and then other times we might have a project or even a worksheet. And different kids may be drawn to those different things. There are some kids for whom games might be really exciting, but there are some kids for whom games might be really stressful, and they would just rather do something else. And that's fine because the point isn't actually playing the game, right?

Mike: I think that's really interesting. I could get so caught up as a teacher sometimes trying to get the mechanics of getting kids out to places and getting kids started and making sure that kids were doing the thing that I would sometimes lose track of, “My point in doing this is to have kids think about structuring 10 or making sense of fractions.” That's a lovely reminder. I really appreciate that. I think that this is a really nice turning point because this question about choice actually plays into one of the other recommendations you had regarding time on task. So, I would love to have you unpack your thinking on this topic, Amy.

Amy: Yeah. Well, you talked about being autobiographical, and this is definitely autobiographical for me because I am very on task. I like to get things done. I like to check things off my list. And that was definitely a force for me when I was teaching. And I think it was something that, one, caused anxiety for me and my kids, and two, limited our opportunities to engage in more playful ways and more joyful ways to follow curiosities because I was so worried about that. And honestly, when it came home to me was when I started teaching university students because I think it is a little harder to clap your hands at 19-year-olds and tell them to get back to work than to do it with 7-year-olds. And what I realized was if I step back and I let my students talk about “The Bachelor” for a minute, they would have the conversation and then they would move on to the mathematical task, and I actually didn't need to intervene. And me intervening would've shifted the emotional tone of the class in a way that would not have been productive for learning, right?

Amy: They would've become resentful or maybe felt self-conscious. And now I have this thing in the way as opposed to just letting them have that break. And I think if we pay attention as adults to how we are in staff meetings or how we are in professional development, we recognize we have a lot of informal conversations around the work we do, and that those informal conversations are not distractions. They're actually, like, building the relationships that let us do the work. And it is similarly true for children. And then I think another thing to remember about particularly young children is language learning, social relationships, all of those are things they actually need to develop. That's part of our work as teachers is to help them grow in those things. And so, giving them the opportunity to build those relationships is, in fact, part of our work.

Mike: I think that's really interesting because I found myself, as you were talking, thinking through my own day, when I log into Zoom to talk to someone across the country. We don't immediately start just working through our agenda. We exchange pleasantries, we tell a joke or two, we talk about what's going on in our world, and we can have an incredibly productive chunk of time. But there are these pieces of social reality that kind of bind us together as people, right? When I'm talking to my friend Nataki in North Carolina, I'm asking her about her son. That might take two minutes out of 55. We've still done a tremendous amount of work and thought deeply about the kind of professional learning we want to provide to teachers. But there's the reality that if we didn't do that, how are we connected? If we're partnering to do this work, there's something about being connected to the other person that we can't schedule out of the experience of working together. Does that make sense?

Amy: Yeah, a hundred percent. And it's true in classroom settings, too. I was thinking the “Batman” movie, the Ben Affleck one was filmed in Detroit, and they happened to be filming right outside the building where I was teaching. And at some point, one of my adult students looked out the window and was, like, there's Ben Affleck. And of course, all my students got up and went to the window. I could have as the teacher been, like, “OK, sit down. We're doing whatever we're doing.” But their minds were all going to be on Ben Affleck out the window. And so instead, we stopped and we watched the movie for a little bit, and that became an experience we came back to as a class over and over in the semester. “Remember when that happened?” And so, yeah, that pressure to be productive I think often interferes with the relationship building that does support good work among adult colleagues and among kids in classrooms. And I would also connect it to the opening conversation on play.

Mike: So, before we close the interview, I'm wondering if you have any recommendations for someone who wants to continue learning about how they could design opportunities for joy in their classrooms. Are there any resources that you would point a listener to?

Amy: I mean, I have a book on play in early mathematics, and that would certainly be a place that someone could start. But, you know, the other thing that I might do is just look at some of the great materials that are out there, both like physical things like Legos and magnet tiles, which often if you don't have at your school, you can get through thrift stores and things. And just bringing them into classrooms and seeing what kids do with them. Oh, the other thing that I always recommend is looking at some of the resources on “soft starts.” And if you just Google this, you'll see videos and articles. And this is often a really, like, nonthreatening way for teachers who are interested in this but haven't done a lot of play in their classrooms, to begin. 

Amy: And the idea is instead of immediately starting with a worksheet or whatever, that you bring in some kind of toy or tool, and maybe children can make some choices about whether they're going to paint or they're going to work on a puzzle, and you just take 15 minutes and that's how you begin the day. And people who have done this, so many people have said it's just been such a lovely culture shift in their classroom, and it also means that children are coming in a little late. It's fine. They can just come in and join, and then everyone's ready to go 15 minutes later, and you really haven't given up that much of your day. So, I think that can be a really, a really smooth entry into this if you're interested.

Mike: Well, I want to thank you so much for joining us, Amy. It really has been a pleasure talking with you.

Amy: Oh, you, too. It was so fun.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

Rounding Up

Season 2 | Episode 3 – Student Engagement

Guest: Dr. Meghan Shaughnessy

Mike Wallus: When we say students are engaged in a discussion or a task, what do we really mean? There are observable behaviors that we often code as engaged, but those are just the things that we can see or hear. What does engagement really mean, particularly for students who may not verbally participate on a regular basis? Today on the podcast, we're talking with Dr. Meghan Shaughnessy about the meaning of engagement and a set of strategies teachers can use to extend opportunities for participation to each and every student.

Mike: Welcome to the podcast, Meghan. We are super excited to have you joining us.

Meghan: I'm excited to be here.

Mike: So, I want to start with a question that I think in the past I would've thought had an obvious answer. So, what does or what can participation look like?

Meghan: So, I think in answering that question, I want to start with thinking about one of the ways that teachers get feedback on participation in their classroom is through administrator observation. And oftentimes those observations are focused on students making whole-group verbal contributions and discussions, particularly with a focus on students sharing their own ideas. Administrators are often looking at how quiet the space is and how engaged students appear to be, which is often determined by looking at students' body language and whether or not that language matches what is often seen as listening body language, such as having your head up, facing the speaker, et cetera. And as I say all of this, I would also say that defining participation in this way for discussions is both a limited and a problematic view of participation. I say limited in the sense that not all participation is going to be verbal, and it certainly won't always include sharing new ideas.

Meghan: So, to give a concrete example, a student might participate by revoicing another student's strategy, which could be really important, providing other students a second chance to hear that strategy. A second example is that a student might create a representation of a strategy being shared verbally by a classmate. And this nonverbal move of creating a representation could be really useful for the class in developing collective understanding of the strategy. The traditional view is problematic, too, in the sense that it assumes that students are not participating when they don't display particular behaviors. To turn to a more equitable approach to conceptualizing and supporting participation, I and my colleagues would argue that this includes learning children's thinking body language, including a focus on written pair talk, and supporting contributions. In other words, moving beyond just having students share their own ideas, having students share what they learned from our classmate.

Mike: Yeah. I want to dig into this a little bit more. Because this idea that my read on a child's behavior influences my understanding of what's happening, but also my practice, is really interesting to me. You've really had me thinking a lot about the way that a teacher’s read on a student's engagement or participation, it has a lot to do with the cultural script for how adults and children are expected to interact, or at least what we've learned about that in our own lived experiences. I'm wondering if you could just talk a little bit about that. 

Meghan: Yeah. One way to start answering that question might be to ask everyone to take a minute to think about how you participate in a discussion. Do you use the sort of listening behaviors that teachers are told matter? Are you always sharing new ideas when you participate in a discussion? You also might want to imagine sitting down with a group of your colleagues and asking them to think about when they engage in a discussion outside of class, what does it look and feel like? Are there lots of people talking at once or people talking one at a time? Is everyone that's participating in the discussion sharing new ideas, or are they participating in other sorts of ways? And further, you might imagine asking those colleagues about their discussions outside of class as a child. What did those discussions look and feel like? One of the challenges of being teachers is that we bring our own experiences and sometimes we don't reflect on what children are experiencing. Children's experiences don't necessarily match our own, and we need to be thinking about changing our expectations or explicitly teaching what it means to participate in particular sorts of ways. Yet another layer of challenge here is a tendency to make assumptions about how students from particular cultural groups engage in discussions. You only know what you know. And teachers need opportunities to learn from their students about how they engage in discussions inside and outside of math class, and to be able to think about the connections and disconnections and the opportunities to leverage.

Mike: So, you really have me deconstructing some of the norms that were unspoken in my own childhood about being a learner, being a good student. And what you have me thinking is, some of those were voiced, some of those were unvoiced, but I'm really reflecting on how that showed up in the way that I read kids. So, I want to ask you to even go a little bit deeper. Can you share some examples of where our read on the meaning of behaviors might lead to an inaccurate understanding of students' cognitive engagement or the contributions that they might make to discourse?

Meghan: Yeah. Some of it can be thinking about sort of traditional behavior reads in a traditional sense. Oftentimes, when children have their heads down or their eyes closed or they're not looking at the speaker, the child is seen as not engaging or participating. But if we think about it, people have lots of different thinking postures, and for some people having their heads down or closing their eyes is actually the way in which they're thinking deeply about the ideas that are being shared in the discussion. And so, engagement might look for them. They may be carefully tracking and thinking about the ideas, but the way that that gets expressed may not be the way that we traditionally think about what engagement should look like in classrooms.

Mike: It feels like there's two pieces to this question about reading behavior and interpretation. One piece that you talked about there was just this idea that we need to have conversations with children. The other piece that I kept thinking about is, how might an educator interrogate their own cultural script around participation? Are there questions that educators could ask themselves or practices that they might engage in with colleagues that would help them take these things that are subconscious and unspoken and maybe raise them up? So, if you have an awareness of them, it's easy to recognize how that's influencing your read or your instructional moves.

Meghan: Yeah, I think there are kind of two pieces to this. So, one goes back to the idea that I shared about the importance of recognizing our own experiences in school as a student and our experiences out of school, both as a child and as an adult in discussions and trying to think about what are we bringing to our work as a teacher that we might need to interrogate because it may be different than the experiences of children? And at the same time, we need to be having conversations with children about what it looks like to participate in discussions in different sorts of spaces so that we can learn more about what children's experiences are outside of school. The big idea is to recognize that children's experiences are often very different from our own, and we have to be careful at the same time not to make assumptions that all children from particular communities experience participation and discussion in the same way. This can be highly variable.

Mike: I think what's really interesting about the work that you and your colleagues have done is, there's an element of it that's really about taking a step back and recognizing these ideas like cultural scripts that we have about participation and really trying to interrogate our own understandings that we've come to, and then how do we interact with kids. But on the other hand, you all have some really practical strategies and suggestions for educators on how they can use an expanded understanding of participation to create more opportunity for kids. So, I'm wondering if we can talk a little bit about some of those things.

Meghan: Absolutely. So, I have a set of four different strategies that my colleagues and I have been working on over time. So, I'm going to start by talking about task selection. Sometimes students’ cultural backgrounds and experiences in schools may be at odds, particularly around the work of critiquing the ideas of others. And this can in particular be a challenge when the critiquing is about critiquing the teacher's ideas. So, it leads to this question of, “How can we support students in learning to critique in ways that don't dismiss their own culture and experience?” So, our practical solution to working in this space is that we've used written critique tasks. So, when working with students, we'll show a fictitious person's response to a mathematics task and ask students to do three sorts of things. So, one is to describe the student's strategy in their own words. A second thing is to think about and write down the questions that they have about the student's strategy. And then the third piece is for students to think about and record what suggestions they have for the student and how they would convince the student to use those suggestions.

Meghan: So, how does this support participation? Well, it can explicitly support the work of critiquing. It's written, and it allows students to think carefully rather than needing to think on the spot. And thirdly, the student is not a classmate, which can reduce the feeling of confrontation that some students feel when engaging in critique. So, one thing that I want to name with this particular strategy around task selection and using a written critique task, is that we've recognized that the way that critiquing is often worked on in mathematics classrooms may be at odds with some students' experiences with critique outside of school. And so, we're not trying to say that students shouldn't be supported in learning to critique mathematical ideas. That's an important part of mathematical work. But rather we're trying to design a structure that's going to not dismiss students' experiences outside of school, but at the same time give them experiences with the mathematical work of critiquing.

Mike: Yeah, the questions themselves are powerful, but it seems like the choice to use a fictitious person is really critical to this task design.

Meghan: Absolutely. And as a teacher, too, it really does give us a little bit more control in terms of what is the critique that's going to unfold in that particular classroom.

Mike: It strikes me that they're able to engage in the task of critique without that feeling of conflict.

Meghan: Absolutely. It really opens up space for students to engage in that critiquing work and takes a lot of that pressure off of them.

Mike: Let's talk about the second idea.

Meghan: Alright. So, the second strategy is to use a deliberate turn and talk. In discussions, some students are ready to share their ideas right away, but other students need a chance to practice verbalizing the ideas that they're about to share. Sometimes students' ideas are not completely formed, and they need to learn how others hear the ideas to refine their arguments. Further, in multilingual classrooms, sometimes students need opportunities to refine their thinking in their home language, and importantly, they also need opportunities to develop academic language in their home language. So, in a deliberate turn and talk, a teacher deliberately pairs students to share their thinking with a partner, and the partner asks clarifying questions. The pairs might be made based on knowledge of students' home language use, their mathematical understandings, or some other important thing the teacher is thinking about as they engage in that pairing. So how might using deliberately paired turn and talks broaden participation in a discussion?

Meghan: Well, first, all students are being asked to participate and have the opportunity to refine their own mathematical argument and consider someone else's ideas. In a whole-class discussion, it's not the case that every student is likely to have that opportunity. So, turn and talks provide that opportunity. Second, turn and talks can support a broader range of students in feeling ready and willing to share their thinking in a whole group. Third, these pairs can also set up students who are not yet comfortable sharing their own ideas in whole group to be able to share someone else's idea. So, a way for them to still share ideas in whole group, even though it's not necessarily their own idea that's being shared.

Mike: So, what I'm thinking about is, if you and I were engaged in a deliberate turn and talk, what might it look like if I'm a student, you're a student and we've engaged in the norms of the deliberate turn and talk as you described them? Let's just walk through that for a second. What would it look like?

Meghan: So, in a pair turn and talk, it really has the structure of partner A, sharing their thinking, and then partner B being responsible for asking questions about the ideas that they just heard in order to further their own understanding of partner's ideas, but also to provide partner A with some feedback about the ways in which they've been expressing their ideas. So, that's pretty different than what often happens in classrooms where kids are invited to share in a discussion and they actually haven't tried verbalizing it yet, right? And they have no way of thinking about, or limited ways of thinking about, how other people might hear those ideas that they're about to share.

Mike: I think the other thing that pops up to me is that another scenario that often occurs in turn and talk is it's really turn and tell. Because one person is essentially sharing their thinking and the norms aren't necessarily that they respond, it's just that they share in kind, right? So, this idea that you're actually engaging with someone's idea feels like an important piece of what it looks like to do a deliberate turn and talk versus some of the other iterations that we've just been describing.

Meghan: Absolutely.

Mike: Well, I'm excited to hear about the third strategy.

Meghan: Alright. Our third strategy focuses on supporting participation through connection-making. So, when you think about a typical discussion in a classroom, opportunities for individual students to make explicit connections between ideas shared, are often pretty limited—or at least their opportunities to verbalize or to record in some other way. Often, only one or two students are able to share the connections. And so, a question for us has been how can we provide opportunities for students who are not yet ready to share those connections in whole group or might not have the opportunity? When you think about the fact that 28 students are not going to be able to share connections on a given day to be able to engage in the making of those connections. So, we have two different structures that we have been exploring. The first structure is really a pair share. Students are paired, if possible, with a student who used a different strategy, who has a different solution.

Meghan: Each partner explains their strategy, and then together they look for connections between their thinking. So again, this moves beyond the traditional turn and talk because in addition to sharing your thinking, there's a task that the partners are doing about thinking about the connections between those two strategies. A second sort of structure is really using a stop and jot. In this instance, the teacher selects one strategy for students to be thinking about making a connection to, and then each student jots a connection between their strategy or solution and the strategy that the teacher has selected. And they do this in their notebook or in some other written form in the classroom. And so, these two different structures can support participation by having all students have an opportunity to share their own thinking, either verbally with a partner or by recording it in written form. And all students at the same time are having an opportunity to make connections in the classroom.

Mike: I think what's interesting about that is to compare that one with the initial idea around critique. In this particular case, I'm going to make a guess that part of the reason that in this one you might actually use students from the classroom versus a fictitious student, is that connecting versus critiquing our two really different kind of social practices. Is that sensible?

Meghan: That is sensible. And I would argue that if you're going to be engaging in critique work just to say it, that part of critiquing actually is recognizing, too, what is similar and different about strategies.

Mike: Gotcha.

Meghan: Right? So, there is that piece in addition to put that out there.

Mike: Gotcha. Let's talk about the fourth one.

Meghan: Alright. So, the fourth strategy really focuses on broadening participation in the conclusion of a discussion. So, as we all know in a discussion, students hear lots of different ideas, but they don't all get to share their thinking in a discussion, nor do they all get to share what they are thinking at the end of the discussion. But we also know that students need space to consolidate their own thinking and the questions that they have about the ideas that have been shared. At the same time, teachers need access to students' thinking to plan for the next day, particularly when a discussion is not finished at the end of a given math lesson. With all of this, the challenge is that time is often tight at the end of a discussion. So, one structure that we've used has been a note to self. And in a note to self, students write a note to themselves about how they are currently thinking about a particular sort of problem at the end of a discussion. And a note to self allows students to take stock of where they are with respect to particular ideas, similar to a stop and jot. It can create a record of thinking that can be accessed on a subsequent day by students. If those notes yourself are recorded in a notebook. Again, support students and tracking on their own questions and how their thinking is changing over time, and it can provide the teacher with a window into all students' thinking.

Mike: Can you talk about the experience of watching the note to self and just seeing the impact that it had?

Meghan: So, it was day one of our mathematics program, and we had done a discussion around an unequally partitioned rectangle task, and students were being asked to figure out what fraction of the hole was shaded. And there clearly wasn't enough time that day to really explore all the different sorts of ideas. And so, Darius Robinson, who was one of the co-teachers, invited students to share some of their initial ideas about the task. And the way that Darius then ended up deciding to conclude things that day was saying to students, “I think we're going to do this thing that I'm going to call a note to self.” And he invited the students to open up their notebooks and to record how they were thinking about the different ideas that had gotten shared thus far in the discussion. There was some modeling of what that might look like, something along the lines of, “I agree with … because,” but it really opened up that space then for students to begin to record how they were thinking about otherwise ideas in math class. So, how might using a note to self-broaden participation in a discussion? Well, first of all, students have the opportunity to participate. All students are being asked to write a note to themselves. It creates space for students to engage with others' ideas that doesn't necessarily require talk, right? So, this is an opportunity to privilege other ways of participating, and it also allows for thinking and processing time for all students.

Mike: I think the other piece that jumps out for me is this idea that it's normal and to be expected that you're going to have some unfinished thinking or understanding at the end of a particular lesson or what have you, right? That partial understanding or growing understanding is a norm. That's the other thing that really jumps out about this practice is it allows kids to say, “This is where I am now,” with the understanding that they have room to grow or they have room to continue refining their thinking. I really love that about that.

Meghan: I think it's so important, right? And oftentimes, we read curriculum materials, we read through a lesson for a particular day and get the sense that everything is going to be tied off with a bow at the end of the lesson, and that we're expecting everybody to have a particular sort of understanding at the end of Section 3.5. But as we all know, that's not the reality in classrooms, right? Sometimes discussions take longer because there are really rich ideas that are being shared, and it's just not feasible to get to a particular place of consensus on a particular day. So, it is for teachers to have access to where students are. But at the same time to feel empowered, to be able to say, “I'm going to pick this up the next day, and that doesn't need to be finished on Monday, but that these ideas that we're working on Monday can flow nicely into Tuesday. And as students, your responsibility is to think about, ‘How are you thinking about the task right now?’ Jot some notes so when we come back to it tomorrow, we can pick that up together.”

Mike: Well, I think that's the other lovely piece about it, too, is that they're engaging in that self-reflection, but they've got an artifact of sorts that they can come back to and say, “Oh yeah, that's where I was, or that's how I was thinking about it.” That allows for a smoother re-engagement with this or that idea.

Meghan: Absolutely. And you can add on the pieces of notation that students might choose to do the next day as well, where they might choose to annotate their notes with notes that said, “Yesterday I was thinking this, but now I think this” as a way to further record the ideas that thinking changes over time.

Mike: So, I think before we close this interview, I want to say to you that I watched you do your presentation in Los Angeles at NCTM, and it was really eye-opening for me, and I found myself stuck on this for some time. And I suspect that there are people who are going to listen to this podcast who are going to think the same thing. So, what I want to ask you is, if someone's a listener, and this is a new set of ideas for them, do you have any recommendations for where they might go to kind of deepen their understanding of these ideas we've been talking about?

Meghan: Sure. I want to give three different sorts of suggestions. So, one suggestion is to look at the fabulous books that have been put together by Amy Lucenta and Grace Kelemanic, who are the authors of “Routines for Reasoning and Thinking for Teaching.” And I would argue that many of the routines that they have developed and that they share in those resources are ones that are really supportive of thinking about, “How do you broaden participation in mathematics discourse?” A second resource that someone might be interested in exploring is a research article that was written in 2017 by Cathy O'Connor, Sarah Michaels, Suzanne Chapin, and Alan (G.) Harbaugh that focuses on the silent and the vocal participation in learning in whole-class discussion, where they carefully looked at learning outcomes for students who were vocally expressing ideas and discussion as well as the silent participants in the discussion, and really found that there was no difference in the learning outcomes for those two groups of students. And so that's important, I think, for us to think about as teachers. At the same time, I want to be clear in acknowledging that all of what we do as teachers needs to be in relation to the learning goals that we have for students. So, sometimes our learning goals are that we want students to be able to share ideas and discussions. And if that's the case, then we actually do need to make sure that we build in opportunities for students to share their ideas verbally in addition to participating in other sorts of ways.

Mike: I'm really glad you said that because what I hear you saying is, “This isn't a binary. We're not talking about …

Meghan: Correct.

Mike: … verbal participation and other forms of participation and saying you have to choose.” I think what I hear you saying is, “If you've only thought about participation from a verbal perspective, these are ways that you can broaden access and also access your students' thinking at the same time.”

Meghan: Absolutely. The third thing to share, which has been a theme across this podcast, has really been the importance of learning from our students and talking with the children with whom we're working about their experiences, participating in discussions both in school and outside of school.

Mike: Megan, thank you so much for joining us. It really was a pleasure.

Meghan: Thank you, Mike, for the opportunity to really share all of these ideas that my colleagues and I have been working on. I want to acknowledge my colleagues, Nicole Garcia, Aileen Kennison, and Darius Robinson, who all played really important roles in developing the ideas that I shared with you today.

Mike: Fabulous. Thank you so much.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

Within the field of mathematics, there is a broad under-representation of many groups of people, especially LGBT+, people of color, undocumented people, people with disabilities, and other marginalized groups. Listen in as Sac State mathematics alumna Brianna Davis leads a group of math undergraduates through a discussion of how current students perceive this issue. The episode highlights some of the work already done at Sac State and what changes would make mathematics accessible and inclusive to all groups.

Find the transcript here: www.csus.edu/crisj/buildingjustice

Rounding Up

Season 2 | Episode 2 – Empathy Interviews

Guest: Dr. Kara Imm

Mike Wallus: 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. Kara 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.

Mike: Well, welcome to the podcast, Kara. We're excited to have you join us.

Kara Imm: Thanks, Mike. Happy to be here.

Mike: 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: 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.

Kara: 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 resumé, 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: 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 paraphrased, 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: 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: 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: Sounds great.

Mike: 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.

Kara: Great. Leanna is a third-grader. She goes to an all-girls school. I've worked in Leanna’s 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: Alright, let's give it a listen.

Leanna: Hi, I'm Leanna, and I'm 8 years old.

Kara: Hi, 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?

Leanna: Yeah.

Kara: Tell me what happened in that moment.

Leanna: Um, they said, “We have a math genius in our class.” And then they all started pointing at me.

Kara: And what was that like for you?

Leanna: It was … like, maybe, like, it was nice, but also it was kind of like, all the pressure was on me.

Kara: 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: A mathematician …

Kara: Yeah.

Leanna: … 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: 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: Maybe.

Kara: Are you a math genius? Do think, what does that even mean?

Leanna: Like, I'm really good at math.

Kara: Yeah. Do you think that's a true statement?

Leanna: Yeah, a little bit.

Kara: A little bit? Do you love math?

Leanna: Yeah.

Kara: Yeah. Have you always loved math?

Leanna: Yeah.

Kara: And so, it might be true that, like, is a math genius the same as a mathematician?

Leanna: No.

Kara: OK. Can you say how they're different?

Leanna: Like, a mathematician is, like … Like, when you're a math genius, you don't always want to be a mathematician when you grow up. A math genius is when you just are really good at math, but, like, a mathematician is when you really, like, want to be when you grow up.

Kara: Yeah.

Mike: That was fascinating to listen to. So, my first inclination is to say, as you were making meaning of what Leanna was sharing, what were some of the things that were going on for you?

Kara: Yeah, I was thinking about how math has this kind of unearned status, this measure of success in our culture that in this interview, Leanna is kind of pointing to. I was thinking about the mixed emotions she has being positioned as a math genius. It called into mind the model minority myth in which folks of Asian descent and Asian Americans are often positioned as stereotypically being good at math. And people say, “Well, this is such a lovely and respectful stereotype, who cares if it's not true?” But she later in the interview talks about the pressure of living up to this notion of math genius and what means. I think about her status in the classroom and how she has the agency to both take up this idea of math genius, and does she have the agency to also nuance it or reject it? And how that might play out in her classroom? So yeah, those are all the things that kind of come to mind as I listen to her.

Mike: I think you're hitting on some of the themes that jumped out for me; this sense that kids who are participating in particular activities have been positioned, either by their participation or by their kids' perceptions of what participation means. And I thought the most interesting part was when she said, “Well, it's nice”—but there was a long pause there. And then she talked about this sense of pressure. What it's making me think about as a practitioner is that there are perhaps ways that as a teacher, if I'm aware of that, that might change something small, some things big about the way that I choose to engage with Leanna in the classroom; that I choose to help her navigate that space that she finds herself in. There's a lot for me there as a practitioner in that small clip that helps me really see her, understand her, and think about ways that I can support her.

Kara: Yeah. And, like, from a design perspective, I huddled with her teacher later in the day, and we talked about this interview, and we thought about what would it mean to design or redesign a space where Leanna could feel really proud of who she was as a mathematician, but she didn't feel the kind of pressure that this math genius moniker is affording her. And so, ultimately, I want these interviews to be conducted by teachers so that, as you said, practitioners might show up differently for kids or think about what we might need to think more deeply about or design for kids like her. She's certainly not the only one.

Mike: Yeah, absolutely. And I think part of what's hitting me in the face is that the term “empathy interview” really is taking on new meaning, even listening to this first one. Because feeling the feelings that she's sharing with us, feeling what it would be like to be in those shoes, I've had kiddos in my class who have been identified or whose folks have chosen to have them participate in programming. And I have to confess that I don't know that I thought as much about what that positioning meant to them or what it meant about how kids would perceive them. I was just struck by how, in so many subtle ways doing an interview like this, might really shift the way that I showed up for a child.

Kara: Yeah, I think so.

Mike: Well, let's listen to another one.

Kara: OK. Maybe Matthew, should we meet Matthew?

Mike: I think we should meet Matthew.

Kara: Yeah.

Mike: Do you want to set up Matthew and give us a sense of what we might need to know about the context?

Kara: Absolutely. Matthew is a fifth-grader who describes, in my conversation with him, several years of what he calls “not good” years in math. And he doesn't enjoy mathematics. He doesn't think he's good at it. He has internalized, he's really blamed himself and taken most of the responsibility for those “bad“ years of learning. When I meet him, he's a fifth-grader, and he has written a mathography at the invitation of his classroom teacher. This is a practice that's part of this school. And in his mathography as a fifth-grader, he uses the word “evolving,” and he tells the story of how he's evolving as a mathematician. That alone is pretty profound and beautiful that he has the kind of insight to describe this kind of journey with mathematics. And he really just describes a fourth-grade teacher who fundamentally changed his relationship to mathematics, his sense of himself, and how he thinks about learning.

Mike: Let's give it a listen.

Kara: Maybe we'll end, Matthew, with: If people were thinking about you as—and maybe there's other Matthews in their class, right—what kinds of things would've helped you back in kindergarten, first and second grade to just feel like math was for you? It took you until fourth grade, right …

Matthew: Yeah.

Kara: … until you really had any positive emotions about math? I'm wondering what could we have done for younger Matthew?

Matthew: Probably, I think I should have paid a lot more attention.

Kara: But what if it wasn't about you? What if it's the room and the materials and the teacher and the class?

Matthew: I think it was mostly just me, except for some years it was really, really confusing.

Kara: OK.

Matthew: And when … you didn't really want in third grade or second grade, you didn't want to be the kid that's always, like, “Hey, can you help me with this?” or something. So that would be embarrassing for some people.

Kara: OK. You just made air quotes right, when you did embarrassing?

Matthew: Yeah.

Kara: Was it embarrassing to ask for help?

Matthew: It wasn't embarrassing to ask for help, and now I know that. But I would always not ask for help, and I think that's a big reason why I wasn't that good at math.

Kara: Got it. So, you knew in some of these math lessons that it was not making sense?

Matthew: It made no sense.

Kara: It made no sense.

Matthew: And then I was, like, so I was in my head, “I think I should ask, but I also don't want to embarrass myself.”

Kara: Hmm.

Matthew: But also, it's really not that embarrassing.

Kara: OK, but you didn't know that at the time. At the time it was like, “Ooh, we don't ask for help.”

Matthew: Yeah.

Kara: OK. And did that include asking another kid for help? You didn't ask anybody for help?

Matthew: Um, only one of my friends that I knew for a really long time …

Kara: Hmm.

Matthew: He helped me. So, I kind of got past the first stage, but then if he was absent on those days or something, then I'd kind of just be sitting at my desk with a blank sheet.

Kara: Wow, so it sounds like you didn't even know how to get started some days.

Matthew: Yeah, some days I was kind of just, like, “I'm not even going to try.”

Kara: “I’m not” … OK.

Matthew: But now I'm, like, “It's not that big of a deal if I get an answer wrong.”

Kara: Yeah, that's true. Right?

Matthew: “I have a blank sheet. That is a big deal. That's a problem.”

Kara: So having a blank sheet, nothing written down, that is a bigger problem for you than, like, “Oh, whoops, I got the answer wrong. No big deal.”

Matthew: I'd rather just get the answer wrong because handing in a blank sheet would be, that would probably be more embarrassing.

Mike: Oh, my goodness. There is a lot in a little bit of space of time.

Kara: Yeah. These interviews, Mike, are so rich, and I offer them to this space and to teachers with such care and with such a deep sense of responsibility ’cause I feel like these stories are so personal. So, I'm really mindful of, can I use this story in the space of Matthew for a greater purpose? Here, I feel like Matthew is speaking to all the kind of socio-mathematical norms in classrooms. And I didn't know Matthew until this year, but I would guess that a kid like Matthew, who is so quiet and so polite and so respectful, might've flown under the radar for many years. He wasn't asking for help, but he was also not making trouble. It makes me wonder, “How would we redesign a class so that he could know earlier on that asking for help—and that this notion that in this class, mathematics—is meant to make sense, and when it doesn't make sense, we owe it to ourselves and each other to help it make sense?” I think it's an invitation to all of us to think about, “What does it mean to ask for help?” And how he wants deep down mathematics to make sense. And I agree with him, that should be just a norm for all of us.

Mike: I go back to the language that you used at the beginning, particularly listening to Matthew talk, “the stories that we tell ourselves.” The story that he had told himself about what it meant to ask for help or what that meant about him as a person or as a mathematician.

Kara: Yeah. I mean, I am trained as a kind of qualitative researcher. So as part of my dissertation work, I did all kinds of gathering data through interviews and then analyzing them. And one of the ways that is important to me is thinking about kind of narrative analysis. So, when Matthew tells us the things that were in his head, he tells you the voice that his head is saying back to him. Kids will do that. Similarly, later in the interview I said, “What would you say to those kids, those kids who might find it?” And what I was interested in is getting him to articulate in his own voice what he might say to those children. So, when I think about stories, I think about when do we speak in a first person? When do we describe the voices that are in our heads? When do we quote our teachers and our mothers and our cousins? And how that's a powerful form of storytelling, those voices.

Mike: Well, I want to listen to one more, and I'm particularly excited about this one. This is Nia. I want to listen to Nia and have you set her up. And then I think what I want to do after this is talk about impact and how these empathy interviews have the potential to shift practice for educators or even school for that matter. So, let's talk about Nia and then let's talk about that.

Kara: You got it. Nia is in this really giant classroom of almost 40 kids, fifth-graders, and it's co-taught. It's purposely designed as this really collaborative space, and she uses the word “collaboration,” but she also describes how that's a really noisy environment. On occasion, there's a teacher who she describes pulling her into a quieter space so that she can concentrate. And so, I think that's an important backstory for her just in terms of her as a learner. I ask her a lot of questions about how she thinks about herself as a mathematician, and I think that's the clip we're going to listen to.

Mike: Alright, let's listen in.

Nia: No, I haven't heard it, but …

Kara: OK. I wonder what people mean by that, “I'm not a math person.”

Nia: I'm guessing, “I don't do math for fun.”

Kara: “I don’t do math for fun.” Do you do math for fun?

Nia: Yes.

Kara: You do? Like, what's your for-fun math?

Nia: Me and my grandma, when we were in the car, we were writing in the car. We had this pink notebook, and we get pen or a pencil, and she writes down equations for me in the backseat, and I do them and she times me, and we see how many questions I could get right in, like, 50 seconds.

Kara: Oh, my gosh. What's an example of a question your grandma would give you?

Nia: Like, they were just practice questions, like, three times five, five times eight. Well, I don't really do fives because I already know them.

Mike: So, we only played a real tiny snippet of Nia. But I think one of the things that's really sticking out is just how dense these interviews are with information about how kids think or the stories that they've told themselves. What strikes you about what we heard or what struck you as you were having this conversation with Nia at that particular point in time?

Kara: For me, these interviews are about both storytelling and about identity building. And there's that dangerous thinking about two types of people, math people and non-math people. I encounter adults and children who have heard of that phrase. And so, I sometimes offer it in the interview to find out what sense do kids make of that? Kids have told me, “That doesn't make sense.” And other kids have said, “No, no, my mom says that. My mom says she's not a math person.” So, she, I'm playing into it to see what she says. And I love her interpretation that a math person is someone who does math for fun. And truthfully, Mike, I don't know a lot of kids who describe doing math for fun. And so, what I loved about that she, A: She a described a math person's probably a person who, gosh, enjoys it, gets some joy or pleasure from doing mathematics.

Kara: But then the granularity of the story she offers, which is the specific pink notebook that she and her grandmother are passing back and forth in the backseat of the car, tell you about mathematics as a thing that she shares a way of relating to her grandmother. It's been ritualized, and really all they're doing if you listen to it is, her grandmother's kind of quizzing her on multiplication facts. But it's such a different relationship to multiplication facts because she's in relationship to her grandmother. They have this beautiful ongoing ritual. And quite honestly, she's using it as an example to tell us that's the fun part for her. So, she just reminds us that mathematics is this human endeavor, and for her, this one ritual is a way in which she relates and connects to her grandmother, which is pretty cool.

Mike: So, I want to shift a little bit and talk about a couple of different things: the types of questions that you ask, some of the norms that you have in mind when you're going through the process, and then what struck me about listening to these is you're not trying to convince the kids who you're interviewing of anything about their current thinking or their feelings or trying to shift their perspective on their experience. And I'm just wondering if you can think about how you would describe the role you're playing when you're conducting the interview. ’Cause it seems that that's pretty important.

Kara: Yeah. I think the role I'm playing is a deep listener. And I'm trying to create space. And I'm trying to make a very, very, very safe environment for kids to feel like it's OK to tell me a variety of stories about who they are. That's my role. I am not their classroom teacher in these interviews. And so, these interviews probably look and sound differently when the relationship between the interviewer and the interviewee is about teachers and students and/or has a different kind of power differential. I get to be this frequent visitor to their classroom, and so I just get to listen deeply. The tone that I want to convey, the tone that I want teachers to take up is just this fascination with who they are and a deep curiosity about their experience. And I'm positioned in these interviews as not knowing a lot about these children.

Kara: And so, I'm actually beautifully positioned to do what I want teachers to do, which is imagine you didn't know so much. Imagine you didn't have the child's cumulative file. Imagine you didn't know what they were like last year. Imagine you didn't know all that, and you had to ask. And so, when I enter these interviews, I just imagine, “I don't know.” And when I'm not sure, I ask another smaller question. So I'll say, “Can you say more about that?” or “I'm not sure if you and I share the same meaning.” The kinds of questions I ask kids—and I think because I've been doing this work for a while, I have a couple questions that I start with and after that I trust myself to follow the lead of the children in front of me—I often say to kids, “Thank you for sitting down and having a conversation with me today. I'm interested in hearing kids' stories about math and their math journey, and somebody in your life told me you have a particularly interesting story.” And then I'll say to kids sometimes, “Where do you want to start in the story?” And I'll try to give kids agency to say, “Oh, well, we have to go back to kindergarten” or “I guess we should start now in high school” or kids will direct me where they think are the salient moments in their own mathematical journey.

Mike: And when they're sharing that story, what are the types of questions that you might ask along the way to try to get to clarity or to understanding?

Kara: Great question. I'm trying to elicit deep emotion. I'm trying to have kids explain why they're telling me particular stories, like, what was significant about that. Kids are interesting. Some kids in these interviews just talk a lot. And other kids, I've had to really pepper them with questions and that has felt a little kind of invasive, like, this isn't actually the kind of natural conversation that I was hoping for. Sometimes I'll ask, “What is it like for you or how do you think about a particular thing?” I ask about things like math community, I ask about math partners. I ask about, “How do you know you're good at math and do you trust those ways of knowing?” I kind of create spaces where we could have alternative narratives. Although you're absolutely right, that I'm not trying to lead children to a particular point of view. I'm kind of interested in how they make sense.

Mike: One of the things that, you used a line earlier where you said something about humanizing mathematics, and I think what's striking me is that statement you made: “What if you didn't have their cumulative report card?” You didn't have the data that tells one story, but not necessarily their story. And that really is hitting me, and I'm even feeling a little bit autobiographical. I was a kid who was a lot like Matthew, who, at a certain point, I just stopped raising my hand because I thought it meant something about me, and I didn't want people to see that. And I'm just struck by the impact of one, having someone ask you about that story as the learner, but also how much an educator could take from that and bring to the relationship they had with that child while they were working on mathematics together.

Kara: You said a lot there, and you actually connect to how I think about empathy interviews in my practice now. I got to work with Rochelle Gutiérrez this summer, and that's where I learned deeply about her framework, rehumanizing mathematics. When I do these empathy interviews, I'm living in this part of her framework that's about the body and emotions. Sometimes kids in the empathy interview, their body will communicate one thing and their language will communicate something else. And so, that's an interesting moment for me to notice how body and motions even are associated with the doing of mathematics. And the other place where empathy interviews live for me is in the work of “Street Data,” Jamila Dugan and Shane Safir's book, that really call into question this idea that what is measurable and what is quantifiable is really all that matters, and they invite us to flip the data dashboard.

Kara: In mathematics, this is so important ’cause we have all these standardized tests that tell children about who they are mathematically and who they're about to become. And they're so limiting, and they don't tell the full story. So, when they talk about “Street Data,” they actually write about empathy interviews as a way in which to be humanizing. Data can be liberatory, data can be healing. I feel that when I'm doing these interviews, I have this very tangible example of what they mean because it is often the case that at the end of the interview—and I think you might've had this experience just listening to the interview—there's something really beautiful about having a person be that interested in your story and how that might be restorative and might make you feel like, “There's still possibility for me. This isn't the last story.”

Mike: Absolutely. I think you named it for me, which is, the act of telling the story to a person, particularly someone who, like a teacher, might be able to support me being seen in that moment, actually might restore my capacity to feel like, “I could do this” or “My fate as a mathematician is not sealed.” Or I think what I'm taking away from this is, empathy interviews are powerful tools for educators in the sense that we can understand our students at a much deeper level, but it's not just that. It's the experience of being seen through an empathy interview that can also have a profound impact on a child.

Kara: Yes, absolutely. I'm part of a collaboration out of University of California where we have thought about the intersection of disability and mathematics, and really thinking about how using the tools of design thinking, particularly the empathy interview can be really transformative. And what the teachers in our studies have told us is that just doing these empathy interviews—and we're not talking about interviewing all the kids that you teach. We're talking about interviewing a select group of kids with real intention about, “Who's a kid who has been marginalized?” And/or “Who's a kid who I don't really know that much about and/or I don't really have a relationship with?” Or “Who's a kid who I suspect doesn't feel seen by me or doesn't feel, like, a deep sense of belonging in our work together?” Teachers report that just doing a few of these interviews starts to change their relationship to those kids.

Kara: Not a huge surprise. It helped them to name some of the assumptions they made about kids, and it helped them to be in a space of not knowing around kids. I think the other thing it does for teachers that we know is that they describe to do an empathy interview well requires a lot of restraint, restraint in a couple of ways. One, I'm not fixing, I'm not offering advice. I'm also not getting feedback on my teaching. And I also think it's hard for teachers not to insert themselves into the interview with our own narratives. I really try to make sure I'm listening deeply and I'm painting a portrait of this kid, and I'm empathetic in the sense I care deeply and I'm deeply listening, which I think is a sign of respect, but the kids don't need to know about my experience in the interview. That's not the purpose.

Mike: We could keep going for quite a long time. I'm going to make a guess that this podcast is going to have a pretty strong on a lot of folks who are out in the field listening.

Kara: Hmm.

Mike: If someone was interested in learning more about empathy interviews and wanted to explore or understand more about them, do you have any particular recommendations for where someone might go to continue learning?

Kara: Yes, and I wish I had more, but I will take that as an invitation that maybe I need to do a little bit more writing about this work. I think the “Street Data” is an interesting place where the co-authors do reference empathy interviews, and I do think that they have a few videos online that you could see. I think Jamila Dugan has an empathy interview that you could watch and study. People can write me and/or follow me. I'm working on an article right now. My colleagues in California and I have a blog called “Designing4Inclusion,” “4” being the number four, and we've started to document the work of empathy and how it shows up in teachers’ practice there.

Mike: Well, I want to thank you so much for joining us, Kara. It has really been a pleasure talking with you.

Kara: Thank you, Mike. I was really happy to be invited.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

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's a little bit tentative. Like, ‘I have it, but I don't have it.’ That's another part, a natural part of learning. When you're first learning something new, you kind of have it, then you’ve got to try it again, and it takes a while for it to become something you're comfortable doing or knowing.

Mike: So, this is fascinating because you're making me think about this, kind of, challenge that we sometimes find ourselves facing in the field where, at the end of a lesson or a unit, there's this idea that if kids don't have what we would consider mastery, then there's a deficit that exists. And I think what you're making me think is that framing this as either developing understanding or fragile understanding is a lot more productive in that it helps us imagine what pieces have students started to understand and where might we go next? Or like, what might we build on that they've started to understand as opposed to just seeing partial understanding or fragile understanding from a deficit perspective.

Beth: Right. I love this point because I think when we think about mastery, it's all or nothing. But that's not learning either. Maybe on an exam or on a test or on assessment, yes, you have it or you don't have it. You've mastered or you haven't. But again, if we looked at it developmentally that ‘I have some partial understanding or I have it and … I'm inconsistent in that,’ that's OK. I could also think, ‘Well, should I have a task that will keep bringing this up for students so that they can continue to build that rich understanding and move along the trajectory toward what we think of as mastery, which means that I know it now, and I'm never going to have to learn it again?’ I don't know that all things we call mastery are actually mastered at that time. We say they are.

Mike: So, I want to pick up on what you said here because in the book there's something about the role of tasks in strengths-based teaching and learning. And specifically, you talk about ‘the cumulative impact that day-to-day tasks have on what students think mathematics is and how hard and how long they should have to work on ideas so that they make sense.’ That kind of blows me away.

Beth: Well, I want to know more about why it blows you away.

Mike: It blows me away because there's two pieces of the language. One is that the cumulative impact has an effect on what students actually think mathematics is. And I think there's a lot there that I would love to hear you talk about. And then also this second part, it has a cumulative impact on how hard and how long kids believe that they should have to work on ideas in order to have them be sensible.

Beth: OK, thank you so much for talking about that a little bit more. So, there's two ways to think about that. One is, and I've done this with teams of teachers, and that's bring in a week's worth of tasks that you designed and taught for two weeks. And I call this a ‘task autopsy.’ It's a really good way because you've done it. So, bring it in and then let's talk about, do you have mostly conceptual ideas? How much time do students get to think about it? Or are students mimicking a procedure or even a solution strategy that you want them to use or a model? Because if most of the time students are mimicking or repeating or modeling in the way that you've asked them, then they're not necessarily reasoning. And they're building this idea that math means that ‘You tell me what I'm supposed to do, I do it, yay, I did it.’ And then we move on to the next thing. 

Beth: And I think that sometimes we have to really do some self-talk about this. I show what I value and what I believe in those decisions that I'm making on a daily basis. And even if I say, ‘It's so important for you to reason, it's so important for you to make sense of it.’ If all the tasks are, ‘You do this and repeat what I've shown you,’ then students are going to take away from that, that's what math is. And we know this because we ask students, ‘What is math?’ Math is, ‘When the teacher shows me what to do, and I do it, and I make my teacher happy.’ And they say lots of things about teacher pleasing because they want to do what they've been asked to d,o and they want to repeat it and they want to do well, right? Or do they say, ‘Yeah, it's problem-solving. It's solving a problem, it's thinking hard. Sometimes my brain hurts. I talk to other students about what I'm solving. We share our ideas.’ We know that students come away with big impressions about what math means based on the daily work of the math class.

Mike: So, I want to take the second part up now because you also talk about what I would call ‘normalizing productive struggle’ for kids when they're engaged in problems. What does that mean and what might it sound like for an educator on a day-to-day basis?

Beth: So, I happened to be in a classroom yesterday. It was a fifth-grade classroom, and the teacher has been really working on normalizing productive struggle. And it was fabulous. I just happened to stop in, and she stopped everything to say, ‘We want to have this conversation in front of you.’ And I said, ‘All right, go for it.’ And the question was, ‘What does productive struggle feel like to you and why is it important?’ That's what she asked her fifth-graders. And they said, ‘It feels hard at first. And uh, amazing at the end of it. Like, you can't feel amazing unless you've had productive struggle.’ We're taking away that opportunity to feel so joyous about the mathematics that we're learning because we got to the other side. And some of the students said, ‘It doesn't feel so good in the beginning, but I know I have to remember what it's going to feel like if I keep going.’ I was blown away. I mean, they were like little adults in there having this really thoughtful conversation. And I asked her what … she said, ‘We have to stop and have this conversation a lot. We need to acknowledge what it feels like because we're kind of conditioned when we don't feel good that somebody needs to fix it.’

Mike: Yeah, I think what hits me is there's kind of multiple layers we consider as a practitioner. One layer is, do I actually believe in productive struggle? And then part two is, what does that look like, sound like? And I think what I heard from you is, part of it is asking kids to engage with you in thinking about productive struggle, that giving them the opportunity to voice it and think about it is part of normalizing it.

Beth: It's also saying, ‘You might be feeling this way right now. If you're feeling like this,’ like for example, teaching a task and students are working on a task trying to figure out how to solve it and, and it's starting to get a little noisy and hands start coming up, stopping the class for a second and saying, ‘If you're feeling this way, that's an OK way to feel,’ right? ‘And here's some things we might be thinking about. What are some strategies’—like re-sort-of focusing them on how to get out of that instead of me fixing it—like, ‘What are some strategies you could think about? Let's talk about that and then go back to this.’ So, it's the teacher acknowledging. It's allowing the students to talk about it. It's allowing everybody … it's not just making students be in productive struggle, or another piece of that is ‘just try harder.’ That's not real helpful.  Like, OK, ‘I just need you to try harder because I'm making you productively struggle.’ I don't know if anyone has had someone tell them that, but I used to run races and when someone said, ‘Try harder’ to me, I'm like, ‘I'm trying as hard as I can.’ That isn't that helpful. So, it's really about being very explicit about why it's important. Getting students to the other side of it should be the No. 1 goal. And then addressing it. ‘OK, you experienced productive struggle, now you did it. How do you feel now? Why is it worth it?’

Mike: I think what you're talking about feels like things that educators can put into practice really clearly, right? So, there's the fron- end conversation maybe about normalizing. But there's the backend conversation where you come back to kids and say, ‘How do you feel once this has happened? It feels amazing.’ This is why productive struggle is so important because you can't get to this amazingness unless you're actually engaged in this challenge, unless it feels hard on the front end. And helping them kind of recalibrate what the experience is going to feel like.

Beth: Exactly. And another example of this is this idea of … so I had a pre-service teacher teaching a task. She got to teach it twice. She taught it in the morning. Students experienced struggle and were puffed up and running around, so engaged when they solved it. Beyond proud. ‘Can we get the principal in here? Who needs to see this, that we did this?’ And then she got some feedback to reduce the level of productive struggle for the second class based on expectations about the students. And she said the engagement, everything went down. Everything went down, including the level of productive struggle went way down. And so, the excitement and joy went way down, too. And so, she did her little mini-research experiment there.

Mike: So, I want to stay on this topic of what it looks like to enact these practices. And there are a couple practices in the book that really jumped out at me that I'd like to just take one at a time. So, I want to start with this idea of giving kids what you would call a ‘walk-back option.’ What's a walk-back option?

Beth: So, a walk-back option is this opportunity once you've had this conversation—or maybe one-on-one, or it could be class conversation—and a walk-back option is to go look at your work. Is there something else that you'd like to change about it? One of the things that we want to be thinking about in mathematics is that solutions and pathways and models and strategies are all sort of in flux. They're there, but they're not all finished all the time. And after having some conversation or time to reason, is there something that you'd like to think about changing? And really building in some of that mathematical reflection.

Mike: I love that. I want to shift and talk about this next piece, too, which is ‘rough-draft thinking.’ So, the language feels really powerful, but I want to get your take on, what does that mean and how might a teacher use the idea of rough-draft thinking in a classroom?

Beth: So rough-draft thinking is really Mandy Jansen's work that we brought into the strengths work because we saw it as an opportunity to help lift up the strengths that students are exhibiting during rough-draft thinking. So, rough-draft thinking is this idea that most of the time ( chuckles ), our conversations in math as we're thinking through a process is rough, right? We're not sure. We might be making a conjecture here and there. We want to test an idea. So, it's rough, it's not finished and complete. And we want to be able to give students an opportunity to do that talking, that thinking and that reasoning while it is rough, because it builds reasoning, it builds opportunities for students to make those amazing connections. You know, just imagine you're thinking through something, and it clicks for you. That's what we want students to be able to do. So, that's rough-draft thinking and that's what it looks like in the math classroom. It's just lots of student talk and lots of students acknowledging that ‘I don't know if I have this right yet, but here's what I'm thinking. Or I have an idea, can I share this idea?’ I watched a pre-service teacher do a number talk and a student said, ‘I don't know if this is going to work all the time, but can I share my idea?’ Yes, that's rough-draft thinking. ‘Let's hear it. And wow, how brave of you and your strength and risk-taking. Uh, come over here and share it with us.’

Mike: Part of what I'm attracted to is even using that language in a classroom with kids, to some degree it reduces the stakes that we traditionally associate with sharing your thinking in mathematics. And it normalizes this idea that you just described, which is, like, reasoning is in flux, and this is my reasoning at this point in time. That just feels like it really changes the game for kids.

Beth: What you hear is very authentic thinking and very real thinking. And it's amazing because even very young children—young children are very at doing this. But then as you move, students start to feel like their thinking has to be polished before it's shared. And then that gives other students who may be on some other developmental trajectory in their understanding, so much more afraid to share their rough-draft thinking or their thoughts or their ideas because they think it has to be at the polished stage. It's very interesting how this sort of idea has developed that you can't share something that you think in math because it's got to be right and completed. And everything's got to be perfect. And before it gets shared, because, ‘Wait, we might confuse other people.’ But students respond really beautifully to this.

Mike: So, the last strategy that I want to highlight is this one of a ‘math amendment.’ I love the language again. So same question, how does this work? What does it look like?

Beth: OK, so how it works is that you have done some sharing in the class. So, for example, you may have already shared some solutions to a task. Students have been given a task they're sharing, they may be sharing a pair-to-pair share or a group-to-group share, something like that. It could be whole class sharing. And then you say, ‘Hmm, you've heard lots of good ideas today, lots of interesting thinking and different strategies. If you'd like to provide a math amendment, which is a change to your solution in addition, something else that you'd like to do to strengthen it, you can go ahead and do that and you can do it in that lesson right there.’ Or what's really, what we're finding is really powerful, is to bring it back the next day or even a few days later, which connects us back to this idea of what you were saying, which is, ‘Is this mastered? Where am I on the developmental trajectory?’ So, I'm just strengthening my understanding, and I'm also hearing … I'm understanding the point of hearing other people's ideas is to go and try them out and use them. And we're really allowing that. So, this is take, this has been amazing, the math amendments that we're seeing students do, taking someone else's idea or a strategy and then just expanding on their own work. And it's very similar to, like, a writing piece, right? Writing. You get a writing piece and you polish and you polish. You don't do this with every math task that you solve or problem that you solve, but you choose and select to do that.

Mike: Totally makes sense. So, before we go, I have the question for you. You know, for me this was a new idea. And I have to confess that it has caused me to do a lot of reflection on language that I used when I was in the classroom. I can look back now and say there are some things that I think really aligned well with thinking about kids' assets. And I can also say there are points where, gosh, I wish I could wind the clock back because there are some practices that I would do differently. I suspect there's probably a lot of people where this is a new idea that we're talking about today. What are some of the resources that you'd recommend to folks who want to keep learning about strengths-based or asset-based teaching and learning?

Beth: So, if they're interested, there's several … so strengths-based or asset-based is really the first step in building equity. And TODOS, they use the asset-based thinking, which is mathematics for all organization. And it's a wonderful organization that does have an equity tool that would be really helpful.

Mike: Beth, it has been such a pleasure talking to you. Thank you for joining us. 

Beth: Thank you so much. I appreciate it. It was a good time.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

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 Work Place so that they can develop that skill at a deeper level. It helps me during that time to really facilitate my students’ practice; help students make the most of their practice time so that as they're going through the Work Place, it's not just a set of rules and procedures that they're following. That they're really thinking about what they're doing and being strategic with those skills as well. So that's my opportunity to really help and focus in on my small groups and provide the support that students need. Or maybe I want them to advance their skills, go a little bit deeper so that they are working at a little bit different level.

Mike: You know, I'm really interested in this idea that Work Places present an opportunity to listen to students’ thinking in real time. I'm wondering if you can talk about an experience where you were able to tuck in with a small group and listen to their thinking and use what you learned to inform your teaching.

Lori: ( chuckles ) One experience kind of stands out to me more than others just because it helped me understand that I need to not assume that my students are thinking about, or thinking in a specific way. So, there was one student, they were playing the Work Place game in grade 3, Loops & Groups, and she had spun a six and rolled, I think, a six as well. So, her problem was to solve six times six. And this student had actually been in front of the class just a few days before, and several times actually when I had worked with her, had solved a problem similar to this by thinking of it as three times six and three times six, which is a great strategy. But what I really wanted this student to develop was some flexibility. 

Lori: So, I asked her to explain her thinking, and I fully expected her to solve it: “Oh, yeah. I thought of it as three times six and three times six. And when I add those two together, I get 36.” And she totally shocked me. ( laughs ) She said, “Oh, I, I thought of it as five times six, and I know what five times six is. That's 30. And if I just add one more set of six, I get 36. So, she had already developed another strategy, which was not what I was expecting. With that, her partner was a little bit confused and said, “I don't understand how you could do that.” So, I asked this little girl if she could use tile maybe to explain her thinking to her friends. So, we got out the tile. She set it up and she explained this thinking to her partner. And her partner was still a little bit unsure, not really sure she could use that with her own thinking. But what it did was, in the future, just days later, that partner started trying that particular strategy. So, it taught me several things. First of all, don't assume. You don't always know what students are thinking. And also, students are their peers’ best teachers. It really encouraged her partner to try that method just a few days later.

Mike: We kind of zoomed really in on a pair of children and, and kind of the impact. The other thing that it makes me think is, by doing the fly on the wall, you as a teacher get a better sense of kind of the themes around thinking that are happening across the classroom.

Lori: Yeah. You definitely do get that, that perspective. And I think the questioning that you use also will help draw that out. Asking students to explain their thinking: “How did you solve the problem? How could you check your work? Is there a different strategy that you could use that would help you make sure that the answer you came up with, the first strategy you used, was correct?” Those kinds of questions always seem to really help students kind of pull out that thinking and be able to explain what they were doing.

Mike: Lori, thank you so much for joining us today. It has really been a pleasure to have you on the podcast and to be able to talk about this.

Lori: You bet. Thank you for having me. It was fun.

Mike: I want to thank all of you who've listened in during the first season of Rounding Up. We're going on a short break this summer, but we'll be back for Season 2 in September. Before we go, we're wondering what topics you'd like us to explore, what guests you'd like to hear from, and what questions you'd like us to take up in Season 2. This week's episode includes a link you can use to share your ideas with us. Let us know what you're thinking about, and we'll use your ideas to inform the topics we consider in Season 2. 

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

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 as though it becomes a second question about how do you work with children to actually say what we call this, or the way that we name fractions is—that's a different question, as opposed to, “Do you understand equal partitioning, conceptually?”

Juanita: Yeah. So, you're pointing at something that I've found in my research in the past. Oftentimes students will use the word half. And verbally explaining, use the word to mean that they're trying to equally partition a piece of a bar. They'll say, “Well, I cut it into halves.” And then when we look at the document, they're pointing to the lines, the partition lines, that are within the bar. And that's what they're referring to. So, we know that they don't necessarily mean that the part itself is a half, but that the partition is what they're indicating. It means that it's a half. And it's this idea that it's behind … languages really attained to this development over time, where students really think about their prior experiences, as in, “I've cut items before. And those cuts before have been halves.” And so, that particular prior knowledge can transfer into new knowledge. And so, there's this disjuncture, or there's this complexity, within the language communication and those actions. And that's why it's important not just to value the verbal communication—but also nonverbals—because they might mean something else.

Mike: Well, part of what you're making me think about, too, is in practice, particularly the way that you described that, Juanita, was this idea that my prior knowledge, my lived experience led me to call the partitions “half.” And the mathematical piece of that is, like, “I understand equal partitioning. The language that I use to describe partitioning is the language of half.” So, my wondering for you is, what would it look like to value the child's partitioning and value the fact that they used this idea of partitioning when they were thinking about halves—and then also build on that to help them have the language of, “We call this type of a partition a third or a fourth,” or what have you.

Juanita: So, this is one of those conundrums that I've talked to and discussed with other colleagues, and we talk about how sometimes they're just not ready for it. And so, when we are trying, and that's the other thing, right? Honoring what they say and taking it as they're saying it. And sometimes it's OK not to correct that. So, because we as the teachers have that, you know, we're honoring their thinking as it is, and eventually that language will develop. It eventually will become where they're no longer calling the partitions halves, and they're calling them appropriately, and they're using the part instead. So, it takes time for the student to really understand that connection. So, if we just say it and we tell them, it doesn't necessarily mean it's going to transfer and that they're going to pick up on that. So, I often try not to tell them, and I just let them explain how they're thinking and how they're saying.

Juanita: And if I honor their nonverbal ways, then I definitely can see what they mean by halves, that they're not necessarily thinking of the part, they're thinking of the partition itself. And so, that is a very important, nuanced, mathematical evolution in their knowledge. And that sometimes, we as teachers try and say, “Oh, well, we should just tell him how it is.” Or how we should develop the appropriate language. And in some instances, it might be OK. But I think most often I would defer not to do something like that because like I said, I still can access their mathematical thinking even if they don't have that language yet. ( chuckles )

Mike: That's super helpful. I think we could probably do a podcast …

Juanita: On that alone? ( laughs )

Mike: The nuances of thinking about that decision. But I want to ask you before we close about whole group. Let's talk a little bit about whole group and what it looks like to value nonverbal communication in a whole group setting. Tell me your thinking.

Juanita: Yeah, so this one is a fascinating one that I've recently come across in my own work. And I have to say, it takes a lot of effort on the part of the teacher to enact these things in the classroom, but it is possible. And so, I'll share an example of what I came across in my practice. So, if this was a bilingual classroom, and the teacher was asking students to participate silently and in written form to attend to each other's mathematical ideas, and they had examples. They had to solve a multiplication area problem individually, and then the teacher would post the student's solutions on a large poster paper and then ask all of the students to go around the room with a sticky note offering comments to each of their peer solutions. And so, what we found was just fascinating because the students were able to really dive deep into the students’ solutions.

Juanita: So, they were more deeply involved in those mathematical ideas with … when you took out the verbal communication. We had an instance where a student was like, “Well, you solved it this way, and I noticed that you had these little pencil marks on each of those squares.” And the student was saying, “Did you count 25 or did you count 26? I think you missed one.” And so, the gestures and the marks, the pencil marks on the piece of paper, that's how detailed the students were kind of attending to each other's thinking. So, they were students that were offering ideas to other students’ solutions. So, they were saying, “Well, what if you thought about it this way?” And they would write their explanation of that strategy of how they would solve it instead of how the student actually did it. And so, it was just fantastical. We were just amazed by how much richness there was to their explanations. Had the teacher done this particular activity verbally, then I wonder how many students would have actually participated. Right? So that was one of our bigger or larger questions, was noticing how many students participated in the level and the depth of their justifications for each other, versus had the teacher done this verbally with the students and had them communicate in a whole group discussion. How many students would've been able to do this? So, it is just fascinating. ( chuckles )

Mike: You touched on some of the things that were coming to mind as I heard you describe this practice, and I'd love your take on it. One of the things that strikes me about this strategy of posting solutions and then asking kids to use Post-it Notes to capture the comments or capture the noticings: Does it have the potential to break down some of the status dynamics that might show up in a classroom if you're having this conversation verbally? What I mean by that is, kids recognize that when someone speaks who they've perceived as, like, “Well, that person understands it, so I'm going to privilege their ideas.” That kind of goes away, or at least it's minimized, in the structure that you described.

Juanita: That is correct. So, I do a lot of writing on also thinking about culturally sustaining pedagogies in our teaching of practice of math. And some of the things that we find, is that a lot of the students that do participate verbally tend to be white monolinguals. And that oftentimes the teacher or other students privilege their knowledge over the student of color. And so being able to participate in nonverbal ways in this manner really showcases that everybody's knowledge can be privileged. And so, those kind of dynamics within the classroom go away. And so, it really highlights that everybody is valued equally, and that everybody can contribute to these ideas, and that everybody has a voice. That's one of the reasons why this particular piece is just dear to my heart, is because it really showcases to teachers that this can be done in the classroom.

Mike: Yeah, I've said this oftentimes on the podcast. I find myself wanting to step back into my classroom role and try this protocol out. It just feels really powerful. Let me go back to something that I wanted to clarify. So, as we've talked about practices that value nonverbal communication, a question that I've been forming and that I suspect other people might be wondering about is, I don't think you're saying that teachers have to either choose to value verbal or nonverbal communication.

Juanita: Yes, that is correct. So, I often do both. ( laughs ) It's a mixture of both. Students will communicate verbally to some extent in the same strategy and nonverbally at the same time. And valuing all forms of communication is most important. In my practice as a bilingual teacher and teaching bilingual students, I've also understood that language can't be the sole focus. And the nonverbal cues also highlighted in that communication are just as important as the language, as the bilingualism, when we're communicating ideas. And so, as teachers, there's a law that we also have to pay attention to. So, it's not just that it's nonverbal or verbal communication, but it's also how we approach the teaching, right? Because we as teachers can definitely take over students' thinking and not necessarily pay attention to what they're actually saying. So, only valuing verbal communication would be detrimental to the student.

Juanita: So, it has to be a little bit of both and a mixture of everything. I've had students [who] have tried to show me in gestures alone with no written comments on a piece of paper, and that sometimes can work. I've had instances where students can gesture with their hands and say they're pointing, and they're using both hands as, “This is how many I mean, and this is how I'm partitioning with my fingers. I'm doing three partitions, and I'm using three fingers, and I'm showing you three iterations of that with closing and opening my fists.” And so, there's just so much that kids can do with their body. And they're communicating ideas not just in a formal written format, but also using gestures. So, there's lots of ways that students can communicate, and I think teachers should pay attention to all of those ways.

Mike: Yeah. The connection that I'm making is, we've done several podcasts, and I've been thinking a lot about this idea of strengths-based, or asset-based, instruction. And I think what you're saying really connects to that because my interpretation is, gestures, nonverbal communication, using manipulative tools, things that kids have either written or drawn, those are all assets that I need to pay attention to in addition to the things that they might use language to describe.

Juanita: That's right. That's right. So, everything. ( laughs ) The whole student. ( laughs )

Mike: Well, I suspect you've given our listeners a lot to think about. For folks who want to keep learning about the practices that value nonverbal communication, what research or resources would you suggest?

Juanita: Yeah, so I have two articles, one that's particular to bilingual pre-service teachers, and another one that I just explained within a whole group discussion. That's an article, titled, “Attending to others’ mathematical ideas: a semiotic alternative to logocentrism in bilingual classrooms.” So, I can give you both links and you can share those along with the podcast.

Mike: That sounds fantastic. We'll put a link to that up when we publish the podcast. I just want to thank you, Juanita. It was lovely to have you with us. I've learned a lot, and I sure appreciate you joining us.

Juanita: Thank you. Well, thank you for having me.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

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 fluency here over the last three, four years, per se. And one of the biggest things that we have spent a lot of time just grappling and chewing on, is when we have students in second grade and they move to third grade, how do we move students from additive thinking, which is adding of singular units, to multiplicative thinking? So, seeing groups of groups of groups. And so, I think when we start thinking about third grade teachers, I'll go ahead and throw myself under the bus here. Like, as a third-grade teacher, when we start thinking about that idea of multiplication, it becomes skip counting and repeated addition. But then no kids ever really move from skip counting and repeated addition to knowing their multiplication facts. Like, I could sit there and do jumping jacks in class, but kids aren't going to know their facts.

Graham: So, then what I would do is, I would jump to having kids try to memorize their facts. And just because kids can memorize their facts doesn't mean that they can reason multiplicatively and seeing those groups of groups. So, I think, thinking of that, what [are] those big jumps in the progression from grade level to grade level? That's probably one of the ones for me that really stands out that I know I struggled for. And we always look back and say, “What are the things I wish I knew back then that I know now?” And I think that jump from additive thinking to multiplicative thinking is a really big jump that is often overlooked, which is now why we have kids struggling in fourth and fifth grade and middle school. ’Cause they're still stuck in additive, but we want them to think multiplicatively and proportionally. But yeah, that's one of those big jumps in terms of a progression that we want kids to make.

Mike: Yeah, this is a great transition because I think, like, what we've been exploring is, how if I understand what I'm helping kids think about in the context of a larger story rather than a set of discreet things that I need to check a box on, that has impact on my practice. But I almost wanted to ask you, just on a day-to-day basis, what's your sense of, if I'm a teacher who's absorbed this sense of progression either across my grade level or across a larger band of time, how do you think that changes the way someone approaches teaching? Or maybe the way that they set up tasks with students?

Graham: Well, I start thinking about learning objectives as they're handed down, and standards. And a lot of the time standards can become, or learning objectives can become, more of a checklist. And so not necessarily looking at these ideas of learning as a checklist, but how do they connect between the grade levels? And so, I think it's important as much as on the day-to-day practice that we're really down in the trenches and we're doing the work and we're making sure that we're meeting those learning objectives, I think it becomes really important that we provide ourselves that space and grace to zoom back out to that 60,000-foot view and say, “Wait a minute, how are all of these connected?” And I think that's a really big piece that maybe we don't always do when we start thinking, even planning, on a day-to-day or a week or a unit. “Where am I going to be able to zoom out and maybe connect some big ideas around an understanding or around a piece of learning?” And I think it can become cumbersome when we start looking at those learning objectives and they're so granular. But I think when we can zoom out and make connections between them, it lessens a little bit of the burden from having to go ahead. “Well, there's just so much to teach, trying to make those connections.” There is a lot to teach, don't get me wrong here. But I think going ahead and making those connections just lessens that burden for us a little bit.

Mike: It's interesting, because I think part of what is coming to mind for me is this ability to zoom out and zoom back in and be able to say, “In what way is this relatively granular learning objective or learning goal serving to advance this larger set of ideas that I want kids to understand about, say, additive thinking as they're making a shift to multiplicative thinking?” And the other connection I'm making is, in what way can I ask a question in this moment that's going to actually advance that larger goal rather than—again, guilty as charged—rather than what I've done often in the past, which is how can I help them just complete the task or get this particular thing right? And if by them getting it right in the moment, I failed to advance their thinking, that's a place where I'd want to take it back. Does that make sense to you?

Graham: Yeah, absolutely. I think about tasks and really about when I first would start to use problem-based lessons or three act tasks and start thinking about those lessons. Normally it would be, like, “OK, I just taught the task for no rhyme or reason just to see if kids could get the right answer.” And so, for me, the big piece with that is a shift in my own craft, is looking at that task placement. And so, thinking of, “Are you a teacher who learns math to solve problems or are you a teacher who solves problems to learn math?” A little play on words there. And I think by default, many of us were taught to learn math to go ahead and solve the problems. But when I start thinking about this idea of using tasks and why we use tasks, it's to use … well, to quote Dan Meyer, talking about this headache and aspirin analogy where you have a problem that's your headache, and then from that problem, the math serves the headache, that's the aspirin that you need.

Graham: So, when we talk about zooming back out, instead of saving the really good tasks for the end of the unit, what would it look like if we put it on day one of a unit? Knowing that the goal on day one isn't for kids to get the right answer, but it's for us to just pull the veil back and see, “Hey, where are my students thinking?” And what I've realized is that when we don't front-end load or pre-teach things, students will usually fall back to the strategy that they feel safe enough. And if you have a student who, say we're in fourth grade and we're playing with two- by two-digit multiplication, if you have a student on day one of a unit who's doing draw all, count all, great, right? That's what they're doing on day one? But if they're still using that same strategy at the end of the unit, that falls back on me.

Graham: Like, what have I done to be intentional enough about moving that student's thinking forward? So, even in the moment when students might not be getting the right answer, it might be wrong answer, but it might be the right thinking. And I think at that moment I need to zoom back out and say, “They don't have the answer yet, but I've still got three or four weeks to get there.” So, now that I know what students are thinking, how can I be intentional? How can I be purposeful about asking the right questions, presenting the right activities and tasks to continue to move that student's thinking forward to the end goal? The end goal isn't on day one of a unit. So yeah, I think that's such a great question because I think a lot of the times we feel as if we fall short or we failed as a teacher if kids aren't getting the right answer. But so often there's beautiful thinking that's happening, it just might not have the right answer. So yeah, big, big change in my practice.

Mike: We've been talking about the use of the progression videos that you've built, and I think in my mind I've imagined myself as a classroom teacher, as the consumer. And I think that's a really powerful way to use those. My wondering is, if you have any thoughts about how someone who might be an instructional coach or an instructional leader in a building or a district, if you could wave a magic wand, how you wish folks who have that type of role might take and use the things that you've built?

Graham: I can share how I've used them in the past. I don't know, I'm sure there's coaches out there that are probably using the progression videos way better than I'm using them. But many times, I've found that when we start looking at individual standards, it's standards out of context. And granted, the progression videos, if I could go back and redo them, I would love to embed much more context into those progression videos. It would definitely lengthen them, which kind of defeats the original purpose of keeping them short and compact. So, now when we show those videos, what's nice is it's not really a coach in that moment talking with the teachers. The coach can now, after the video, say, “Hey, what was new to you? What was something that, that maybe you didn't recognize?” And also, like, “What are you doing well?” There's so much goodness that's already happening.

Graham: I think as coaches, we have to be really mindful, like, there's great things that [are] happening with teachers, let's support and lift up those great things that are already happening with our teachers that we're supporting, just like teachers do with students as well. So, I think showing the videos and asking, “Hey, what's the same, what are you comfortable with? What doesn't sit well with you?” Thinking about kindergarten teachers when they see five frames, it's like, “Whoa, wait a minute. I've never really thought about using five frames.” So, just different ways of thinking it to kind of be a catalyst for the conversation, just a launch point.

Mike: Totally makes sense. So, I suspect there are some folks who are going to be listening to this who are, like, “Oh my goodness, I want to go check these things out right now. Or I want to think about sharing them with my teammates that I'm working with on a daily basis.” Walk me through how to find these and any kind of advice that you might have for people as they start to initially poke around and look at what's there.

Graham: Well, you can jump on my website, gfletchy.com, with my full name, Graham Fletcher. Just one of those things that we kind of went with growing up. I was called “Fletchy” as a kid. So yeah, at gfletchy.com you can look on progression videos, and then right there you'll see five of them. But as you start poking around, I'm going to harness my inner Brené Brown here and just say, “Vulnerability is the birthplace of professional growth.” And so, no one is ever going to get a new idea and go ahead and try it and then it be successful right on that get-go. So, when you poke around there, give things a try. I love reaching out on Twitter, sharing on Twitter, and just kind of growing in that space. Find a colleague. Or if you are a coach, one of the things I love doing is when coaches ask for ideas, go muck about, find a good task, and then muck about in a third-grade classroom with that task and make yourself vulnerable around the teachers you're supporting.

Graham: And that really helps build and solidify that relationship where, “Hey, we're in this together and I'm trying to fumble through this just like you, let's kind of work here together. Give me feedback and, and in the end, I think kids win.” I'm a firm believer that all of us are smarter than one of us. And so, I love finding new things, testing new things with a friend, and trying not to lock myself in a silo. So, that would kind of be it in terms of poking around there. Yeah, find an idea and go share it with a friend and see how it works and keep on tweaking and revising.

Mike: I love that. Graham, thank you so much for joining us. It's really been a pleasure.

Graham: Yeah, it's been great. I appreciate it. And thanks for the opportunity.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

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, as we're looking at the work, I think the best place to start is, if we're talking as colleagues, “What do you see that the kids know? What are they doing well?” Whether you're talking about one kid or whether you're talking about a group of kids or your class collectively, “What are they doing well?” And for me, even just sitting here across from you saying this, that feels like a much brighter place to start. I'm like, “OK, I'm into this conversation about what my kids know,” and I would then start to say, “OK, and how can we build on what they know?”

Mike: Ooh, I love that. Keep talking about that.

Tisha: So, if we're looking at say, fractions, and we're kind of at the beginning, we could come in and we could say, “Oh, our kids are just not getting it. They don't know anything about fractions.” And that feels very defeating. But if you start with, “OK, well, I can see that they can partition into half, great. OK, so can we get them to fourths? Can we get them to eighths? How about thirds? All right. Can they get it on a rectangle? Can they get it on a circle? Can they get it in this context? Can they get it if it's a sharing situation?” Right? Now, we're brainstorming all of these questions of what can they do next.

Mike: And those are actionable things, right? Like …

Tisha: Right. 

Mike: … in addition to saying, “This is what kids are doing,” thinking about “What I can build from” actually leads to action, it leads me to a path of instruction, and that does feel really different.

Tisha: So, if we are here and we take the perspective that our kids don't get fractions, then that could bleed into our instruction in a different way. So, instead of now thinking about what we can do next and how we can keep building them up, we may be thinking about how do we need to water things down? How do I need to make things easier? And we want to make sure that we are not taking away rich mathematical opportunities from our students because our perspective is that they're not able, they have deficits. We want to instead think about “How do we build them up? How do we still make sure that they're getting these rich mathematical problems and opportunities in class and being able to grow them in that way?”

Mike: Love that. So, one of the things that really just jumped out, and I want to come back to this because I think the language is so darn important: This idea that an asset-based perspective leads to thinking about instruction as “building upon.” That just seems like such a practical, simple thing. But boy, shifting your mindset and approaching it the way you described it, Tisha, that really does feel profoundly different than a lot of the data conversations that I've sat in over the years.

Tisha: At that point, we should be stopping to think, “What do they need next?” But it's hard to make that [determination] based on saying, “Well, they don't know this.” It's much easier to think about what they need next if you're looking for what they do know. And you can say, “Oh, I can make some connections to that and move them maybe even just a little bit to a little bit further, help them take another step.”

Mike: It strikes me that what I don't hear you saying is, “We can't acknowledge that there's sometimes going to be a difference between what kids understand and our ultimate goals for them.” That can still be true, but we're looking at their starting point as the starting point and the next steps, rather than just only saying, like, “The gap is this wide.” And even using the language of “gap” is challenging, right? 

Tisha: Absolutely. 

Mike: Because we're trying to say, like, “Our job is to build, not just to measure.”

Tisha: Well, and when you think about talking about a gap, it almost feels like it's the kids' fault. 

Mike: Uh-hm. 

Tisha: But right now, in our conversation, we are talking about where the responsibility is. 

Mike: Oh! Yeah!

Tisha: And the responsibility is on me to keep thinking about “How do I help this kid grow?” 

Mike: Uh-hm.

Tisha: “How do I keep helping this kid grow in their math understanding?” It is not uncommon in elementary schools to group or classify kids based on their abilities. And coming from the best place, right? Like, we're all wanting to help our students. I believe that everybody wants to help their students grow. 

Mike: This conversation has really got me thinking a lot, and I suspect that anyone who's listening is in the same place. I'm curious, if I'm a person who's new to this conversation, if these ideas are new, I'm wondering if you have any recommendations about where someone could go to keep learning, be it, uh, a book, a website, something along those lines that could keep me thinking about this and exploring these ideas?

Tisha: A good place to start is a book called “The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices.” And that is an NCTM publication.

Mike: I love that one. It's fantastic. In fact, I've read it myself. We'll put a link to that in the podcast notes.

Tisha: That would be great. I think that it's a great resource for thinking about assessment and just equity-based practices in general.

Mike: Fabulous. Tisha, it was lovely having you on. Thank you so much.

Tisha: Oh, it's been so much fun.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.

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

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 about it in relation to the orienting to resources. 

Mike: I want to check my own understanding. I was struck by the way that you talked about the way that  the teacher positions the materials. It seems like a pitfall, I know that I have fallen into at different points  in time is: Using the materials to set a conversation up in a way where children might come away  thinking, ‘Oh, that's the way to do it,’ which is very different from, I think the way I heard you describe it. It was more like, this is a tool that can help us think about for future reference. I just wanted to call that  out because I thought I heard that, but I wasn't exactly sure if I was interpreting that accurately. 

Hala: Thank you for mentioning that. I think what you're really referring to is what often happens,  especially when we use some manipulatives, let's say, or resources or tools. Where the idea becomes  that the tool equates what it means to do or to reason, like, as if the idea is within the tool and/or the  representation, uh, et cetera. And I think the idea that there is a lot of choice. So, one of the things for  example, that we are currently studying is in kindergarten classrooms, the nature of the use of multiple  representations. There's one question, ‘How often can students come up with their own  representations?’ They invent the representations. How often can they go on their own to draw on  certain tools to represent an idea? Those say something when it's actually coming from the student, where you can follow up with questions and say, ‘So, tell me why you use this? Like how do you see it in  this one?’ And that's the work that we saw teachers do often, is that they're orienting the resources but  then they're orienting to resources as supporting reasoning.  

Hala: And there is the question of why, pressing students. There is a nice example that I always love to  think about, especially with kindergarteners using multiple representations and their own choices. Of  course, students come to class with various fluency in academic language, vocabulary, et cetera. So, there was an instance where the teacher was asking the students, ‘If we've been in school for 129 days, in how many days like that number 29 is going to, we are going to get another 10?’ And they were  working with bundling sticks and other things. They focused on the number 9 as nine ones. And how  many more ones till we get another 10? Then the teacher asks the class, ‘Well, is there another way we  can think about how many more days till we get to another 10?’ 

Hala: ‘Can we use the number 29 altogether?’ And a student raises her hand, we call her Gloria, and  actually points to the number line above the whiteboard and says, ‘One twenty-nine, 130.’ And the  teacher says, ‘What do you mean by those two?’ That literally points to it: 129, 130. So, what the teacher  does, she presses Gloria to explain more and says, ‘Tell us a little bit more. What do you mean by 129  and 130?’ Then Gloria actually sees that just looking at the number line as a representation—we call it a  language proxy—to help her really explain her thinking, according to Gloria, wasn't enough for her. She  actually goes back to the hundreds chart. She points at 29, makes a hub, and says, ‘One jump and we get  to 30.’ So, we see this is just as a small example of where the student is really using their agency in  deciding on the representation, and the teacher then helps the class try to see the connection that  Gloria was trying to make between this representation. We think this is important for not only this grade  level, but whenever we use multiple representations. The power of multiple representations is in helping  the students see the conceptual connection between them. So, that's where I would caution all of us  when we are doing this, to try to make sure we are focusing on the conceptual piece that the  representation is allowing us to see.

Mike: I think part of what you had me thinking about is The Math Learning Center and Bridges. We have  kind of hung our hat on this idea that visual representations are a powerful tool. But the caution that I  always feel is, if those visual representations just turn into another version of an algorithm that's more  like geometric or visually laid out, then we are not advancing the kind of classroom culture or discourse  or thinking that we want, right? That it really is to expose the big ideas. And I think that's what I take, particularly from that example is, the visual actually served as, like, a tool that helped them find the  language to describe the concept rather than just as, like, a here's how you do it. Does that make sense? 

Hala: Exactly. I think the tool here is a way for them … the difference is that they're using it not to apply  the reasoning, it's not an application. That's kind of where I see it. Don't just come and show me how  like, like base ten blocks can represent a number. Base ten blocks are used as a way to support a  mathematical idea, not just to apply, like, to show you and show you how something looks like on a hundreds chart. Actually going back to the hundreds chart, to the hub between 29 and 30, was in the  service of really explaining what they meant by 130, 129, 100, there is a hub. That's what they were  talking about in class that when you, you're counting by ones, you're actually now, you got no more 9,  10—9 ones—you actually have one more. And now you could bundle it, and it's your extra 10. So, it's all  couched in the history of working with these representations, like how these students experienced the  work as to not just, ‘Hey, come, let's represent the numbers.’ Or there was more talk about, like, those  key ideas that the students were talking about. 

Mike: What you're making me think about is that there's an overall pattern that I want to explore in the  context of kindergarten, which is that, as a field, in my mind, one of the things that I wonder about is  whether we have almost explicitly thought about communicating our thinking as something that  happens in the verbal realm. And the more that I've been in the profession is, that we need to broaden  that, particularly when we're talking about young children in pre-K and kindergarten. And I'm wondering, in your mind, what broadening out communication might look like, particularly in kindergarten? 

Hala: That's a great question. And I would link it again, like, whenever I think about the norms, the  resources, I see them literally as a triangle with other things working together. Especially critical at this  young age is verbal and non-verbal communication; or really, assets for the students to express their  thinking and communicate with others. And that's where, in a way, the resources become the mediators  of this, with non-verbal—we call them language proxies—is that they become ways of helping the  communication without necessarily waiting for that correct vocabulary or the specific language. And I  think the more we honor various ways of participating and contributing to the learning of the collective,  the more students are going to be able to make improvements, and to make connections, and to show  us what they know, rather than thinking it's too difficult for them to do something maybe because they  don't have that particular, specialized language that someone is looking for. 

Hala: We actually think of kindergartners in the way they're really acquiring this new—not only the  verbal language, so that they become more proficient in it—the academic language. And actually, if you  come to think of it, every student in math class, in a way, is a language learner, especially the idea of  what does it mean to explain one's reasoning? And when we are thinking about certain ways that  schools go, they want to follow, for example, the Common Core standards and what they expect in terms  of providing evidence, supporting it. That's actually a language learning process. And there is actually the  literature about supporting bilingual students and multilingual students in classrooms, helps us a lot  think about how we could support learners in the early childhood span. And most recently I was reading  an opinion piece by Tim Boals at the WIDA at the University of Wisconsin. I just actually highlighted a few things in what he said in his opinion piece, which is basically about what it takes to make sure that  multilingual students encounter opportunities to learn. 

Hala: So, in a parallel way, it makes me think what it takes for opportunities for early childhood learners  and kindergartners to learn. I just highlighted a few elements that might be one of the resources I share  with you in the end, in case someone is interested in them; about what school programs could do to  ensure that multilingual learners have opportunities to learn. One of them is actually the idea that  always encourage the can-do kind of stance, that you can do it. It's not too difficult for you, like, even in  the choice of tasks. How this guides us for kindergartners is tha,t let's not just give tasks that allow  kindergartners even to skip count on a number line. Actually using tasks where they can reason and  think about why something is true, would be something they can do. So, thinking about not what they  can't do because they're restricted with what they know with numbers, et cetera, it's actually what they  can do. 

Hala: So, the idea of designing tasks that leverages what they know, that they could really show you the  way they're reading a situation, what they know about the situation, and really leverage the resources  they have to explain their thinking. My favorite in terms of what he lists in terms of opportunities for  multilingual learners, is this idea of building academic identities, where he says that ‘this is much more  than merely teaching content knowledge and skills. It's about learning to communicate and think like  people who work in those academic or vocational areas.’ That's all of this can do. And opening  possibilities for reasoning helps our kindergartners develop really mathematical identities early on that  we know are going to impact their opportunities to learn later. And that's what research shows. 

Mike: So, in the third part of your article, you talk about the idea of narration. And I'm wondering if you  could explain narration in this context and then talk a little bit about why it's particularly helpful for young learners? 

Hala: So, let me explain what we meant by it in that article. It's literally when, because students may not  have that facility to explain their thinking articulately, elaborately, it's when the teacher actually supports  them by recapping what they said to the class. And on top of it, building on it and setting it up for further  

articulation or investigation. So, we try to distinguish here, that's why we're trying to revisit the word  ‘narration’ because, we don't think of it just as revoicing. We think of it as a way where the teacher is  highlighting something the student did and, often, we see it in exchange. It's highlighted not only in  terms of the verbatim words that they used or the actions that they took. Highlighting why this is really  helping in the task that we are working on together, and then follows it. It positions it in a way where, now this is what Gloria did. 

Hala: So, really it positions the student in a way where other students are now listening, are trying to see  what the student is doing and saying, and then it sets the stage for further focus or deeper conceptual  exploration of particular ideas. So, an example of that would be when Gloria went from 129 to 130 and  went down to the hundreds chart and said, ‘You know, there is a hop from 29 to 30.’ So, the teacher may  say, ‘OK, here's what Gloria said so far. She picked those two numbers, she saw that they follow each  other. Actually we're going to get to 130. Then she went down to the hundreds chart to really focus on  that jump of one from 29 to 30.’ And then she would immediately go on with a question to the group.  ‘Now what do we do?’ I think that makes it more ambitious than just simply revoicing or appropriating  something that the student said, or trying to put words that they may not have used. I think positioning  it for further and deeper conceptual work takes us a bit away from that.

Mike: That's really helpful. You started to address the question that I was going to ask next, which is  what's the sweet spot for what you described in the article as narration? It struck me, at least as I was  reading it, that over narrating, if we were defining it as kind of revoicing for kids, might impact kids in  ways that are not productive. But what I hear you saying is, narration is much more than revoicing. 

Hala: Absolutely. And that sweet spot that I think you are getting at is really knowing when do you do it  and when do you hold off. In the sense, I don't think there is a rule, but it all goes to the teacher's ability  to know: ‘Is there a shared language here that the students can access through what a student said?’ So, 

knowing your students in terms of, is this something that I need to further articulate so that now they  could engage productively with someone's idea? And if it's not, then actually it's just highlighting; pulling  from what a student says, the valuable pieces that you think are going to be important for the continued  work of the class, rather than, literally, a student says something, you say verbatim, and then you ask  more questions. It's really tracking what seems to be important for the development of everyone's  thinking, that collective as an intellectual community that's working together. 

Mike: That's really helpful. And I think what I heard are simultaneous things that are happening. One is  attending to the ideas that you want to position as important. And the other thing that really jumps is this idea that we're also positioning the child as the author of the ideas. 

Hala: Yes. And you know, in later grades we've seen teachers being able to do this in grades 1 and 2, is  often—especially when we are working early on to build that classroom talk community, that math talk  community—is encouraging students as listeners to someone to say, ‘Did you hear something that you  think is important for the way we are really working on this problem in what Mike said? So, let's listen.  Was there something you have a question about, you're not certain about?’ Also, distributing the work of the narration, if we want to call it that way, so it's distributed. It's not just about me, but now the class  is listening and trying to pull what's important and worthy of focusing on. 

Mike: I love that. Particularly that idea that you can in fact distribute the idea of narration to the class, and it doesn't just live with the teacher. It also advances that broader cultural goal that you have, which  is that the students are actually sense makers, which is the thing from the very beginning of this  conversation. 

Hala: Again, it goes back to the way I think about all the practices that we've talked about, to be very  interconnected. It's not like we know you set up norms, you put them on a chart. You know, norms are  reinforced, are renegotiated with your students through the work that you do. And there's a lot of  socializing that you're doing while you're working on content. It reinforces certain ideas, it reintroduces  certain ideas for others to see how they're able to access them and be part of them. So yes, I agree with  you. They're all connected in that way. 

Mike: Well, Hala, before we close the podcast, I'm wondering if you could share some resources with  listeners who might be encountering some of the ideas we're talking about for the first time. Is there  anything that you might suggest for a listener who just wants to keep thinking about this and perhaps  learn more? 

Hala: So, if they're interested in thinking a little bit more about representations, there is a recent article that I published with Dr. Eric Siy, who is currently at Boston University, in relation to what multiple  representations mean. And how different they are from just using different representations.

Mike: Yep. We could absolutely put a link to that on the podcast notes. 

Hala: Yeah. And I find the work of Dr. Amy Parks at Michigan State University. You know, she has this  book called ‘Exploring Mathematics Through Play in the Early Childhood Classroom.’ [It] has wonderful  pieces that really could support this work in relation to the idea of reasoning in kindergarten, discourse  in kindergarten. And it could happen during play. It doesn't have to happen necessarily only during  academic tasks that are, like, problem-solving situations or worth problems. 

Mike: We could absolutely add a link to that. And I think that's probably another great podcast that we  should do relatively soon. 

Hala: Yes, I find you really connecting wonderful, cohesive dots together here, which I think is really  going to be helpful to the listener. 

Mike: Well, I want to thank you so much for joining us, Hala. It's really been a pleasure talking with you. 

Hala: Thank you very much. And it's been a great opportunity to talk about these ideas with you, and the  questions are on target in terms of the things that we have to pay attention to. 

Mike: Oh, thank you so much.  

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence  and ability.

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

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 a little bit of a twofer in that one. The first is this idea that what you're doing is purposely setting up a future idea, right? It's kind of like saying, ‘I'm going to build this prior knowledge about ten-ness, and then I'm going to have kids think about the relationship between 10 and nine.’ So, like, the care in this work is actually really understanding those relationships and how you're going to leverage them. The other thing that really jumps out from what you said, this has long-term implications for students thinking. It's not just fact acquisition, it's what you said, research shows that this has implications for how kids are thinking further down the road. Am I understanding that right? 

Jenny: That's absolutely correct. So just that strategy alone. Let's say they're adding 29 plus 39. And they're like, ‘Oh hey, both of those numbers are right next to the next benchmark. So instead of 29 plus 39, I'm going to add 30 plus 40, 70. And I got, I went up two, so I'm going to come back down two. And I know that two less than a benchmark's going to land on an eight to that.’ Again, it's coming back to this relationship of how far apart numbers are, what's right there within a set of 10, helps then to generalize within 10s or within 100s. And by the way, how about fractions? 

Mike: Hmm. Talk about that. 

Jenny: ( laughs ) It generalizes to fractions. So, let's take that same idea of adding. Let's just say it's like, two and seven-eighths plus two and seven-eighths. So, if we just pretended those were both threes because they're both super close to three, then you'd have six, and then you added on two-eighths too much. So, you come back two-eighths, or a fourth, and you have your answer. You don't have to do the regrouping with fractions and all the mess that really gets bogged down. And it's a much more efficient method that, again, you set students up for when they understand these number relationships. When you get into fractions, you're thinking about, like, how close are you to the next whole number maybe, instead of to the next 10s number. 

Mike: It strikes me that if you have a group of teachers who have a common understanding of this approach to facts, and everyone's kind of playing the long game and thinking about how what they're doing is going to support what's next, it just creates a system that's much more intentional in helping kids not only acquire the facts, but build a set of ways of thinking. 

Jenny: Mike, that's exactly it. I mean, here we are, we're trying to make up for lost time. We never have enough time in the classroom. We want an efficient way to make sure our kids get the most learning in. And so, to me that is about investing early in the fact strategies. Because then actually when you get up to those other things that you're adding or subtracting or multiplying or whatever you're doing, you benefit from the fact that you took time early to learn those strategies. Because those strategies are now very useful for all this other math that you're doing. And then students are more successful in making good choices about how they're going to solve those problems that are, oftentimes—especially when, I like to mention fractions and decimals at least once in a basic facts talk because we get back, by the time we get into fractions and decimals—we're back to just sometimes only showing one way. The sort of standard algorithm way. When, in fact, those basic facts strategies absolutely apply to almost-always-more-efficient strategies for working with fractions and decimals. 

Mike: I want to shift a little bit. One of the things that was really helpful for me in growing my understanding is, the way that you talk about a set of facts that you would describe as ‘foundational’ facts and another set of facts that you would describe as ‘derived’ facts. And I'm wondering if you can unpack what those two subsets are and how they're related to one another. 

Jenny: Yeah. So, the foundational facts are ones where automaticity is needed in order to enact a strategy. So, to me, the foundational fact strategies are, they're names. Like the doubling strategy or double and double again, some people call it. Or add a group for multiplication, and the addition ones of making 10s and pretend-to-10 strategies. And in those strategies, you can solve lots of different facts. But there's too much going on ( laughs ) in your brain if you don't have automaticity with the facts you need. So, for example, if you have your six facts, and you're trying to get your six facts down. And you already know your fives, like, automaticity with your fives. Then that becomes a useful way to get your sixes. So, if you have six times eight, and you know five times eight is 40, then you're like, ‘I got one more 8, 48.’ 

Jenny: That's an added group strategy. But if you're not automatic with your fives, this is how this sounds when you're interviewing a child. They're going to use add a group strategy, but they don't know their fives. So, then they're like, ‘Let's see, five times eight is 5, 10, 15, 20, 25, 30, 40. Now, what was I doing?’ Like, they can't finish it because they were skip-counting with their fives. They lose track of what they're doing, is my point. So, the key is that they just know those facts that they need in order to use a strategy. And that, going back to, like, the pretend-to-10, they got to know 10-and-some-more facts to be successful. They have to know nine’s one less than 10 to be successful. So, that's the idea is, if they reach automaticity with the foundational fact sets, then their brain is freed up to go through those reasoning strategies. 

Mike: That totally makes sense. I want to shift a little bit now. One of the things that I really appreciated about the article was that you made what I think is a very strong, unambiguous case for ending many of the past practices used for fact acquisition—worksheets and timed tests, in particular. This can be a tough sell because this is often what is associated with elementary mathematics, and families kind of expect this kind of practice. How would you help an educator explain the shift away from these practices to folks who are out in the larger community? What is it that we might help say to folks to help them understand this shift? 

Jenny: That's a great question, and the real answer is it depends, again, on audience. So, who is your audience? Even if the audience is parents, what do those parents prioritize and want for their children? So, I feel like [there are] lots of reasons to do it, but to really speak to what matters to them. So, I'm going to give a very generic answer here. But for everyone, they want their child to be successful. So, I feel that that opportunity to show, to give a problem like 29 plus 29, and ask how parents might add that problem. And if they think 30 plus 30 and subtract two to get to the answer, whatever, then that gives this case to say, ‘Well this is how we're going to work on basic facts. We're building up so that your child is ready to use these strategies. We're going to start right with the basic facts, learning these strategies. These really matter.’

Jenny: And the example I gave could be whatever fits with the level of their kid. So, it could be like 302 minus 299. It's a classic one where you don't want your child to implement an algorithm there, you want them to notice those numbers are three apart. And so, there's this work that begins early. So, I think that's part of it. I think another part of it is helping people just reflect on their own learning experiences. What were your learning experiences with basic facts? And even if they liked the speed drills, they oftentimes recognize that it was not well-liked by most people. And also, then they really didn't learn strategies. So, I feel like we have to be showing that we're not taking something away, we're adding something in. They are going to become automatic with their facts. They're not going to forget them because we're not doing this memorizing that leads to a lot of forgetting. And bonus, they're going to have these strategies that are super useful going forward. So, to me, those are some of the really strong speaking points. I like to play a game and then just stop and pause for a minute and just say, ‘Did you see how hard it was for me to get you quiet? Do you see how much fun you were having?’ And then I just hold up a worksheet ( laughs ). I'm like, ‘And how about this?’ You know, again, that emotional connection to the experience and the outcomes. 

Mike: That is wonderful. Since you brought it up, let's talk about replacements for worksheets and timed tests. 

Jenny: Um-hm. 

Mike: So, you advocate for games as you said, and for an activity-based approach. I think that what I want to try to do is get really specific so that if I'm a classroom teacher, and I can't see a picture of that yet, can you help paint a picture? Like what might that look like? 

Jenny: I love that question because [there are] lots of good games and lots of places. But again, like I said earlier, this thinking really deeply about what game I'm choosing and for what. What do my students need to practice? And then being very intentional about game choice is really important. So, for example, if students are working on their 10-and-some-more facts, then you want to play a game where all the facts are 10-and-some-more facts. That's what they're working on. And then maybe you mix in some that aren't. Or you play a game with that and then they sort cards and find all the solve the 10 and more, or [there are] lots of things they can do. They can play concentration, where the fact is hidden and the answer is hidden and things like that. So, you can be very focused. And then when you get to the strategies, you want to have a game that allows for students to say, allow their strategies. 

Jenny: So, I'm a big fan of, like, sentence frames, for example. So, [there are] games that we have in our ‘Math Fact Fluency’ book that are in other places that specifically work on a strategy. So, for example, if I'm working on the pretend-to-10 strategy, I like to play the game fixed-addend war, which is the classic game of war, except, there's an addend in the middle, and it's a nine, to start. And then each of the two players turns up a card. So, Mike, if you turn up a seven, then you're going to explain how you're going to use the pretend-to-10 strategy to add it. And I turned up a six, so I'm going to, I'm going to do this then I'll, you can do it. So, I turned up a six. So, I'm going to say, ‘Well, 10 and six is 16, so nine and six is one less, 15.’ I've just explained the pretend-to-10 strategy. And then you get your turn. 

Mike: And I'd say, ‘Well seven and 10, I know seven and 10 is 17, so seven and nine has to be one less, and that's 16.

Jenny: Yeah. So, your total's higher than mine, you win those two cards, you put them in your deck, and we move on. So, that's a way to just practice thinking through that strategy. Notice there's no time factor in that. You have a different card than I have. You have as much time, and we're doing think-aloud. These are all high-leverage practices. Then we get to the games where it's like, you might turn up a six and a five where you're not going to use the pretend-to-10 strategy for that. You've got to think, ‘Oh that doesn't really fit that strategy because neither one of those numbers is really close to 10. Oh hey, it's near a double, I'm going to use my double.’ So, you sequence these games to, if you start with one of those open-ended games, it might be too big of a jump because students aren't ready to choose between their strategies. They have to first, be adept at using their strategies. And once they're adept at using them, then they're ready to play games where they get to choose among the strategies. 

Mike: So, you're making me think a couple things, Jenny. One is, it's not just that we're shifting to using games as a venue to practice to get to automaticity. You're actually saying that when we think about the games, we really need to think about, ‘What are the strategies that we're after for kids?’ And then make sure that the way that the game is structured, like, when you're talking about the pretend-to-10, with the fixed addend. That's designed to elicit that strategy and have kids work on developing their language and their thinking around that particularly. So, there's a level of intent around the game choice and the connection to the strategies that kids are thinking about. Am I understanding that right? 

Jenny: That's it. That's exactly right. That's exactly right. And a huge, a lot of intentionality so that they have that opportunity and a no-pressure, a low-stress, think through the strategy. If they make a mistake, they're peer or themselves usually correct it in the moment, and they get so much practice in. I mean, imagine going through half a deck of cards playing that game. 

Mike: Yeah. 

Jenny: That's 26 facts. And then picture those 26 facts on a page of paper. And then, and again, in the game that you've got the added benefit of think-aloud, and then you're hearing what your peer has said. 

Mike: You know, one of the things that strikes me is, if I'm a teacher, I might be thinking like, ‘This is awesome, I'm super excited about it. Holy mackerel, do I have to figure these games out myself?’ And I think the good news is, there's a lot of work that's been done on this. I know you've done some. Do you have any recommendations for folks? There's of course curriculum. But do you have recommendations for resources that you think, help a teacher think about this or help a teacher see some of the games that we're talking about? 

Jenny: Well, I'm going to start with my ‘Math Fact Fluency’ book because that is where we go through each of these strategies, each of the foundational facts sets and the strategies, and for each one supply a game. And then from those games they're easily adaptable to other settings. And some of the games are classic games. So, there's a game, for example, called ‘Square Deal.’ And the idea is that you're covering a game board, and you're trying to make a square. So, you get a two-by-two grid taken, and you score a point or five points or whatever you want to score. Well, we have that game housed under the 10-and-some-more facts. So, all the answers are like 19, 16, 15, and the students turn over a 10 card and another card, and if it's a 10 and a five, they get to claim a 15 spot on the game board. 

Jenny: Well, that game board can be easily adapted to any multiplication fact sets, any other addition. I like to do a Square Deal with 10 and some more, and then I like to do Square Deal with nine and some more. There's my effort, again, to come back to either pretend-to-10 or making 10. Where they're like, ‘Oh, I just played 10 and some more. Now we're doing the same game, but it's nine and some more.’ So, I feel like there's a lot of games there. And there is a free companion website that has about half of the games ready to download in English and in Spanish. 

Mike: Any chance you'd be willing to share it? 

Jenny: Yeah, absolutely. So, you can just Google it. The Kentucky Center for Mathematics created it during Covid, actually, as a gift to the math community. And so, if you type in ‘Kentucky Center for Math’ or ‘KCM math fact fluency companion website,’ it will pop up. 

Mike: That's awesome. I want to ask you about one more thing before we close because we've really talked about the replacement for worksheets, the replacements for timed tests. But there is a piece of this where people think about ‘How do I know?’ right? ‘How can I tell that kids have started to build this automaticity?’ And you make a pretty strong case for interviewing students to understand their thinking. I'm wondering if you could just talk again about the ‘why’ behind it and a little bit about what it might look like. 

Jenny: So, first of all, timed tests are definitely a mistake for many reasons. And one of the reasons— beyond the anxiety they cause—they're just very poor assessment tools. So, you can't see if the student is skip-counting or not, for example, for multiplication facts. You can't see if they're counting by ones for the addition facts. You can't see that when they're doing the test, and you can't assume that they're working at a constant rate; that they're just solving one every, you know, couple of seconds, which is the way those tests are designed. Because I can spend a lot of time on one and less time on the other. So, they're just not, they're just not effective as an assessment tool. So, if you flip that. Let's say they're playing the game we were talking about earlier, and you just want to know can they use the pretend-to- 10 strategy? 

Jenny: That's your assessment question of the day. Well, you just wander around with a little checklist ( chuckles ), you know? Yes, they can. No, they can't. And so, a checklist can get at the strategies, and a checklist can also get at the facts like how well are they doing with their facts? So, once they do some of those games that are more open-ended, you can just observe and listen to them and get a feel for that. If they're playing Square Deal with whatever fact, you know. So, what happens is you're, like, ‘I wonder how they're doing with their fours. We've really been working with their fours a lot.’ Well, you can play Square Deal or a number of other games where that day you're working on fours. The fixed-addend war can become fixed-factor war, and you put a four in the middle. So adaptable games and then you're just listening and watching. 

Jenny: And if you're not comfortable with that approach, then they can be playing those games, and you can have students channeling through where you do a little mini-interview. It only takes a few questions to get a feel for whether a student knows their facts. And you can really see who's automatic and who's still thinking. So, for example, a student who's working on their fours, if you give them four times seven, they might say, ‘Twenty-eight.’ I call that automatic. Or they might, they might do four times seven, and they pause, and they're like, ‘Twenty-eight.’ Then I'm like, ‘How did you think about that?’ And they're like, ‘Well, I doubled and doubled again.’ ‘Great.’ So, I can mark off that they are using a strategy, but they're not automatic yet. So that to me is a check, not a star. And if I ask, ‘How did you do it?’ And they say, ‘Well, I skip-counted.’ Well then, I'm marking down the skip-counted. Because that means they need a strategy to help them move toward automaticity.

Mike: I think what strikes me about that, too, is, when you understand where they're at on their journey to automaticity, you can actually do something about it as opposed to just looking at the quantity that you might see on a timed test. What's actionable about that? I'm not sure, but I think what you're suggesting really makes the case that I can do something with data that I observe or data that I hear in an interview or see in an interview. 

Jenny: Absolutely. I mean this whole different positioning of the teacher as coaching the student toward their growth; helping them grow in their math proficiency, their math fluency. You see where they're at and then you're monitoring that in order to move them forward instead of just marking them right or wrong on a timed test. I think that's a great way to synthesize that. 

Mike: Well, I have to say, it has been a pleasure talking with you. Thank you so much for joining us today. 

Jenny: Thank you so much. I am again thrilled to be invited and always happy to talk about this topic. 

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. 

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