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.

Dr. Amanda Jansen is a professor in the School of Education at the University of Delaware. She is a mathematics educator who conducts research on students' engagement in mathematics classrooms and teachers' learning from their reflections on their own practice. She is committed to honoring students' voices through her research on students' motivation and engagement. Her most recent book, "Rough Draft Math: Revising to Learn," was published by Stenhouse Publishers in March 2020.

Before working in academia, Dr. Jansen was a junior high mathematics teacher. At UD, she is invested in continually improving UD's elementary mathematics teacher education courses through research and development work.

Book - Rough Draft Math: Revising to Learn

MTMS article - Rough-Draft Talk in Mathematics Classrooms

MTLT article - Rough-Draft Thinking and Revising in Mathematics

NCTM article - Mathematics Teaching in the Middle School

Website with free resources

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mike: Amen.

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

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

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

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

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

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

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

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

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

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

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

Professor Kim Hartweg teaches mathematics education at Western Illinois University.

NCTM Classroom Conversations

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

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

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

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

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

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

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

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

Kim: Exactly. Yes.

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

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

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

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

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

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

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

Kim: It is, and it's fun to watch because hear some of them say, "1, 2, 3, 4, 5. Five, there's five." [laughs] So they kind of get that idea. But yeah, that cardinality of number is a key principle and leads into the conservation of numbers.

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

Kim: Yeah, so the conservation of number, this is my favorite task when I have young kids around. I want to see if they can serve number or not. So I might first do the sugar [packet] thing or whatever and see if they can tell you how many there are. But the real fun is, do they conserve that number?

So I think back to a friend of mine who brought her daughter over one time, and I had these toy matchbox cards on my table, and her name was Katie. And I said, "Katie, how many cars are there?" And she counted, "1, 2, 3, 4, 5. There's five toy cars." And I moved them around and I said, "Now how many cars are there?" And she counted, "1, 2, 3, 4, 5. There's five toy cars." So she has cardinality of number. However, I kept moving those cars into different positions, never adding or taking any away. That's all that were there the whole time. And after about seven or eight times, I said, "Now how many cars are there?" And her mom finally jumped in and said, "Katie, you've counted those already. There's five cars." [laughs] And I said, "No, no, no. This is just whether she conserves number or not, it's a developmental-type thing."

But you know they conserve number when you ask them, "Well, now how many cars are there?" And they look at you and like, "Well, why would you ask that again? There's five." [chuckles] So then they can conserve number. So It's real fun to do that with elementary students who are getting their number sense going and even before they enter school. However, there will be some that may not get that conservation of number until they're 5 or 6 years old.

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

Kim: Yeah, so subitizing, I think that's interesting. You know, we work so hard getting kids to count and learn the language and have one-to-one correspondence, and then be able to eventually conserve number. But then we want them to just be able to recognize a set of numbers without counting. And that's when they're really starting to develop some number sense.

I think of dice. And if you roll a single [die], we want students to just know that when there's an arrangement of four dice, they know it's four without having to count "1, 2, 3, 4." So the subitizing idea, a lot of dice games, maybe some 10-frame cards, dot cards, lots of things like that can help students develop a little bit more of that subitizing, or recognizing a set of items without having to count those.

Mike: So when I look at a set of three dots, I can just say, "That's three," as opposed to an earlier point where a child might actually say, "1, 2, 3. That's three."

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

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

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

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

Kim: Yeah, and we want to give kids the idea that counting collections and things, it's a fun thing to do. And I know there may be teachers that have seashells or rocks or different types of collections they might count, and we want students to count those and then discuss how they counted them, arrange them.

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

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

Kim: The idea here is that we want students to get good at counting. And the research is showing that students who maybe don't show one-to-one correspondence when they count out, maybe, eight counters, might show one-to-one correspondence when they count out 31 pennies, which seems like it shouldn't happen. But there's research out there that over 70 percent of them did better counting 31 pennies than they did with eight counters.

So, I think what you count makes a big difference for kids—and to not hold them back, to not think that "OK, we've got to get one-to-one correspondence before we count this collection of 50 items." I don't think that's the case. And the research is even showing that these ideas that we've talked about all develop concurrently. It's not a linear process. But this counting collections is kind of a big deal with that. And having students count, again, collections that they're interested in, writing number sentences about their collections, comparing what they counted with another partner, and then turning it into problem-solving questions where they're actually doing, "What happens if you lost five of yours?" or "What happens if you combined your collection with somebody else?" And turning it into where they're actually doing addition and subtraction, but not actually the formal process of that.

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

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

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

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

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

Kim: Yes. I think again, it goes to that idea that we have these 10 fingers that are great manipulatives, that we shouldn't stop students from doing that. And I know there was a time when teachers would say, "Don't use your fingers," "Don't count on your fingers." And I get the idea that we want students to start to subitize eventually and make combinations and not have to count on their fingers, but to stop them from doing it when they need that would be very detrimental to them.

And I actually have a story. When I was supervising student teachers, one teacher was telling a student, "Don't count with their fingers." And I saw them nodding their head, and I went over and I said, "What are you doing?" He said, "Well, I can't count my fingers, so I'm using my tongue, and I'm counting my teeth." [laughs] So coming up with a problem that way, still using a manipulative, but it wasn't their fingers.

Mike: That's pretty creative.

Kim: [laughs] Yeah.

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

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

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

Kim: Yeah, I'll just mention a few that we use a lot of. We do the two-color counters a lot where one side is yellow and one is red. But we do a lot of dot cards, where again, there are arrangements of dots on a card that you could just flash to a student, kind of like I've already explained. There's lots of resources on the National Council of Teachers of Mathematics website. That has 10-frame activities.

And if you haven't used rekenreks before, I think those are pretty amazing as well—along with hundreds charts. And just being able to have students create some of their own manipulatives and their own numbers makes a huge difference for kids.

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

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

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

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

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

Tatyana Kleyn is a professor in the Bilingual Education and TESOL programs and founding faculty advisor to the Dream Team at The City College of New York. Her dissertation,focused on the intersections of bilingual and multicultural education, earned an Outstanding Dissertation Award from the National Association for Bilingual Education (NABE).

CUNY New York State Initiative on Emergent Bilinguals

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

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

Tatyana Kleyn: Thank you. This is very exciting.

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

Tatyana: Yes, absolutely. I think for many of us in education, we don't randomly end up teaching in the areas that we're teaching in or doing the work that we're doing. So I always like to share my story so people know why I'm doing this work and where I'm coming from.

So my personal story, I work a lot at the intersection of language, migration, and education, and those are all three aspects that have been critical in bringing me here. So, I was actually born in what was the Soviet Union many, many years ago, and my family immigrated to the United States as political refugees, and I was just 5½ years old. So I actually never went to school in the Soviet Union. Russian was my home language, and I quickly started speaking English, but my literacy was not quick at all, and it was quite painful because I never learned to read in my home language. I never had that foundation. So, when I was learning to read in English, it wasn't meaning making, it was just making sounds. It was kind of painful. I once heard somebody say, "For some people, reading is like this escape and this pure joy, and for other people it's like cleaning the toilet. You get in and you get out." And I was like, "That's me. I'm the toilet cleaner." [laughs]

So, that was how reading was for me. I always left my home language at the door when I came into school, and I wanted it that way because I, as a young child, got this strong message that English was the language that mattered in this country. So, for example, instead of going by Tatyana, I went by Tanya. So, I always kind of kept this secret that I spoke this other language. I had this other culture, and it wasn't until sixth grade where my sixth-grade teacher, Ms. Chang, invited my mom to speak about our immigration history. And I don't know why, but I thought that was so embarrassing. I think in middle school, it's not really cool to have your parents around. So, I was like, "Oh my God, this is going to be horrible." But then I realized my peers were really interested—and in a good way—and I was like, "Wait, this is a good thing?" So, I started thinking, "OK, we should be proud of who we are and let just people be who they are." And when you let people be who they are, they thrive in math, in science, in social studies, instead of trying so hard to be someone they're not and then focusing on that instead of everything else that they should be focusing on as students.

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

Tatyana: Sure. Well, I'm so glad to be talking about translanguaging in this space specifically, because often when we talk about translanguaging, it's in bilingual education or English as a second language or is a new language, and it's important in those settings, right? But it's important in all settings. So, I think you're not the only one, especially if we're talking about math educators or general elementary educators, it's like, "Oh, translanguaging, I haven't heard of that," right?

So, it is not something brand new, but it is a concept that Ofelia García and some of her colleagues really brought forth to the field in the early 2000s—around 2009. And what it does is instead of saying, "English should be the center of everything, and everyone who doesn't just speak English is peripheral," it's saying, "Instead of putting English at the center, let's put our students' home language practices at the center. And what would that look like?" So, that wouldn't mean everything has to be in English. It wouldn't mean the teacher's language practices are front and center, and the students have to adapt to that. But it's about centering the students and then the teacher adapting to the languages and the language practices that the students bring. Teachers are there to have students use all the languages at their resource—whatever language it is, whatever variety it is. And all those resources will help them learn. The more you can use, when we're talking about math, well, if we're teaching a concept and there are manipulatives there that will help students use them, why should we hide them? Why not bring them in and say, "OK, use this." And once you have that concept, we can now scaffold and take things away little by little until you have it on your own.

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

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

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

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

Tatyana: Yes, absolutely. Ofelia García, Kate Seltzer and Susana Johnson talk about a translanguaging stance. So translanguaging is not just a practice or a pedagogy like, "Oh, let me switch this up," or "Let me say this in this language." Yes, that's helpful, but it's how you approach who students are and what they bring. So if you don't come from a stance of valuing multilingualism, it's not really going to cut it, right? It's something, but it's really about the stance.

So something that's really important is to change the culture of classrooms. So just because you tell somebody like, "Oh, you can say this in your home language" or "You can read this book side by side in Spanish and in English if it'll help you understand it." Some students may not want to because they will think their peers will look down on them for doing it, or they'll think it means they're not smart enough. So, it's really about centering multilingualism in your classroom and celebrating it. And then as that stance changes the culture of the classroom, I can see students just saying, "Ah, no, no, no, I'm good in English." Even though they may not fully feel comfortable in English yet, but because of the perception of what it means to be bilingual.

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

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

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

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

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

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

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

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

And I'll give you one really powerful example. This is a million years ago, but it stays with me from my dissertation. It was in a Haitian Creole bilingual classroom. They were taking a standardized test, and the word problem was where it was like three gumballs, two gumballs, this color, what [is] the probability of a blue gumball coming out of this gumball machine? And this student just got stuck on "gumball machine" because in Haiti people sell gum, not machines, and it was irrelevant to the whole problem.

So language matters, but culture matters, too, right? So giving students the opportunity to see things side by side and thinking about, "Are there any things here that might trip them up that I could explain to them?" So I think it's starting small. It's taking risks. It's letting go of control and centering the students.

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

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

And language is fun. We can play with language, we can put language side by side. So then if you're labeling or if you have a math word wall, why not put key terms in all the languages that the students speak in the class and then they could teach each other those languages? So I think you have to start little. You have to expect some resistance. But over time, if you keep pushing away at this, I think it will be good for not only your multilingual students, but all your students to say, like, "Oh, wait a minute, there's all these languages in the world, but they're not just in the world. They're right here by my friend to the left and my friend to the right" and open up that space.

Mike: So I want to ask another question. What I'm thinking about is participation. And we've done an episode in the past around not privileging verbal communication as the only way that kids can communicate their ideas. We were speaking to someone who—their focus really was elementary years mathematics, but specifically, with multilingual learners. And the point that they were making was, kids' gestures, the way that they use their hands, the way that they move manipulatives, their drawings—all of those things are sources of communication that we don't have to only say, "Kids understand things if they can articulate it in a particular way." That there are other things that they do that are legitimate forms of participation.

The thing that was in my head was, it seems really reasonable to say that if you have kids who could share an explanation or a strategy that they've come up with or a solution to a problem in their home language in front of the group, that would be perfectly legitimate. Having them actually explain their thinking in their home language is accomplishing the goal that we're after, which is, "Can you justify your mathematical thinking?" I guess I just wanted to check in and say, does that actually seem like a reasonable logic to follow? That that's actually a productive practice for a teacher, but also a productive practice for a kid to engage in?

Tatyana: That makes a lot of sense. So I would say for every lesson you, you may have a math objective, you may have a language objective, and you may have both. If your objective is to get kids to understand a concept in math or to explain something in math, who cares what language they do it in? It's about learning math. And if you're only allowing them to do it in a language that they are still developing in, they will always be about English and not about math.

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, etc.

Mike: Well, and what you made me think, too, is for that peer, it's actually a great opportunity for them to, like, 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, right? 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 [hyphen] 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. And 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, Tatyana. 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

If you ask an educator to share some of the most important ideas in elementary mathematics, most would include place value on that list. But what does it mean to understand place value, really? And what types of language, practices, and tools support students as they build their understanding? On this week's podcast, we discuss these questions and more with Dr. Eric Sisofo from the University of Delaware.

Dr. Eric Sisofo is an assistant professor specializing in mathematics education in the School of Education at the University of Delaware. He earned his EdD in educational leadership, curriculum, and instruction; his MEd in curriculum and instruction with a concentration in mathematics education; and his BS in mathematical sciences from the University of Delaware. With a focus on mathematics education, Dr. Sisofo's education leadership portfolio, the capstone project for his doctoral degree, investigated the effects of lesson study on teacher candidates' use of student thinking when planning and reflecting on their teaching.

https://www.nctm.org/Get-Involved/

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

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

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

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

Eric: Yeah, that's a really great question.

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

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

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

So this happens with decimal numbers too. With our novice teachers, we spend a lot of time trying to get them to name those units and not just say, for example, "3.4" miles when they're talking about a numeral. We wouldn't want them to say "3.4." We instead want them to say "three and four-tenths," or "three ones and four-tenths miles." So saying the numeral "3.4" focuses mostly just on the face value of those digits and removes some of the mathematics that's embedded in the numeral. So instead saying the numerals "three ones and four-tenths" or "three and four-tenths" really requires you to think about the face value and the place value of each digit.

So those are the three big ideas that we discuss often with our novice teachers at the University of Delaware, and we hope that this helps them develop their conceptual understanding of those ideas so that they're better prepared to help their future students make sense of those same ideas.

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

Eric: I think the same was true when I was growing up. That math was often taught to be a bunch of procedures or memorized kinds of things that my teacher taught me, that I didn't really understand the meaning behind what I was doing. And so, mathematics became more of just doing what I was told and memorizing things and not really understanding the reasoning why I was doing it.

Talking about relationships between things I think helps kids develop number sense. And so, when you talk about how 10 tenths are required to make one one, and knowing that that's how many of those one-tenths are needed to make one one, and that same pattern happens for every unit connected to the next larger unit. Seeing that in decimal numbers helps kids develop number sense about place value. And then when they start to need to operate on those numerals or on those numbers, if they need to add two decimal numbers together and they get more than 10 tenths when they add down the columns or something like that in a procedure—if you're doing it vertically. If they have more of a conceptual understanding of the relationship, maybe they'll say, "Oh, I have more than 10 tenths, so 10 of those tenths will allow me to get one one, and I'll leave the others in the tens place," or something like that. So, it helps you to make sense of the regrouping that's going on and develop number sense so that when you operate and solve problems with these numbers, you actually understand the reasoning behind what you're doing as opposed to just memorizing a bunch of rules or steps.

Mike: Yeah. I will also say, just as an aside, I taught kindergarten and first grade for a long time, and just that idea of 10 ones and one ten, simultaneously, is such a big deal. And I think that idea of being able to say, "This unit is comprised of these equal-sized units," how challenging that can be for educators to help build that understanding. But how rich and how worthwhile the payoff is when kids do understand that level of equivalence between different sets of units.

Eric: Absolutely, and it starts at a young age with children. And getting them to visualize those connections and that equivalence that a 10, one ten, can be broken up into these 10 ones or 10 ones can create one ten. And seeing that visually, multiple times, in lots of different situations really does pay off because that pattern will continue to show up throughout the grades. When you're going into second, third grade, like I said before, you've got to realize that 10 of these things we call "tens," then we'll make a new unit called one "hundred." Or 10 of these one hundreds will then make a unit that is called a "thousand." And a thousand is equivalent to 10 hundreds. So these ideas are really critical pieces of students' understanding about place value when they go ahead and try to add or subtract with these using different strategies or the standard algorithm. They're able to break numbers up, or decompose numbers into pieces that make sense to them. And their understanding of the mathematical relationships or ideas can just continue to grow and flourish.

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

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

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

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

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

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

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

Mike: That totally makes sense.

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

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

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

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

Mike: I think 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 kind of 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 ten 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.

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 ten 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 ten 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. 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 ten numeral, we've got to break all the stuff into groupings that base ten 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.

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 ten blocks in kindergarten or first grade, to be honest. When students are just learning about tens and ones, because the long in a base ten 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 ten blocks, and so you're kind of removing some of the thinking. And so that's why I would choose Unifix cubes over base ten 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 ten 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-addition 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-sized 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-sized 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.

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 one 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, "3 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."

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 one 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. 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, you know, 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 that'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

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

Kim Morrow-Leong is an experienced mathematics education specialist and consultant with a demonstrated history of working productively with teachers in the area of mathematics education leadership. Skilled in lesson planning, educational technology, instructional design, leadership, and curriculum development, Kim holds a PhD and MEd in mathematics education leadership from George Mason University.

Children's Mathematics: Cognitively Guided Instruction

Warning Signs!

Never Say Anything a Kid Can Say!

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

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

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

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

Kim: So, I'm going to rephrase your question a little bit and change the emphasis to say, "What is right ?" And the reason I want to change the emphasis of that is because we often talk about what is wrong. And so rather than talking about what is wrong, let's talk about what's right. When we look at student work, it's a picture. It's a snapshot of where they are at that particular moment. And the greater honesty that we can bring to that situation to understand what their thinking is, the better off we're going to be.

So, there's a lot of talk lately about asset-based instruction, asset-based assessment. And I think it's a great initiative, and it really gets us thinking about how we can think about what students are good at and what they bring to the table or what they bring to the classroom culture. But we don't often talk a lot about how we do that, how we break the mold. Because many of our metaphors and our language about learning are linear, and they indicate that students are moving from somewhere to achieve a goal somewhere down the path, somewhere down the line.

How do you switch that around? Well, rather than looking at what they're missing and what part of the path they haven't achieved yet, we can look at where they are at the moment because that reflects everything they've learned up to that moment. So, one of the ways we can do this is to unpack our standards a little more carefully. And I think a lot of people are very good at looking at what the skills are and what our students need to be able to do by the end of the year. But a lot of what's behind a standard are concepts. What are some big ideas that must be in place for students to be successful with the skills?

So, I'm going to give a very specific example. This one happens to be about a fourth-grade question that we've asked before in a district I used to work at. The task is to sketch as many rectangles as you can that are 48 square units. There's some skills behind this, but understanding what the concepts are is going to give us a little more insight into student thinking. So, one of the skills is to understand that there are many ways to make 48: to take two factors and multiply them together, and only two factors, and to make a product of 48 or to get the area. But a concept behind that is that 48 is the product of two numbers. It's what happens when you multiply one dimension by the other dimension. It's not the measure of one of the dimensions. That's a huge conceptual idea for students to sort out what area is and what perimeter is, and we want to look for evidence of what they understand about the differences between what the answer to an area problem is and what the answer to, for example, a perimeter problem is. Another concept is that area indicates that a space is covered by squares. The other big concept here is that this particular question is going to have more than one answer. You're going to have 48 as a product, but you could have 6 times 8 and 4 times 12 and many others. So that's a lot of things going into this one, admittedly very rich, task for students to take in.

One of the things I've been thinking a lot about lately is this idea of a listening stance. So, a listening stance describes what you're listening for. It describes how you're listening. Are you listening for the right answer? Are you listening to understand students' thinking? Are you listening to respond or are you listening to hear more—and asking for more information from your student or really from any listener? So, one of the ways we could think about that, and perhaps this sounds familiar to you, is you could have what we call an evaluative listening stance.

An evaluative listening stance is listening for the right answer. As you listen to what students say, you're listening for the student who gives you the answer that you're looking for. So, here's an example of something you might see. Perhaps a student covers their space and has dimensions for the rectangle of 7 times 6, and they tell you that this is a space that has an area of 48 square units. There's something right about that. They are really close. Because you can look at their paper and you can see squares on their paper and they're arranged in an array. And you can see the dimensions on this side and the dimensions on that side, and you can see that there's almost 48 square units.

I know we all can see what's wrong about that answer, but that's not what we're thinking about right now. We're thinking about what's right. And what's right is they covered that space with an area that is something by 6. This is a great place to start with this student to figure out where they got that answer. If you're listening evaluatively, that's a wrong answer and there's nowhere else to go. So, when we look at what is right in student work, we're looking for the starting point. We're looking for what they know so that we can begin there and make a plan to move forward with them. You can't change where students are unless you meet them where they are and help them move forward.

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

Kim: Think about ways that you might be listening to a student's answer and very quickly say, "Oh, they got it," and you move on. And you grab the next student's paper or the next student comes up to your desk and you take their work and you say, "Tell me what you're thinking." And they tell you something. You say, "That's good," and you move to the next one.

Sometimes you can take the time to linger and listen and ask for more and ask for more and ask for more information. Teachers are very good at gathering information at a glance. We can look at a stack of papers and in 30 seconds get a good snapshot of what's happening in that classroom. But in that efficiency we lose some details. We lose information about specifics, about what students understand, that we can only get by digging in and asking more questions.

Someone once told me that every time a student gives an answer, you should follow it with, "How do you know?" And somebody raised their hand and said, "Well, what if it's the right answer?" And the presenter said, "Oh, you still ask it. As a matter of fact, that's the best one to ask. When you ask, 'How do you know?' you don't know what you're going to hear. You have no idea what's going to happen." And sometimes those are the most delightful surprises, is to hear some fantastical, creative way to solve a problem that you never would've thought about. Unless you ask, you won't hear these wonderful things.

Sometimes you find out that a correct answer has some flawed reasoning behind it. Maybe it's reasoning that only works for that particular problem, but it won't work for something else in the future. You definitely want to know that information so that you can help that student rethink their reasoning so that the next time it always works. Sometimes you find out the wrong answers are accidents. They're just a wrong computation. Everything was perfect up until the last moment and they said, "3 times 2 is 5," and then they have a wrong answer. If you don't ask more, either in writing or verbally, you have incorrect information about that student's progress, their understandings, their conceptual development, and even their skills. That kind of thing happens to everyone because we're human. By asking for more information, you're really getting at what is important in terms of student errors and what is not important, what is just easily fixable.

I worked with a group of teachers once to create some open-ended tasks that require extended answers, and we sat down one time to create rubrics. And we did this with student work, so we laid them all out and someone held up a paper and said, "This is it! This student gets it." And so, we all took a copy of this work and we looked at it. And we were trying to figure out what exactly does this answer communicate that makes sense to us that seems to be an exemplar. And so, what we did was we focused on exactly what the student said. We focused on the evidence in front of us. This one was placing decimal numbers on a number line. We noted that the representation was accurate, that the position of the point on the number line was correct. We noticed that the label on the point matched the numbers in the problem, so that made sense. But then all of a sudden somebody said, "Well, wait a minute. There's an answer here, but I don't know how this answer got here." Something happened, and there's no evidence on the page that this student added this or subtracted this, but magically the right answer was there. And it really drove home for this group—and for me, it really stuck with me—the idea that you can see a correct answer but not know the thinking behind it. And so, we learned from that point on to always focus on the evidence in front of us and to make declarative statements about what we saw, what we observed, and to hold off on making inferences. We saved our inferences for the end.

After I had this experience with the rubric grading and with this group of teachers and coaches, I read something about over-attribution and under-attribution. And it really resonated with me. Over-attribution is when you make the claim that a student understands something when there really isn't enough evidence to make that statement. It doesn't mean that's true or not true; it means that you don't have enough information in front of you. You don't have enough evidence to make that statement. You over-attribute what it is they understand based on what's in front of you. Similarly, you get under-attribution. You have a student who brings to you a drawing or a sketch or a representation of some sort that you don't understand because you've never seen anybody solve a problem this way before. You might come to the assumption that this student doesn't understand the math task at hand. That could be under attribution. It could be that you have never seen this before and you have not yet made sense of it.

And so, focusing on evidence really gets us to stop short of making broad, general claims about what students understand, making broad inferences about what we see. It asks us to cite evidence to be grounded in what the student actually put on the paper. For some students, this is challenging because they mechanically have difficulties putting things on paper. But we call a student up to our desk and say, "Can you tell me more about what you've done here? I'm not following your logic." And that's really the solution is to ask more questions. I know, you can't do this all the time. But you can do it once in a while, and you can check yourself if you are assigning too much credit for understanding to a student without evidence. And you can also check yourself and say, "Hmm, am I not asking enough questions of this student? Is there something here that I don't understand that I need to ask more about?"

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

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

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

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

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

There's something out there called a learning trajectory that you've mentioned. A learning trajectory comes out of people who really dig in and understand student thinking on a fine-grain level, how students will learn developmentally. What are some ideas they will develop before they develop other ideas? That's the nature of a learning trajectory. And sometimes those are reflected in our standards. The way that kindergartners are asked to rote count before they're asked to really understand one-to-one correspondence. We only expect one-to-one correspondence up to 20 in kindergarten, but we expect rote counting up to 100 because we acknowledge that that doesn't come at the same time. So, a learning trajectory, to some degree, is built into your standards. But as we talked about earlier, there are pieces and parts that aren't outlined in your standards.

One of the things we know about students and their interactions with grids and arrays is that a student might be able to recognize an array that is 6-by-8, but they may not yet be able to draw it. The spatial structuring that's required to create a certain number of lines going vertically and a certain number of lines going horizontally may not be in place. At the same time, they are reading arrays and understanding what they mean. So, the skill of structuring the space around you takes time. The task where we ask them to draw these arrays is asking something that some kids may not yet be able to do, to draw these grids out. If we know that, we can give them practice making arrays, we can give them tools to make arrays, we can give them blocks to make arrays, and we can scaffold this and help them move forward. What we don't want to assume is that a student who cannot yet make a 6-by-8 array can't do any of it because that's not true. There's parts they can do. So, our job as teachers is to look at what they do, look carefully at the evidence of what they do, and then make a plan. Use all of that skill and experience that's on our teams. Even if you're a new teacher, all those people on your teams know a lot more than they're letting on, and then you can make a plan to move forward and help that student make these smaller steps so they can reach the standard.

Mike: When we talked earlier, one of the things that you really shifted for me was some of the language that I found myself using. So, 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 [laughs] 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, like, 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[: Cognitively Guided Instruction ]. 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 listened to what students were saying.

And the second piece is called "Warning Signs! " And this one is one of my favorites. And in this [article], they give three warning signs that you as the 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 and to 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

There's something magical about getting lost in a great story. Whether you're reading a book, watching a movie or listening to a friend, stories impart meaning and capture our imagination. Dr. John Staley thinks a lot about stories. On this episode of Rounding Up, we'll talk with John about the ways he thinks that the concept of story can impact our approach to the content we teach and the practices we engage in to support our students.

Dr. John Staley is currently in his 27th year with Baltimore County Public Schools where he had the opportunity to teach middle and high school mathematics, serve as the coordinator of secondary mathematics and director of mathematics PreK–12. He currently works on special projects that involve supporting building administrators and their leadership teams and coordinating the work of our system improvement teams to improve student achievement by "raising the bar, closing gaps, and preparing every student for the future."

Throughout his career of more than 30 years, he taught in schools in Pennsylvania, Virginia, and Maryland at the secondary and post-secondary levels. In addition, he has presented at state, national, and international conferences; served on many committees and tasks forces; facilitated workshops and professional development sessions on a variety of topics; received the Presidential Award for Excellence in Teaching Mathematics and Science; and served as President for NCSM, the mathematics education leadership organization (2015–2017), and chair of the U.S. National Commission on Mathematics Instruction (2018–2020).

Building Thinking Classrooms in Mathematics: Grades K–12: 14 Teaching Practices for Enhancing Learning

The Impact of Identity and K–8 Mathematics: Rethinking Equity-Based Practices

Humanizing Disability in Mathematics Education: Forging New Paths

Motivated: Designing Mathematics Classrooms Where Students Want to Join In

Mike Wallus: There's something magical about getting lost in a great story. Whether you're reading a book, watching a movie, or listening to a friend, stories impart meaning, and they capture our imagination. Dr. John Staley thinks a lot about stories. On this episode of Rounding Up, we'll talk with John about the ways that he thinks that the concept of story can impact our approach to the content we teach and the practices we engage in to support our students.

Well, John, welcome to the podcast. We're really excited to talk with you today.

John Staley: I'm glad to be here. Thank you for the invitation, and thank you for having me.

Mike: So when we spoke earlier this year, you were sharing a story with me that I think really sets up the whole interview. And it was the story of how you and your kids had engaged with the themes and the ideas that lived in the Harry Potter universe. And I'm wondering if you could just start by sharing that story again, this time with the audience.

John: OK. When I was preparing to present for a set of students over at Towson University and talking to them about the importance of teaching and it being a story. So the story of Harry Potter really began for me with our family—my wife, Karen, and our three children—back in '97 when the first book came out. Our son Jonathan was nine at that time and being a reader and us being a reading family, we came together. He would read some, myself and my wife would read some, and our daughter Alexis was five, our daughter Mariah was three. So we began reading Harry Potter. And so that

really began our journey into Harry Potter. Then when the movies came out, of course we went to see the movies and watch some of those on TV, and then sometimes we listened to the audio books. And then as our children grew, because Harry Potter took, what, 10 years to develop the actual book series itself, he's 19 now, finally reading the final book. By then our three-year-old has picked them up and she's begun reading them and we're reading. So we're through the cycle of reading with them.

But what they actually did with Harry Potter, when you think about it, is really branch it out from just books to more than books. And that right there had me thinking. I was going in to talk to teachers about the importance of the story in the mathematics classroom and what you do there. So that's how Harry Potter came into the math world for me, [chuckles] I guess you can say.

Mike: There's a ton about this that I think is going to become clear as we talk a little bit more.

One of the things that really struck me was how this experience shaped your thinking about the ways that educators can understand their role when it comes to math content and also instructional practice and then creating equitable systems and structures. I'm wondering if we can start with the way that you think this experience can inform an educator's understanding for content. So in this case, the concepts and ideas in mathematics. Can you talk about that, John?

John: Yeah, let's really talk about the idea of what happens in a math classroom being a story. The teaching and learning of mathematics is a story that, what we want to do is connect lesson to lesson and chapter to chapter and year to year.

So when you think about students' stories, and let's start pre-K. When students start coming in pre-K and learning pre-K math, and they're engaging in the work they do in math with counting and cardinality initially, and as they grow across the years, especially in elementary, and they're getting the foundation, it's still about a story. And so how do we help the topics that we're taught, the grade level content become a story? And so that's the connection to Harry Potter for me, and that's what helped me elevate and think about Harry Potter because when you think about what Harry Potter and the whole series did, they've got the written books. So that's one mode of learning for people for engaging in Harry Potter.

Then they went from written books to audiobooks, and then they went from audiobooks to movies. And so some of them start to overlap, right? So you got written books, you got audiobooks, you got movies—three modes of input for a learner or for an audience or for me, the individual interested in Harry Potter, that could be interested in it. And then they went to additional podcasts, Harry Potter and the Sacred Text and things like that. And then they went to this one big place called Universal Studios where they have Harry Potter World. That's immersive. That I can step in; I can put on the robes; I can put the wand in my hand. I can ride on, I can taste, so my senses can really come to play because I'm interactive and engaged in this story.

When you take that into the math classroom, how do we help that story come to life for our students? Let's talk one grade. So it feels like the content that I'm learning in a grade, especially around number, around algebraic thinking, around geometry, and around measurement and data. Those topics are connected within the grade, how they connect across the grade and how it grows. So the parallel to Harry Potter's story—there's, what, seven books there? And so you have seven books, and they start off with this little young

guy called Harry, and he's age 11. By the time the story ends, he's seven years later, 18 years old. So just think about what he has learned across the years and how what they did there at Hogwarts and the educators and all that kind of stuff has some consistency to it. Common courses across grade levels, thinking, in my mind, common sets of core ideas in math: number, algebra, thinking, geometry, measurement of data. They grow across each year. We just keep adding on.

So think about number. You're thinking with base ten. You then think about how fractions show up as numbers, and you're thinking about operations with whole numbers, base ten, and fractions. You think about decimals and then in some cases going into, depending if you're K–8 or K–5, you might even think about how this plays into integers. But you think about how that's all connected going across and the idea of, "What's the story that I need to tell you so that you understand how math is a story that's connected?" It's not these individual little pieces that don't connect to each other, but they connect somehow in some manner and build off of each other.

Mike: So there are a couple of things I want to pick up on here that are interesting. When you first started talking about this, one of the things that jumped out for me is this idea that there's a story, but we're not necessarily constrained to a particular medium. The story was first articulated via book, but there are all of these ways that you can engage with the story. And you talked about the immersive experience that led to a level of engagement.

John: Mm-hmm.

Mike: And I think that is helping me make sense of this analogy—that there's not necessarily one mode of building students' understanding. We actually need to think about multiple modes. Am I picking up on that right?

John: That's exactly right. So what do I put in my tool kit as an educator that allows me to help tap into my students' strengths, to help them understand the content that they need to understand that I'm presenting that day, that week, that month, that I'm helping build their learning around? And in the

sense of thinking about the different ways Harry Potter can come at you—with movies, with audio, with video—I think about that from the math perspective. What do I need to have in my tool kit when it comes to my instructional practices, the types of routines I establish in the classroom?

Just think about the idea of the mathematical tools you might use. How do the tools that you use play themselves out across the years? So students working with the different manipulatives that they might be using, the different mathematical tools, a tool that they use in first grade, where does that tool go in second grade, third grade, fourth grade, as they continue to work with whole numbers, especially with doing operations, with whatever the tool might be? Then what do you use with fractions? What tools do you use with decimals? We need to think about what we bring into the classroom to help our students understand the story of the mathematics that they're learning and see it as a story. Is my student in a more concrete stage? Do they need to touch it, feel it, move it around? Are they okay visually? They need to see it now, they're at that stage. They're more representational so they can work with it in a different manner or they're more abstract. Hmm. Oh, OK. And so how do we help put all of that into the setting? And how are we prepared as classroom teachers to have the instructional practices to meet a diverse set of students that are sitting in our classrooms?

Mike: You know, the other thing you're making me think about, John, is this idea of concepts and content as a story. And what I'm struck by is how different that is than the way I was taught to think about what I was doing in

my classroom, where it felt more like a checklist or a list of things that I was tracking. And oftentimes those things felt disconnected even within the span of a year.

But I have to admit, I didn't find myself thinking a lot about what was happening to grade levels beyond mine or really thinking about how what I was doing around building kindergartners' understanding of the structure of number or ten-ness.

John: Mm-hmm.

Mike: How that was going to play out in, say, fifth grade or high school or what have you. You're really causing me to think how different it is to think about this work we're doing as story rather than a discrete set of things that are kind of within a grade level.

John: When you say that, it also gets me thinking of how we quite often see our content as being this mile-wide set of content that we have to teach for a grade level. And what I would offer in the space is that when you think about the big ideas of what you really need to teach this year, let's just work with number. Number base ten, or, if you're in the upper elementary, number base ten and fractions. If you think about the big ideas that you want students to walk away with that year, those big ideas continue to cycle around, and those are the ones that you're going to spend a chunk of your time on. Those are the ones you're going to keep bringing back. Those are the ones you're going to keep exposing students to in multiple ways to have them make sense of what they're doing.

And the key part of all of that is the understanding, the importance of the vertical nature as to what is it I want all of my students sitting in my

classroom to know and be able to do, have confidence in, have their sense of agency. Like, "Man, I can show you. I can do it, I can do it." What do we want them to walk away with that year? So that idea of the vertical nature of it, and understanding your learning progressions, and understanding how number grows for students across the years is important. Why do I build student understanding with a number line early? So that when we get the fractions, they can see fractions as numbers. So later on when we get the decimals, they can see decimals as numbers, and I can work with it. So the vertical nature of where the math is going, the learning progression that sits behind it, helps us tell the story so that students, when they begin and you are thinking about their prior knowledge, activate that prior knowledge and build it, but build it as part of the story.

The story piece also helps us think about how we elevate and value our students in the classroom themselves. So that idea of seeing our students as little beings, little people, really, versus just us teaching content. When you think about the story of Harry Potter, I believe he survived across his time at Hogwarts because of relationships. Our students make it through the math journey from year to year to year to year because of relationships. And where they have strong relationships from year to year to year to year, their journey is a whole lot better.

Mike: Let's make a small shift in our conversation and talk a little bit about this idea of instructional practice.

John: OK.

Mike: I'm wondering how this lived experience with your family around the Harry Potter universe, how you think that would inform the way that an educator would think about their own practice?

John: I think about it in this way. As I think about myself being in the classroom—and I taught middle school, then high school—I'm always thinking about what's in my tool kit. I think about the tools that I use and the various manipulatives, the various visual representations that I need to have at my fingertips. So part of what my question would be, and I think about it, is what are those instructional strategies that I will be using and how do I fine-tune those? What are my practices I'm using in my routines to help it feel like, "OK, I'm entering into a story"?

Harry Potter, when you look at those books, across the books, they had some instructional routines happening, some things that happen every single year. You knew there was going to be a quidditch match. You knew they were going to have some kind of holiday type of gathering or party or something like that. You knew there was going to be some kind of competition that happened within each book that really, that competition required them to apply the knowledge and skills from their various courses that they learned. They had a set of core courses that they took, and so it wasn't like in each individual course that they really got to apply. They did in some cases, they would try it out, they'd mess up and somebody's nose would get big, ears would get big, you know, change a different color. But really, when they went into some of those competitions, that's when the collection of what they were learning from their different courses, that's when the collection of the content. So how do we think about providing space for students to show what they know in new settings, new types of problems? Especially in elementary, maybe it's science application type problems, maybe they're doing something with their social studies and they're learning a little bit about that.

As an educator, I'm also thinking about, "Where am I when it comes to my procedural, the conceptual development, and the ability to think through and apply the applications?" And so I say that part because I have to think about students coming in, and how do I really build this? How do I strike this balance of conceptual and procedural? When do I go conceptual? When do I go procedural? How do I value both of them? How do I elevate that? And how do I come to understand it myself? Because quite often the default becomes procedural when my confidence as a teacher is not real deep with building it conceptually. I'm not comfortable, maybe, or I don't have the set of questions that go around the lesson and everything. So I've got to really think through how I go about building that out.

Mike: That is interesting, John, because I think you put your finger on something. I know there have been points in time during my career when I was teaching even young children where we'd get to a particular idea or concept, and my perception was, "Something's going on here and the kids aren't getting it." But what you're causing me to think is often in those moments, the thing that had changed is that I didn't have a depth of understanding of what I was trying to do. Not to say that I didn't understand the concept myself or the mathematics, but I didn't have the right questions to draw out the big ideas, or I didn't have a sense of, "How might students initially think about this and how might their thinking progress over time?"

So you're making me think about this idea that if I'm having that moment where I'm feeling frustrated, kids aren't understanding, it might be a point in time where I need to think to myself, "OK, where am I in this? How much of this is me wanting to think back and say, what are the big ideas that I'm trying to accomplish? What are the questions that I might need to ask?" And those might be things that I can discover through reflection or trying to make

more sense of the mathematics or the concept. But it also might be an opportunity for me to say, "What do my colleagues know? Are there ways that my colleagues are thinking about this that I can draw on rather than feeling like I'm on an island by myself?"

John: You just said the key point there. I would encourage you to get connected to someone somehow. As you go through this journey together, there are other teachers out there that are walking through what they're walking through, teaching the grade level content. And that's when you are able to talk deeply about math.

Mike: The other thing you're making me think about is that you're suggesting that educators just step back from whether kids are succeeding or partially succeeding or struggling with a task and really step back and saying, like, "OK, what's the larger set of mathematics that we're trying to build here? What are the big ideas?" And then analyzing what's happening through that lens rather than trying to think about, "How do I get kids to success on this particular thing?" Does that make sense? Tell me more about what you're thinking.

John: So when I think about that one little thing, I have to step back and ask myself the question, "How and where does that one thing fit in the whole story of the unit?" The whole story of the grade level. And when I say the grade level, I'm thinking about those big ideas that sit into the big content domains, the big idea number. How does this one thing fit into that content domain?

Mike: That was lovely. And it really does help me have a clearer picture of the way in which concepts and ideas mirror the structures of stories in that, like,

there are threads and connections that I can draw on from my previous experience to understand what's happening now. You're starting to go there.

So let's just talk about where you see parallels to equitable systems and structures in the experience that you had with Harry Potter when you were in that world with your family.

John: First, let's think about this idea of grouping structures. And so when you think about the idea of groups and the way groups are used within the classroom, and you think about the equitable nature of homogeneous, heterogeneous, random groupings, truly really thinking about that collectively. And I say collectively in this sense, when you think about the parallel to the Harry Potter story, they had a grouping structure in place. They had a random sorting. Now who knows how random it was sometimes, right? But they had a random sorting the minute the students stepped into the school. And they got put into one of the four houses. But even though they had that random sorting then, and they had the houses structured, those groups, those students still had opportunities as they did a variety of things—other than the quidditch tournaments and some other tournaments—they had the opportunity where as a collection of students coming from the various houses, if they didn't come together, they might not have survived that challenge, that competition, whatever it was. So the idea of grouping and grouping structures and how we as educators need to think about, "What is it really doing for our students when we put them in fixed groups? And how is that not of a benefit to our students? And how can we really go about using the more random grouping?"

One of the books that I'm reading is Building Thinking Classrooms [in Mathematics: Grades K–12: 14 Teaching Practices for Enhancing Learning] .

And so I'm reading Peter [Liljedahl]'s book and I'm thinking through it in the chapter when he talks about grouping. I think I read that chapter and highlighted and tapped every single page in it multiple times because it really made me think about what's really happening for our students when we think about grouping. So one structure and one part to think about is, "What's happening when we think we're doing our grouping that's not really getting students engaged in the lesson, keeping them engaged, and benefiting them from learning?"

Another part, and I don't know if this is a part of equitable systems and structures or just when I think about equity work: One of the courses that they had to take at Hogwarts was about the history of wizarding. I bring that up in this space because they learned about the history of what went on with wizards and what went on with people. And to me, in my mindset, that's setting up and showing the importance of us sharing the history and bringing the history of our students—their culture, their backgrounds, in some cases their lived experiences—into the classroom. So that's us connecting with our students' culture and being culturally responsive and bringing that into the classroom. So as far as an equitable structure, the question I would ask you to think about is, "Do my students see themselves in my mathematics classroom?"

And I say it that way versus "in the mathematics," because some people will look at the problems in the math book and say, "Oh, I don't see them there. I don't see, oh, their names, their culture, their type of foods." Some of those things aren't in the written work in front of you. But what I would offer is the ability for me as the educator to use visuals in my classroom, the ability for me to connect with the families in my classroom and learn some of their stories, learn some of their backgrounds—not necessarily learn their stories, but learn

about them and bring that in to the space—that's for me to do. I don't need a textbook series that will do that for me. And as a matter of fact, I'm not sure if a textbook series can do that for you, for all the students that you have in your

classroom or for the variety of students that you have in your classroom, when we think about their backgrounds, their culture, where they might come from. So thinking about that idea of cultural responsiveness, and really, if you think about the parallel in the Harry Potter series, the history of wizarding and the interaction, when you think about the interaction piece between wizards and what they call Muggles, right? That's the interactions between our students, learning about other students, learning about other cultures, learning about diverse voices. That's teaching students how to engage with and understand others and learn about others and come to value that others have voice also.

Mike: I was just thinking, John, if I were to critique Hogwarts, I do wonder about the houses. Because in my head, there is a single story that the reader comes to think about anyone who is in Harry's house versus, say, like Slytherin house.

John: Yes.

Mike: And it flattens anyone who's in Slytherin house into bad guys, right? John : Mm-hmm.

Mike: And so it makes me think there's that element of grouping where as an educator, I might tell a single story about a particular group, especially if that group is fixed and it doesn't change. But there's also, like, what does that do internally to the student who's in that group? What does that signal to them about their own identity? Does that make sense?

John: That does make sense. And so when you think about the idea of grouping there at Hogwarts, and you think about these four fixed groups, because they were living in these houses, and once you got in that house, I don't think anybody moved houses. Think about the impact on students. If you put them in a group and they stay in that group and they never change groups, you will have students who realize that the way you did your groups and the way you named your groups and the way they see others in other groups getting more, doing different, and things like that. That's a nice caution to say the labels we put on our groups. Our kids come to internalize them and they come to, in some cases, live up to the level of expectations that we set for "just that group." So if you're using fixed groups or thinking about fixed groups, really I'd offer that you really get into some of the research around groups and think, "What does it do for students?"

And not only what does it do for students in your grade, but how does that play out for students across grades? If that student was in the group that you identified as the "low group" in grade 2, [exhales] what group did they show up in grade 3? How did that play with their mindset? Because you might not have said those words in front of students, but our students pick up on being in a fixed group and watching and seeing what their peers can do and what their peers can't do, what their group members can do and what their group can't do. As our students grow from grades 2 to 3, 4, 5, that really has an impact. There's somewhere between grade 3 and 5 where students' confidence starts to really shake. And I wonder how much of it is because of the grouping and types of grouping that is being used in the classroom that has me in a group of, "Oh, I am a strong doer [of mathematics]" or, "Oh, I'm not a good doer of mathematics." And that, how much of that just starts to resonate with students, and they start to pick that up and carry that with

them, an unexpected consequence because we thought we were doing a good thing when we put 'em in this group. Because I can pull them together, small group them, this and that. I can target what I need to do with them in that moment. Yeah, target what you need to do in that moment, but mix them up in groups.

Mike: Just to go back and touch on the point that you started with. Building Thinking Classrooms has a lot to say about that particular topic among others, and it's definitely a book that, for my money, has really caused me to think about a lot of the practices that I used to engage in because I believed that they were the right thing to do. It's a powerful read. For anyone who hasn't read that yet, I would absolutely recommend it.

John: And one last structure that I think we can speak to. I've already spoken to supports for students, but the idea of a coherent curriculum is I think an equitable structure that systems put in place that we need to put in place that you need to have in place for your students. And when I say a coherent curriculum, I'm thinking not just your one grade, but how does that grow across the grades? It's something for me, the teacher, to say, "I need to do it my way, this way…". But it's more to say, "Here's the role I play in their pre-K to 12 journey." Here's the chapter I'm going to read to them this year to help them get their deep understanding of whichever chapter it was, whichever book it happened to be of.

In the case of the parallel of Harry Potter, here's the chapter I'm doing. I'm the third grade chapter, I'm the fourth grade chapter, I'm the fifth grade chapter. And the idea of that coherent curriculum allows the handoff to the next and the entry from the prior to be smoother. Many of the curriculums, when you look at them, a K–5 curriculum series will have those coherent pieces

designed in it—similar types of tools, similar types of manipulatives, similar types of question prompts, similar types of routines—and that helps students build their confidence as they grow from year to year. And so to that point, it's about this idea of really thinking about how a coherent curriculum helps support equity because you know your students are getting the benefit of a teacher who is building from their prior knowledge because they've paid attention to what came before in this curriculum series and preparing them for where they're going. And that's quite often what the power of a coherent curriculum will do.

The parallel in the Harry Potter series, they had about five to seven core courses they had to take. I think about the development of those courses. Boom. If I think about those courses as a strand of becoming a wizard, [laughs] how did I grow from year to year to year to year in those strands that I was moving across?

Mike: Okay, I have two thoughts. One, I fully expect that when this podcast comes out, there's going to be a large bump in whoever is tracking the sale of the Harry Potter series on Amazon or wherever it is.

John: [laughs]

Mike: But the other question I wanted to ask you is what are some books outside of the Harry Potter universe that you feel like you'd recommend to an educator who's wanting to think about their practice in terms of content or instructional practices or the ways that they build equitable structure?

John: When I think about the works around equitable structure, I think about The Impact of Identity and K–8 Mathematics: Rethinking Equity-Based Practices by Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin as being

one to help step back and think about how am I thinking about what I do and how it shows up in the classroom with my students.

Another book that I just finished reading: Humanizing Disability in Mathematics Education[: Forging New Paths] . And my reason for reading it was I continue to think about what else can we do to help our students who are identified, who receive special education services? Why do we see so many of our students who sit in an inclusive environment—they're in the classroom on a regular basis; they don't have an IEP that has a math disability listed or anything along those lines—but they significantly underperform or they don't perform as well as their peers that don't receive special education services. So that's a book that got me just thinking and reading in that space.

Another book that I'm reading now, or rereading, and I'll probably reread this one at least once a year, is Motivated[: Designing Mathematics Classrooms Where Students Want to Join In] by Ilana [Seidel] Horn. And the reason for this one is the book itself, when you read it, is written with middle schools' case stories. Part of what this book is tackling is what happens to students as they transition into middle school. And the reason why I mentioned this, especially if you're elementary, is somewhere between third grade and fifth grade, that process of students' self-confidence decreasing their beliefs in themselves as doers of math starts to fall apart. They start to take the chips in the armor. And so this book, Motivated itself, really does not speak to this idea of intrinsic motivation. "Oh, my students are motivated." It speaks to this idea of by the time the students get to a certain age, that upper fifth grade, sixth grade timeframe, what shifts is their K, 1, 2, 3, "I'm doing everything to please my teacher." By [grades] 4 or 5, I'm realizing, "I need to be able to show up for my peers. I need to be able to look like I can do for my peers." And so if I can't, I'm backing out. I'm not sharing, I'm not volunteering, I'm not "engaging."

So that's why I bring it into this elementary space because it talks about five pieces of a motivational framework that you can really push in on, and not that you push in on all five at one time. [chuckles] But you pick one, like meaningfulness, and you push in on that one, and you really go at, "How do I make the mathematics more meaningful for my students, and what does it look like? How do I create that safe space for them?" That's what you got to think about.

Mike: Thanks. That's a great place to stop. John Staley, thank you so much for joining us. It's really been a pleasure.

John: 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 all individuals to discover and develop their mathematical confidence and ability.

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

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

Susan B. Empson is a professor in the Department of Learning, Teaching, and Curriculum and the Richard Miller endowed chair of mathematics education at the University of Missouri. Her research on children's thinking about fractions is the topic of her 2011 book, Extending Children's Mathematics: Fractions and Decimals, co-authored with Linda Levi. She has published widely in refereed journals, including Cognition and Instruction, Journal for Research in Mathematics Education , Educational Studies in Mathematics, Teaching Children Mathematics , and Journal of Mathematics Teacher Education. She has been a researcher of Cognitively Guided Instruction since 1989 and is a co-author of Children's Mathematics: Cognitively Guided Instruction (1st and 2nd editions).

Extending Children's Mathematics: Fractions and Decimals

Mathematics Teacher: Learning and Teaching PK-12

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

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

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

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

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

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

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

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

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

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

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

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

Mike: Got it.

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

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

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

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

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

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

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

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

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

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

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

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

Susan: Yes. We see lots of parallels. J as children's understandings of whole numbers develop, they're able to use these understandings to solve multidigit operations problems by composing and decomposing numbers. So, for example—to take an easy addition—to add 37 + 8, , a child might say, "I don't know what that is, but I do know how to get from 37 to 40 with 3." So, they take 3 from the 8, add it to the 37 get to 40, and then once at 40 they might say, "I know that 40 plus 5 more is 45." So, in other words, they decompose the 8 in a way that helps them use what they understand about decade numbers. Operations with fractions work similarly, but children often do not think about the similarities because they don't understand fractions are numbers, too—versus two numbers, one on top of the other.

If children understand that fractions can be composed and decomposed just as whole numbers can be composed and decomposed, then they can use these understandings to add, subtract, multiply, and divide fractions. For example, to add 1 ⅘ + ⅗ , a child might say, "I know how to get up to 2 from 1 ⅘. I need one more fifth. And then I have two more fifths still to add from the ⅗.So, it's 2 ⅖." So, in other words, just as they decompose the 8 into 3 and 5 to add 8 to 37, they decompose the ⅗ into ⅕ and ⅖ to add it to 1 ⅘.

Mike: I could imagine a problem like 1 ½ + ⅝. I could say, "Well, I know I need to get a half up. Five-eighths is really ⁴⁄₈ and ⅛, and ⁴⁄₈ is a half."

Susan: Yep.

Mike: "So, I'm actually going from 1 ½ + ⁴⁄₈. OK. That gets me to 2, and then I've got one more eighth left. So, it's 2 ⅛."

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

Mike: Well, I have a final question for you, Susan. 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 fifthgrade children's strategies for equal sharing problems. We specifically focused on over 1,500 drawing-based strategies used by children in a written assessment at the end of the school year. We've been surprised both by the variety of details in these strategies—so, for example, how children represent items, how they decide to distribute pieces to people—and also by the percentages of children using these drawing-based strategies. For each of grades three, four, and five, over 50 percent of children used a drawing-based strategy. There are also, of course, other kinds of strategies that don't depend on drawings that children use, but by far the majority of children were using these strategies.

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

Susan: I think it speaks to the length of time and the number of experiences that children need to really build meaning for fractions that they can then use in more 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 in 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 whole 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 in 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 10 ones, 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 equations.

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 1 = ⅓ + ⅓ + 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

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.

Anderson Norton's research is driven by a desire to understand how humans have access to knowledge as powerful and reliable as mathematics. Throughout his career, building upon Jean Piaget's genetic epistemology, he has learned that many philosophical questions about the nature of mathematics have psychological answers. He grounds his research in psychological models of students' mathematics, and collaborates with psychologists and neuroscientists to find these answers.

In 2022, he authored a related book, published by Routledge: The Psychology of Mathematics: A Journey of Personal Mathematical Power for Educators and Curious Minds .

Developing Fractions Knowledge

Teaching Children Mathematics

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

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

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

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

Andy: Yeah, definitely something larger, and it doesn't come down to strategies. Students can solve multiplication tasks—what, to us, look like multiplication tasks—using additive reasoning. And they often do. I think they get through a lot of elementary school using, for example, repeated addition. If I gave a task like, "What is 4 times 5?", then they might just say, "That's 5 and 5 and 5 and 5," which is fine. They're solving a multiplication problem, but their method for solving it is repeated addition, so it's basically additive reasoning. But it starts to catch up to them in later grades where that kind of additive reasoning requires them to do more and more sophisticated or complicated strategies that maybe their teachers can teach them, but it starts to add up, especially when they get to fractions or algebra.

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

Andy: The main distinction is with additive reasoning, you're working within one level of unit. So, for example, if I want to know—going back to that 4 times 5 example—really what I'm doing is I'm working with ones. So, I say I have 5 ones and 5 ones and 5 ones and 5 ones, and that's 20 ones. But in a multiplication problem, you're really transforming across units. If I want to understand 4 times 5 as a multiplication problem, what I'm saying is, "If I measure a quantity with a unit of 5, the measure is 4."

Just to make it a little more concrete, suppose my unit of measure is like a stick that's 5 feet long, and then I say, "OK, I measured this length, and it was four of these sticks. So, it's four of these 5-foot sticks. But I want to know what it is in just feet." So, I've changed my unit. I'm saying, "I measured this thing in one unit, this stick length, but I want to understand its measure in a different unit, a unit of ones." So, you're transforming between this one kind of unit into another kind of unit, and it's a 5-to-1 transformation. So, I'm not just doing 5 plus 5 plus 5 plus 5, I'm saying every one of that stick length contains 5 feet, five of these 1-foot measures. And so, it's a transformation from one unit into another, one unit for measuring into a different unit for measuring.

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

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

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

Andy: Uh-huh.

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

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

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

Andy: Hmm.

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

Andy: Yeah, that's a really good question, because it goes back to this idea that students are already solving the kinds of tasks that should involve multiplicative reasoning, but they might be using additive strategies to do it. Those strategies get more and more complicated, and we as teachers facilitate students just, sort of, doing something more procedural instead of really struggling with the issue. And what the issue should be is opportunities to work with multiple levels of units and then to reflect on their activity and working with them.

So, for example, one task I like to give students is, I'll cut out a piece of construction paper and I'll hand it to the student. And I'll have hidden what I'm going to label a whole, and I'll have hidden what I'm going to label to be the unit fraction that might be appropriate for measuring this thing I gave them. So, I'll give them this piece of construction paper and I'll say, "Hey, this is five-sevenths of my whole." Now what I've given them as a rectangular strip of paper …

Mike: Mm-hmm

Andy: …without any partitions in it. I've hidden the whole from which I created this five-sevenths. I've hidden one-seventh, and I've put them away, maybe inside of envelopes. So, it becomes like a game: "Can you guess what I have in this envelope? I just gave you five-sevenths. Can you guess, what is this five of? What is the unit that this is five of?" and "What is the whole this five-sevenths fraction is?" So, it's getting them thinking about two different levels of units at once. They've been given this one measurement, but they don't know the unit in which it's measured, and they don't even have visually present for them what the whole unit would be.

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

Mike: I think that example is really helpful. I was picturing it in my head, and I could see the opportunities that that affords for, kind of, pressing on some of those big ideas.

One of the things that you made me think about is the idea of manipulatives, or even if we broaden it out a little bit, visual models. Because the question I was going to ask is, "What role might a visual model or a manipulative play in supporting a shift from additive to multiplicative thinking?" I'm curious about how you would respond to that initially. And then I think I have a follow-up question for you as well.

Andy: OK. I can think of two important roles for visual models—or at least two for manipulatives—and at least one works with visual models as well.

But before answering that, the bigger answer is, no one manipulative is going to be the silver bullet. It's how we use them. We can use manipulatives in ways where students are just following our procedures. We can use visual models where students are just doing what we tell them to do and reading off the answer on paper. That really isn't qualitatively any different than when we just teach them an algorithm. They don't know what they're doing. They get the answer, they read it off the paper. You could consider that to be a visual model, what they're doing on their paper or even a manipulative; they're just following a procedure.

What manipulatives should afford is opportunities for students to manipulate. They should be able to carry out their mental actions. So, maybe when they're trying to partition something and then iterate it, or they're thinking about different units. That's too much for them to keep in mind in their visual imagination. So, a visual model or a manipulative gives them a way to carry those actions out to see how they work with each other, to notice the effects of those actions.

So, if the manipulative is used truly as a manipulative, then it's an opportunity for them to carry out their mental actions to coordinate them with a physical material and to see what happens. And visual models could be similar, gives them a way to sort of carry out their mental actions, maybe a little more abstractly, because they're just using representations rather than the actual manipulative, but maybe gives them a way to keep track of what would happen if I partitioned this into three parts and then took one of those parts and partitioned into five. How would that compare to the whole? So, it's their actions that have to be afforded by the manipulative or the visual model. And to decide what is an appropriate manipulative or an appropriate task, we need to think about, "OK, what can they already do without it?" And I'm trying to push them to do the next thing where it helps them coordinate at a level they can't just do in their imagination, and then to reflect on that activity by looking at what they wrote or looking at what they did.

So, it's always that: Carrying out actions in slightly more powerful ways than they could do in their mind. That's sort of the sense in which mathematics builds on itself. After they've reflected on what they've done and they've seen the results, now maybe that's something that they can take as an object, as something that's just there for them in [their] imagination so they can do the next thing: adding complexity.

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

Andy: Mm-hmm.

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

Andy: And I'll just add one thing, not to use too many fractions examples, but that is where most of my empirical research has been, was working with elementary and middle-school children with fractions. But I have to make these decisions based on the child. So, sometimes I'll use these Cuisenaire Rods, the old fraction rods, the colored fraction rods. Sometimes I'll use those with students because then it sort of simplifies the idea. They don't have to wonder whether a piece fits in exactly a certain number of times. The rods are made to fit exactly. And maybe I'm not as concerned about them cutting a construction paper into equal parts or whatever. So, the rods are already formed. But other times I feel like they might be relying too much on the rods, where they start to see the brown rod as a 4. They're not even really comparing the red rod, which fits into it twice. They're just, "Oh, the red is a 2; the brown is a 4. I know it's in there twice because two and two is four." So, you start to think about them, whole numbers. And so sometimes I'll use the rods because I want them to manipulate them in certain ways, and then other times I'll switch to the construction paper to sort of productively frustrate this idea that they're just going to work with whole numbers. I actually want them to create parts and to see the measurements and actually measure things out.

So, it all depends on what kind of mental action I want them to carry out that would determine [which] manipulative as well. Because manipulatives have certain affordances and certain constraints. So, sometimes Cuisenaire Rods have the affordances I want, and other times they have constraints that I want to go beyond with, say, construction paper.

Mike: Absolutely.

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

Andy: Yeah. So, one big idea that comes up starting in middle school—but becomes more and more important as they move into algebra and calculus, any kind of engineering problem—is a rate of change. So, a rate of change is describing a relationship between units. It's like, take a simple example of speed. It's taking units of distance and units of time and transforming them into a third level of unit that is speed. So, it's that intensive relationship that's defining a new unit. When I talk about units coordination, I'm not usually talking about physical units like distance, time, and speed. I'm just talking about different numerical units that students might have to coordinate.

But to get really practical, when we talk about the sciences, units coordinations have to happen all the time. So, students are able to be successful with their additive reasoning up to a point, and I would argue that point is probably around where they first see improper fractions. (chuckles) They're able to work with them up to a point, and then after that, things [are] going to be less and less sensible if they're just relying on these additive sort of strategies that each have a separate rule for a different task instead of being able to think more generally in terms of multiplicative relationships.

Mike: Well, I will say from a former K–12 math curriculum director, thank you for making a very persuasive case for why it's important to …

Andy: (chuckles)

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

Andy: Yeah. So, there are a bunch of us developing ideas and trying to even create resources that teachers can pick up and use. Selfishly, I'll mention one called " 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. (chuckles) You'll find a lot of papers there. Some of them have been written in teacher journals as well, like journals published by the National Council of Teachers of Mathematics, like 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

At their best, programs with titles such as "gifted and talented" seek to provide enrichment to a subset of learners. That said, these initiatives can 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.

Tisha Jones is the senior manager of assessment at The Math Learning Center and a dedicated educator with more than 11 years of experience teaching elementary students. Early in her career, she discovered her passion for teaching elementary math and has since worked to develop better methods of teaching math to children. She strongly believes that all children can learn and enjoy math.

Concept Quests

Open Middle

NRICH

YouCubed

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.

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 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 [The] 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, "a) Would you rather: 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. There's 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 maybe 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

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 young mathematics learners.

Amy Noelle Parks is interested in young children's mathematical experiences, both in and out of schools. She is particularly concerned with representing the experiences of children from marginalized groups in the research literature and with promoting humane schooling practices for all children. Her current projects include investigations of the role of play in mathematical learning, the resources parents draw on when supporting their children in mathematics, connections between emotional relationships and content learning in primary classrooms, and the mathematical engagements that are possible in informal spaces.

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 the 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? They 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.

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.

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 like, "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 about 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, right? 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.

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.

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 10s, 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?

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 of 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 LEGO [bricks] 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.

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 [if] 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