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

When we say students are engaged in a discussion or a task, what do we really mean? There are observable behaviors that we often code as engaged, but those are just the things we can see or hear. What does engagement really mean, particularly for students who may not verbally participate on a regular basis?

Supporting Elementary Mathematics Teachers

Routines for Reasoning

The silent and the vocal: Participation and learning in whole-class discussion

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

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

Meghan: I'm excited to be here.

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

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

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

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

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

Yet another layer of challenge here is a tendency to make assumptions about how students from particular cultural groups engage in discussions. You only know what you know. And teachers need opportunities to learn from their students about how they engage in discussions inside and outside of math class, and to be able to think about the connections and disconnections and the opportunities to leverage.

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

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

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

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

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

Meghan: Absolutely. So, I have a set of four different strategies that my colleagues and I have been working on over time. 

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

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

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

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

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

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

Mike: Let's talk about the second idea.

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

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

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

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

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

Meghan: Absolutely.

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

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

The first structure is really a pair share. Students are paired, if possible, with a student who used a different strategy or has a different solution. Each partner explains their strategy, and then together they look for connections between their thinking. So again, this moves beyond the traditional turn and talk because in addition to sharing your thinking, there's a task that the partners are doing about thinking about the connections between those two strategies. 

A second sort of structure is really using a stop and jot. In this instance, the teacher selects one strategy for students to be thinking about making a connection to, and then each student jots a connection between their strategy or solution and the strategy that the teacher has selected. And they do this in their notebook or in some other written form in the classroom. And so, these two different structures can support participation by having all students have an opportunity to share their own thinking, either verbally with a partner or by recording it in written form. And all students at the same time are having an opportunity to make connections in the classroom.

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

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

Mike: Gotcha. 

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

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

Meghan: Alright. So, the fourth strategy really focuses on broadening participation in the conclusion of a discussion. So, as we all know in a discussion, students hear lots of different ideas, but they don't all get to share their thinking in a discussion, nor do they all get to share what they are thinking at the end of the discussion. But we also know that students need space to consolidate their own thinking and the questions that they have about the ideas that have been shared. At the same time, teachers need access to students' thinking to plan for the next day, particularly when a discussion is not finished at the end of a given math lesson. 

With all of this, the challenge is that time is often tight at the end of a discussion. So, one structure that we've used has been a note to self. And in a note to self, students write a note to themselves about how they are currently thinking about a particular sort of problem at the end of a discussion. And a note to self allows students to take stock of where they are with respect to particular ideas, similar to a stop and jot. It can create a record of thinking that can be accessed on a subsequent day by students if those notes to self are recorded in a notebook. Again, support students in tracking on their own questions and how their thinking is changing over time, and it can provide the teacher with a window into all students' thinking.

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

Meghan: So, it was day one of our mathematics program, and we had done a discussion around an unequally partitioned rectangle task, and students were being asked to figure out what fraction of the hole was shaded. And there clearly wasn't enough time that day to really explore all the different sorts of ideas. And so, Darrius Robinson, who was one of the co-teachers, invited students to share some of their initial ideas about the task. And the way that Darrius then ended up deciding to conclude things that day was saying to students, “I think we're going to do this thing that I'm going to call a note to self.” And he invited the students to open up their notebooks and to record how they were thinking about the different ideas that had gotten shared thus far in the discussion. There was some modeling of what that might look like, something along the lines of, “I agree with … because …,” but it really opened up that space then for students to begin to record how they were thinking about otherwise ideas in math class. 

So, how might using a note to self broaden participation in a discussion? Well, first of all, students have the opportunity to participate. All students are being asked to write a note to themselves. It creates space for students to engage with others' ideas that doesn't necessarily require talk, right? So, this is an opportunity to privilege other ways of participating, and it also allows for thinking and processing time for all students.

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

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

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

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

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

Meghan: Sure. I want to give three different sorts of suggestions. So, one suggestion is to look at the fabulous books that have been put together by Amy Lucenta and Grace Kelemanik, who are the authors of Routines for Reasoning and Thinking for Teaching .  And I would argue that many of the routines that they have developed and that they share in those resources are ones that are really supportive of thinking about, “How do you broaden participation in mathematics discourse?” 

A second resource that someone might be interested in exploring is a research article that was written in 2017 by Cathy [Catherine] O'Connor, Sarah Michaels, Suzanne Chapin, and Alan [G.] Harbaugh that focuses on the silent and the vocal participation in learning in whole-class discussion, where they carefully looked at learning outcomes for students who were vocally expressing ideas and discussion as well as the silent participants in the discussion. And really found that there was no difference in the learning outcomes for those two groups of students. And so that's important, I think, for us to think about as teachers. 

At the same time, I want to be clear in acknowledging that all of what we do as teachers needs to be in relation to the learning goals that we have for students. So, sometimes our learning goals are that we want students to be able to share ideas and discussions. And if that's the case, then we actually do need to make sure that we build in opportunities for students to share their ideas verbally in addition to participating in other sorts of ways.

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

Meghan: Correct.

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

Meghan: Absolutely. 

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

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

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

Mike: Fabulous. Thank you so much. 

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

If there were a list of social skills we hope to foster in children, empathy is likely close to the top. Empathy matters. It helps us understand how others are feeling so we can respond appropriately, and it can help teachers understand the way their students are experiencing school. Today on a podcast, we talk with Dr. Kara Imm about a practice referred to as an empathy interview. We'll discuss the ways empathy interviews can help educators understand their students' lived experience with mathematics and make productive adaptations to instructional practice.

Mike Wallus: If there were a list of social skills we hope to foster in children, empathy is likely close to the top. Empathy matters. It helps us understand how others are feeling so we can respond appropriately, and it can help teachers understand the way their students are experiencing school. Today on a podcast, we talk with Dr. Kara Imm about a practice referred to as an empathy interview. We'll discuss the ways empathy interviews can help educators understand their students' lived experience with mathematics and make productive adaptations to instructional practice. 

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

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

Mike: So, I have to confess that the language of an empathy interview was new to me when I started reading about this, and I'm wondering if you could just take a moment and unpack, what is an empathy interview, for folks who are new to the idea?

Kara: Yeah, sure. I think I came to understand empathy interviews in my work with design thinking as a former teacher, classroom teacher, and now teacher-educator. I've always thought of myself as a designer. So, when I came to understand that there was this whole field around design thinking, I got very intrigued. And the central feature of design thinking is that designers, who are essentially thinking about creating new products, services, interactions, ways of being for someone else, have to start with empathy because we have to get out of our own minds and our own experiences and make sure we're not making assumptions about somebody else's lived experience. So, an empathy interview, as I know it now, is first and foremost a conversation. It's meant to be as natural a conversation as possible. When I do empathy interviews, I have a set of questions in mind, but I often abandon those questions and follow the child in front of me or the teacher, depending on who I'm interviewing.

Kara: And the goal of an empathy interview is to elicit stories; really granular, important stories, the kind of stories that we tell ourselves that get reiterated and retold, and the kinds of stories that cumulatively make up our identities. So, I'm not trying to get a resumé, I'm not interested in the facts of the person, the biography of the person. I'm interested in the stories people tell about themselves. And in my context, the stories that kids tell themselves about their own learning and their own relationship to school, their classrooms, and to mathematics. I'm also trying to elicit emotions. So, designers are particularly listening for what they might call unmet needs, where as a designer we would then use the empathy interview to think about the unmet needs of this particular person and think about designing something uniquely and specifically for them—with the idea that if I designed something for them, it would probably have utility and purpose for other people who are experiencing that thing. So, what happened more recently is that I started to think, “Could empathy interviews change teachers' relationship to their students? Could it change leaders' relationships to the teachers?” And so far, we're learning that it's a different kind of conversation, and it's helping people move out of deficit thinking around children and really asking important questions about, what does it mean to be a kid in a math class?

Mike: There's some language that you've used that really stands out for me. And I'm wondering if you could talk a little bit more about it. You said “the stories that we tell about ourselves”; or, maybe paraphrased, the stories that kids tell themselves. And then you had this other bit of language that I'd like to come back to: “the cumulative impact of those stories on our identity.” Can you unpack those terms of phrase you used and talk a little bit about them specifically, as you said, when it comes to children and how they think about their identity with relation to mathematics?

Kara: Sure. I love that kind of phrase, “the story we tell ourselves.” That's been a pivotal phrase for me. I think stories kind of define and refine our existence. Stories capture this relationship between who we are and who we want to become. But when I'm thinking about stories in this way, I imagine as an interviewer that I'm trying to paint a portrait of a child, typically. And so, I'm trying to interact with this child in such a way that I can elicit these stories, painting a unique picture of this kid, not only as a learner but also as a human. What inevitably happens when you do these interviews is that I'm interested in their experience in math class. When I listen to kids, they have internalized, “I'm good at math, and here's why” or “I'm bad at math, and here's why. I just know it.” But when you dig a little bit deeper, the stories they tell are a little more nuanced, and they kind of live in the space of gray. And I'm interested in that space, not the space of testing and measurement that would land you in a particular identity as meant for math or not meant for math.

Mike: I think what I was going to suggest is, why don't we listen to a few, because you shared a couple clips before we got ready for the interview, and I was fascinated by the approach that you had in chatting with these children and just how much information I could glean from even a minute or two of the interview slices that you shared. Why don't we start and get to know a few of these kiddos and see what we can learn together.

Kara: Sounds great.

Mike: We've got a clip that I'm going to invite you to set it up and give us as much context as you want to, and then we'll play the clip and then we can talk a little bit about it. I would love to start with our friend Leanna.

Kara: Great. Leanna is a third-grader. She goes to an all-girls school. I've worked in Leanna’s school over multiple years. I know her teacher well. I'm a part of that community. Leanna was kind of a new mathematician to me. Earlier in the day I had been in Leanna's classroom, and the interview starts with a moment that really struck me, which I won't say much more about. And I invited Leanna to join me after school so we could talk about this particular moment. And I really wanted to know how she made sense of what happened. So, I think we'll leave it at that and we'll listen to what happened.

Mike: Alright, let's give it a listen.

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

Kara: Hi, Leanna. Today when I was in your class, something interesting happened where I think the kids said to me, and they said, “Do you know we have a math genius in our class?” Do you remember that moment? 

Leanna: Yeah. 

Kara: Tell me what happened in that moment.

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

Kara: And what was that like for you?

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

Kara: Yeah, I was wondering about that. Why do you think the girls today—I mean, I'm a visitor, right?—why do you think they use the word “math genius”? And why did they choose you? What do you think they think of you?

Leanna: A mathematician …

Kara: Yeah.

Leanna: … because I go to this thing every Wednesday. They ask me what I want to be when I grow up, and I always say, a mathematician. So, they think that I am a math genius.

Kara: Gotcha. Do you think all the girls in your class know that you want to be a mathematician when you grow up? But do they mean something else? They didn't say, “We have a mathematician in our class.” They said, “We have a math genius.”

Leanna: Maybe.

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

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

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

Leanna: Yeah, a little bit.

Kara: A little bit? Do you love math? 

Leanna: Yeah. 

Kara: Yeah. Have you always loved math? 

Leanna: Yeah. 

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

Leanna: No. 

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

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

Kara: Yeah.

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

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

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

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

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

Kara: Yeah, I think so.

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

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

Mike: I think we should meet Matthew. 

Kara: Yeah. 

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

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

Mike: Let's give it a listen. 

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

Matthew: Yeah.

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

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

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

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

Kara: OK.

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

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

Matthew: Yeah.

Kara: Was it embarrassing to ask for help?

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

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

Matthew: It made no sense.

Kara: It made no sense. 

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

Kara: Hmm.

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

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

Matthew: Yeah. 

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

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

Kara: Hmm.

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

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

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

Kara: “I’m not” … OK.

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

Kara: Yeah , that's true. Right? 

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

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

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

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

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

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

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

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

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

Mike: Alright, let's listen in.

Nia: No, I haven't heard it, but …

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

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

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

Nia: Yes. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Kara: Hmm.

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

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

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

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

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

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

What if it were possible to capture all of the words teachers said or thought about students and put them in word clouds that hovered over each student throughout the day? What impact might the words in the cloud have on the student’s learning experience? This is the question that Beth Kobett and Karen Karp pose to start their book about strength-based teaching and learning. Today on the podcast, we’re talking about practices that support strength-based teaching and learning, and ways educators can implement them in their classrooms.  

https://corwin-connect.com/author/bethkobettandkarenkarp/

Strengths-Based Teaching and Learning in Mathematics: 5 Teaching Turnarounds for Grades K-6 

TODOS Math

Mike Wallus: What if it were possible to capture all of the words teachers said or thought about students and put them in word clouds that hovered over each student throughout the day? What impact might the words in the clouds have on students’ learning experience? This is the question that Beth Kobett and Karen Karp pose to start their book about strengths-based teaching and learning. Today on the podcast, we're talking about practices that support strengths-based teaching and learning, and ways educators can implement them in their classrooms. 

Mike: Hey, Beth, welcome to the podcast.

Beth Kobett: Thank you so much. I'm so excited to be here, Mike.

Mike: So, there's a paragraph at the start of the book that you wrote with Karen Karp. You said: “As teachers of mathematics, we've been taught that our role is to diagnose, eradicate, and erase students' misconceptions. We've been taught to focus on the challenges in students' work rather than recognizing the knowledge and expertise that exist within the learner.” This really stopped me in my tracks, and it had me thinking about how I viewed my role as a classroom teacher and how I saw my students’ work. I think I just want to start with the question, “Why start there, Beth?”

Beth: Well, I think it has a lot to do with our identity as teachers, that we are fixers and changers and that students come to us, and we have to do something. And we have to change them and make sure that they learn a body of knowledge, which is absolutely important. But within that, if we dig a little bit deeper, is this notion of fixing this idea that, “Oh my goodness, they don't know this.” And we have to really attend to the ways in which we talk about it, right? For example, “My students aren't ready. My students don't know this.” And what we began noticing was all this deficit language for what was really very normal. When you show up in second grade, guess what? There's lots of things you know, and lots of things you're going to learn. And that's absolutely the job of a teacher and a student to navigate. So, that really helped us think about the ways in which we were entering into conversations with all kinds of people—teachers, families, leadership, and so on—so that we could attend to that. And it would help us think about our teaching in different ways.

Mike: So, let's help listeners build a counternarrative. How would you describe what it means to take a strengths-based approach to teaching and learning? And what might that mean in someone's daily practice?

Beth: So, we can look at it globally or instructionally. Like, I'm getting ready to teach this particular lesson in this class. And the counternarrative is, “What do they know? What have they been showing me?” So, for example, I'm getting ready to teach place value to second graders, and I want to think about all the things that they've already done that I know that they've done. They've been grouping and counting and probably making lots of collections of 10 and so on. And so, I want to think about drawing on their experiences, A. Or B, going in and providing an experience that will reactivate all those prior experiences that they've had and enable students to say, “Oh yeah, I've done this before. I've made sets or groups of 10 before.” So, let's talk about what that is, what the [name] of it [is], why it's so important, and let's identify tasks that will just really engage them in ways that help them understand that they do bring a lot of knowledge into it. And sometimes we say things so well intentioned, like, “This is going to be hard, and you probably haven't thought about this yet.” And so, we sort of set everybody on edge in ways that set it's going to be hard, which means, “That's bad.” It's going to be hard, which means, “You don't know this yet.” Well, why don't we turn that on its edge and say, “You've done lots of things that are going to help you understand this and make sense of this. And that's what our job is right now, is to make sense of what we're doing.”?

Mike: There's a lot there. One of the things that I think is jumping out for me is this idea is multifaceted. And part of what we're asking ourselves is, “What do kids know?” But the other piece that I want to just kind of shine a flashlight on is there's also this idea of “What experiences have they had—either in their home life or in their learning life at school—that can connect to this content or these ideas that you're trying to pull out?” That, to me, actually feels like another way to think about this. Like, “Oh my gosh, we've done partitioning, we've done grouping,” and all of those experiences. If we can connect back to them, it can actually build up a kid's sense of, like, “Oh, OK.”

Beth: I love that. And I love the way that you just described that. It's almost like positioning the student to make those connections, to be ready to do that, to be thinking about that and providing a task or a lesson that allows them to say, “Oh!” You know, fractions are a perfect example. I mean, we all love to use food, but do we talk about sharing? Do we talk about when we've divided something up? Have we talked about, “Hey, you both have to use the same piece of paper, and I need to make sure that you each have an equal space.”? I've seen that many times in a classroom. Just tweak that a little bit. Talk about [how] when you did that, you actually were thinking about equal parts. So, helping students … we don't need to make all those connections all the time because they're there for students, and children naturally make connections. That's their job (chuckles). It really is their job, and they want to do that.

Mike: So, the other bit that I want to pick up on is the subtle way that language plays into this. And one example that really stood out for me was when you examined the word “misconception.” So, talk about this particular bit of language and how you might tweak it or reframe it when it comes to student learning.

Beth: Well, thank you for bringing this up. This is a conversation that I am having consistently right now. Because this idea of misconception positions the student. “You're wrong, you don't understand something.” And again, let's go back to that [...] “I've got to fix it.” 

But what if learning is pretty natural and normal to, for example, think about Piaget’s conservation ideas—the idea that a young child can or can't conserve based on [...] the arrangement. So, you put [...] you know, five counters out, they count them and then you move them, spread them out and say, “Are they the same, more, or less?” We wouldn't say that that's a misconception of a child because it's developmental. It's where they are in their trajectory of learning. And so, we are using the word “misconception” for lots of things that are just natural, the natural part of learning. And we're assuming that the student has created a misunderstanding along the way when that misunderstanding or that that idea of that learning is very, very normal.

Place value is a perfect example of it. Fractions are, too. Let's say they're trying to order fractions on a number line, and they're just looking at the largest value wherever it falls—numerator, denominator, I'm just throwing it down. You know, those are big numbers. So, those are going to go at the end of a number line. But what if we said, “Just get some fraction pieces out”? That's not a misconception 'cause that's normal. I'm using what I've already learned about value of number, and I'm throwing it down on a number line (chuckles). [...] So, it changes the way we think about how we're going to design our instruction when we think about what's the natural way that students do that. So, we also call it “fragile understanding.” So, fragile understanding is when it's a little bit tentative. Like, “I have it, but I don't have it.” That's another part, a natural part of learning. When you're first learning something new, you kind of have it, then you’ve got to try it again, and it takes a while for it to become something you're comfortable doing or knowing.

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

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

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

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

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

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

And then we move on to the next thing. 

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

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

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

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

Beth: It's also saying, “You might be feeling this way right now. If you're feeling like this”—like, for example, teaching a task and students are working on a task, and trying to figure out how to solve it, and it's starting to get a little noisy, and hands start coming up, stopping the class for a second and saying, “If you're feeling this way, that's an OK way to feel,” right? And, “Here's some things we might be thinking about. What are some strategies?”—like [sort of refocusing] them on how to get out of that instead of me fixing it—like, “What are some strategies you could think about? Let's talk about that and then go back to this.” 

So, it's the teacher acknowledging. It's allowing the students to talk about it. It's allowing everybody … It's not just making students be in productive struggle, or another piece of that is ‘Just try harder.” That's not real helpful. Like, OK, “I just need you to try harder because I'm making you productively struggle.” I don't know if anyone has had someone tell them that, but I used to run races and when someone said, “Try harder” to me, I'm like, “I'm trying as hard as I can.” That isn't that helpful. So, it's really about being very explicit about why it's important. Getting students to the other side of it should be the No. 1 goal. And then addressing it: “OK, you experienced productive struggle, now you did it. How do you feel now? Why is it worth it?”

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

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

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

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

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

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

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

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

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

Beth: OK, so how it works is that you have done some sharing in the class. So, for example, you may have already shared some solutions to a task. Students have been given a task they're sharing; they may be sharing a pair-to-pair share or a group-to-group share, something like that. It could be whole-class sharing. And then you say, “Hmm, you've heard lots of good ideas today, lots of interesting thinking and different strategies. If you'd like to provide a math amendment, which is a change to your solution in addition, something else that you'd like to do to strengthen it, you can go ahead and do that.” 

And you can do it in that lesson right there, or [...] what we're finding is really powerful is to bring it back the next day or even a few days later, which connects us back to this idea of what you were saying, which is, “Is this mastered?” “Where am I on the developmental trajectory?” So, I'm just strengthening my understanding, and I'm also hearing … I'm understanding the point of hearing other people's ideas is to go and try them out and use them. And we're really allowing that. So, [...] this has been amazing; the math amendments that we're seeing students do, taking someone else's idea or a strategy and then just expanding on their own work. And it's very similar to, like, a writing piece, right? You get a writing piece and you polish and you polish. You don't do this with every math task that you solve or problem that you solve, but you choose and select to do that.

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

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

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

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

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

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

Many elementary educators recognize that timed memorization activities have the potential to seriously derail a child’s identity as a doer of mathematics. But if we want to build fluency, what alternatives are there? Today on the podcast, we talk with Lori Bluemel, a curriculum consultant at The Math Learning Center, about a play-based alternative for building fluency from conceptual understanding.   

Mike Wallus: When I meet someone new at a gathering and tell them that I work in math education, one of the most common responses I hear is, “I was never good at math in school.” When I probe a bit further, this belief often originated in the person's experience memorizing basic facts. How can we build students' fluency with facts, encourage flexible thinking, and foster students' confidence? That's the topic we'll explore in this episode of Rounding Up. 

Mike: One of the challenges that we face in education can be letting go of a practice—even if the results are questionable—when the alternative is unclear. In elementary math, this challenge often arises around building computational fluency. We know that speed tests, drill and kill, and worksheets—those are all ineffective practices. And even worse, they can impact students' math identity. So today, we're going to spend some time unpacking an alternative, a component of the Bridges in Mathematics curriculum called Work Places. We're doing this not to promote the curriculum, but to articulate an alternative vision for ways that students can develop computational fluency. To do that, we're joined by Lori Bluemel, a curriculum consultant for The Math Learning Center. 

Mike: Lori, welcome to the podcast. It's great to have you with us.

Lori: Thank you. It's good to be here.

Mike: Well, let's just start with a basic question: If I'm a listener who's new to the Bridge's curriculum, can you describe what a Work Place is?

Lori: The simple answer would be that it's math activities or games that are directly focusing on the skills or the ideas and concepts that students are working on during Problems & Investigations. The best aspect, or the feature about Work Places, is that teachers have an opportunity to be like a fly on the wall as they're listening in to their students and learning about what strategies they're using and the thinking process that they're going through.

Mike: How do you think practicing using a Work Place differs from the version of practice that children have done in the past? What changes for the child or for the learner?

Lori: Well, I always felt like a piece of paper was pretty static. There wasn't a lot of interaction. You could run through it so quickly and be finished with it without really doing a lot of thinking and processing—and with absolutely no talking. Whereas during Work Places, you're discussing what you're doing. You're talking to your partner. You're listening to your partner. You're hearing about what they're doing and the different methods or strategies that they're using. And [there’s] nothing at all static about it because you're actively working together to work through this game or this activity.

Mike: That is so fascinating. It makes me think of a book that I was reading recently about thinking classrooms, and one of the things that they noted was, there's data that suggests that the more talk that's happening in a classroom, the more learning that's actually happening. It really connects me to what you just said about Work Places.

Lori: Yeah, and I feel like that's the big difference between Work Places and doing a worksheet on your own. You can do it completely isolated without any outside interaction; whereas Work Places, it's very interactive, very collaborative.

Mike: Yeah. So, as a former classroom teacher who used Work Places on a daily basis, how did you set up norms and routines to make them successful for students?

Lori: Well, I actually went through several different methods, or routines, before I landed on one that really worked well for me. One that worked best for me is, at the beginning of the year when we first started doing Work Places, I would take that very first Work Place time, and we would just have a class meeting and talk about what we're doing in Work Places. Why would we even have Work Places? We would create an anchor chart, and we'd have one [column] that would say “Students.” The other side would say “Teachers.” And then we would talk about the expectations. And the students would come up with those. Then we would talk about me as the teacher, what do they think I should be doing? And again, that would come up with all different ideas. And then we always came back to that final thought of, “We need to be having fun.”

Mike: Hmm. 

Lori: Math needs to be fun during Work Places. And then we would start in, and students would go to Work Places. They would choose their partner, and then they would get started. And that first few times we did Work Places, I always just kind of watched and listened and walked around. And if I felt like things needed to be slightly different, maybe they weren't talking about math or they weren't really playing the Work Place, then we would call a class meeting. And everyone would freeze, and we'd go to our meeting spot, and we would talk about what I saw. And we would also talk about what was going well and what they personally could do to improve. And then we'd go back to Work Places and try it again. Needless to say, a lot of times those first few times at Work Places they didn't play the games a lot because we were setting up expectations. But in the long run, it made Work Places run very smoothly throughout the rest of the year.

Mike: Yeah. The word that comes to mind as I listen to you talk, Lori, is investment. 

Lori: Um-hm. 

Mike: Investing the time to help set the norms, set the routines, give kids a vision of what things look like, and the payoff is productive math talk.

Lori: Exactly. And that was definitely the payoff. They needed reminders on occasion, but for the most part, they really understood what was expected.

Mike: I think it's fascinating that you talked about your role and asked the kids to talk about that. I would love [it] if you could say more about why you asked them to think about your role when it came to Work Places.

Lori: I wanted them to realize that I was there to help them. But at the same time, I was there to help their peers as well. So, if I was working with a small group, I wanted them to understand that they might need to go to another resource to help them answer a question. They needed to make sure that I was giving my attention to the, the small group or the individual that I was working with at that time. So, by talking about what was expected from me , my hope was that they would understand that there were times when they might have to wait a minute, or they might go to another resource to find an answer to their question, or to help them with the situation that they were in. And that seemed to be the case. I think I alleviated a lot of those interruptions just by talking about expectations.

Mike: So, I want to return to something that you said earlier, Lori, ’cause I think it's really important. I can imagine that there might be some folks who are listening who are wondering, “What exactly is the teacher doing while students are engaged in Work Places?” 

Lori: Um-hm. 

Mike: And I wanted to give you an opportunity to really help us understand how you thought about what your main focus was during that time. So, children are out, they're engaged with the Work Places. How do you think about what you want to do with that time?

Lori: OK. So, I often look at the needs of my students and, and think about “What have I seen during Problems & Investigations? What have I seen during Work Places previously? And where do I focus my time?” And then I kind of gravitate towards those students that I want to listen in on. So, I want to, again, be like that fly on the wall and just listen to them, maybe ask a few questions, some clarifying questions about what they're doing, get an idea of what strategies or the thinking that they're going through as they're processing the problem. And then from there, I can start focusing on small groups, maybe adjust the Work Place so that they can develop that skill at a deeper level. It helps me during that time to really facilitate my students’ practice; help students make the most of their practice time so that as they're going through the Work Place, it's not just a set of rules and procedures that they're following—that they're really thinking about what they're doing and being strategic with those skills as well. So that's my opportunity to really help and focus in on my small groups and provide the support that students need. Or maybe I want them to advance their skills, go a little bit deeper so that they are working at a little bit different level.

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

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

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

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

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

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

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

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

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

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

Participation is an important part of learning to make sense of mathematics. 

But ask yourself, “What counts as participation?”

In this episode, we talk with Dr. Juanita Silva from Texas State University about an expanded definition of participation—and what it might mean for how we engage with and value our students’ thinking.   

Todos

Attending to others’ mathematical ideas: a semiotic alternative to logocentrism in bilingual classrooms

Mike Wallus: Participation is an important part of learning to make sense of mathematics. But stop and ask yourself, “What counts as participation?” In this episode, we'll talk with Dr. Juanita Silva from Texas State University about an expanded definition of participation and what it might mean for how we engage with and value our students' thinking. 

Welcome, Juanita. Thanks for joining us on the podcast.

Juanita Silva: Hi. Thank you for inviting me. I'm excited to talk about this topic.

Mike: I think I'd like to start by asking you to just talk about the meaning of participation. What is it and what forms can participation take in an elementary math classroom?

Juanita: Well, there's a mixture of nonverbal and verbal communication. And you can add in there gestures [as a] form of communication, not just in an interconnected space, but also thinking about students’ respect. And it's not just bidirectional, but there’s a lot of things that are kind of added in that space.

Mike: So, it strikes me that when I was a classroom teacher, when I look back, I probably overemphasized verbal communication when I was assessing my students' understanding of math concepts. And I have a feeling that I'm not alone in that. And I'm wondering if you could talk about the way that we've traditionally thought about participation and how that might have impacted student learning?

Juanita: Yes, this is a great question. In thinking about, “What does this look like?”, “How to participate in the classroom?” Mostly teachers think about this as whole-group discussions or in small-group discussions. And I emphasize the word their discussions, where students can share verbally how they thought about the problem. So, for example, if a student is solving a fraction word problem, the teacher may ask, “OK, so how did you solve this problem? Can you share your strategy with the class? What does that look like?” And so, the student sometimes will say, “If I'm solving a fraction word problem about four parts or four chocolate bars, then I can cut those leftovers into four parts.” So that's usually what we think of, as in our teaching and practice in elementary schooling. We think of that as verbal communication and verbal participation, but there are others. (laughs)

Mike: Let's talk about that. I think part of what you have pushed me to think about is that a student's verbal communication of their thinking, it really only offers a partial window into their actual thinking. What I'd like to do is just talk about what it might look like to consciously value participation that's nonverbal in an elementary classroom. Like, what are the norms and the routines that a teacher could use to value nonverbal communication, maybe in a one-to-one conversation in a small-group or even in a whole-group discussion?

Juanita: Yes. So, I can share a little bit for each one of those. For example, in a one-to-one environment, the teacher and student can more effectively actually communicate ideas if the teacher attends to that child's thinking in nonverbal ways as well. So, for instance, I've had a student before in the past where he would love to explain his thinking using Unifix cubes and to share his thinking on a multiplication problem that was about three sets of cookies. And those sets were in groups of seven. So, there were seven cookies in each bag. And I asked him, “Well, how would you share? Could you explain your thinking to me?” And so, he showed me three sets of seven Unifix cubes, and he pointed to each of the seven linking cubes and then wrote on his paper the number sentence, “7 plus 7 plus 7 is 21.” And when I asked him if the seven represented the cookies, he simply nodded yes and pointed to his paper, saying and writing the words “21 total.”

So, I didn't ask him to further explain anything else to me verbally because I had completely understood how he thought of the problem. And in this example, I'm showing that a student's gestures and a student's explanation on a piece of paper should be valued enough. And we don't necessarily need to engage in a verbal communication of mathematical ideas because this honors his ways of thinking. But at the same time, I could clearly understand how this child thought of the problem. So, I think that's one way to think about how we can privilege a nonverbal communication in a one-to-one setting.

Mike: That's really helpful. I think that part of the example that you shared that jumps out for me is attending to the ways that a child might be using manipulative tools as well, right? 

Juanita: Correct. 

Mike: So, it was kind of this interaction of the student's written work—their manipulative tools, the way that they gestured to indicate their thinking— … that gave you a picture of how this child was thinking. And you didn't really need to go further than that. You had an understanding as an educator that would help you think about what you might do next with that child.

Juanita: Absolutely. And that is one of the tools that I find to be super useful, is to not just have students explain their thinking, but also just listen to their nonverbal cues. And so, paying attention to those and also valuing those is extremely important in our practice. I can share one of my favorites, which is a small-group example. And this one is kind of foundational to think of the practice when we're teaching in our elementary math classrooms. It's not just that interactions between student and teacher, but the interactions between students and students can be very powerful. So, that's why this is one of my favorite examples. I had two students at one point in my practice. And this was Marco and José, and they were in fourth grade. They were having a hard time communicating verbally with one another, and José was trying to convince Marco of his strategy to split the leftovers of an equal-sharing problem into three parts instead of halves.

But his verbal communication of these ideas were not clear to Marco. And José explains to Marco, “You have to cut it into halves.” And Marco would say, “Yes, that is what I did.” Like, frustrated, as if, like, “You have to cut this into halves.” And José would say, and Marco was like, “Yes, that's exactly what I did.” So, this exchange of verbal communication was not really helping both of them showcase how they were trying to communicate. So, then José started to insist, and he said, “No, look.” And then he showed Marco his strategy on his paper. And in his paper, he had split the bar into three parts. And then Marco looked at José and said, “Ah, OK.” Had José not shown this strategy on his paper, then Marco would have never really understood what he meant by “You have to cut it into halves.” And so, I share this example because it really showcases that sometimes what we're trying to say and communicate might come across differently verbally, but we mean something else when we showcase it nonverbally. So, in this instance, José was trying to explain that, but he couldn't figure out how to tell that to Marco. And so, in this instance, I feel like it really showcases the power of the nonverbal communication among students.

Mike: I think what's fascinating about that is, conceptually the strategy was right there. It was kind of like, “I'm going to equally partition into three parts.” The issue at hand was the language choice. I'm essentially referring to this equal partition as a half, this second equal partition as a half, and this third equal partition as a half. That's a question of helping figure out what is the language that we might use to describe those partitions. But if we step back and say, “Mathematically, does the child actually understand the idea of equal partitioning?” Yes. And then it seems as though it becomes a second question about, “How do you work with children to actually say what we call this?” or the way that we name fractions is—that's a different question, as opposed to, “Do you understand equal partitioning, conceptually?”

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

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

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

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

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

Juanita: On that alone? (laughs)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Participation is an important part of learning to make sense of mathematics. 

But ask yourself, “What counts as participation?”

In this episode, we talk with Dr. Juanita Silva from Texas State University about an expanded definition of participation—and what it might mean for how we engage with and value our students’ thinking.   

Todos

Attending to others’ mathematical ideas: a semiotic alternative to logocentrism in bilingual classrooms

Mike Wallus: Participation is an important part of learning to make sense of mathematics. But stop and ask yourself, “What counts as participation?” In this episode, we'll talk with Dr. Juanita Silva from Texas State University about an expanded definition of participation and what it might mean for how we engage with and value our students' thinking. 

Welcome, Juanita. Thanks for joining us on the podcast.

Juanita Silva: Hi. Thank you for inviting me. I'm excited to talk about this topic.

Mike: I think I'd like to start by asking you to just talk about the meaning of participation. What is it and what forms can participation take in an elementary math classroom?

Juanita: Well, there's a mixture of nonverbal and verbal communication. And you can add in there gestures [as a] form of communication, not just in an interconnected space, but also thinking about students’ respect. And it's not just bidirectional, but there’s a lot of things that are kind of added in that space.

Mike: So, it strikes me that when I was a classroom teacher, when I look back, I probably overemphasized verbal communication when I was assessing my students' understanding of math concepts. And I have a feeling that I'm not alone in that. And I'm wondering if you could talk about the way that we've traditionally thought about participation and how that might have impacted student learning?

Juanita: Yes, this is a great question. In thinking about, “What does this look like?”, “How to participate in the classroom?” Mostly teachers think about this as whole-group discussions or in small-group discussions. And I emphasize the word their discussions, where students can share verbally how they thought about the problem. So, for example, if a student is solving a fraction word problem, the teacher may ask, “OK, so how did you solve this problem? Can you share your strategy with the class? What does that look like?” And so, the student sometimes will say, “If I'm solving a fraction word problem about four parts or four chocolate bars, then I can cut those leftovers into four parts.” So that's usually what we think of, as in our teaching and practice in elementary schooling. We think of that as verbal communication and verbal participation, but there are others. (laughs)

Mike: Let's talk about that. I think part of what you have pushed me to think about is that a student's verbal communication of their thinking, it really only offers a partial window into their actual thinking. What I'd like to do is just talk about what it might look like to consciously value participation that's nonverbal in an elementary classroom. Like, what are the norms and the routines that a teacher could use to value nonverbal communication, maybe in a one-to-one conversation in a small-group or even in a whole-group discussion?

Juanita: Yes. So, I can share a little bit for each one of those. For example, in a one-to-one environment, the teacher and student can more effectively actually communicate ideas if the teacher attends to that child's thinking in nonverbal ways as well. So, for instance, I've had a student before in the past where he would love to explain his thinking using Unifix cubes and to share his thinking on a multiplication problem that was about three sets of cookies. And those sets were in groups of seven. So, there were seven cookies in each bag. And I asked him, “Well, how would you share? Could you explain your thinking to me?” And so, he showed me three sets of seven Unifix cubes, and he pointed to each of the seven linking cubes and then wrote on his paper the number sentence, “7 plus 7 plus 7 is 21.” And when I asked him if the seven represented the cookies, he simply nodded yes and pointed to his paper, saying and writing the words “21 total.”

So, I didn't ask him to further explain anything else to me verbally because I had completely understood how he thought of the problem. And in this example, I'm showing that a student's gestures and a student's explanation on a piece of paper should be valued enough. And we don't necessarily need to engage in a verbal communication of mathematical ideas because this honors his ways of thinking. But at the same time, I could clearly understand how this child thought of the problem. So, I think that's one way to think about how we can privilege a nonverbal communication in a one-to-one setting.

Mike: That's really helpful. I think that part of the example that you shared that jumps out for me is attending to the ways that a child might be using manipulative tools as well, right? 

Juanita: Correct. 

Mike: So, it was kind of this interaction of the student's written work—their manipulative tools, the way that they gestured to indicate their thinking— … that gave you a picture of how this child was thinking. And you didn't really need to go further than that. You had an understanding as an educator that would help you think about what you might do next with that child.

Juanita: Absolutely. And that is one of the tools that I find to be super useful, is to not just have students explain their thinking, but also just listen to their nonverbal cues. And so, paying attention to those and also valuing those is extremely important in our practice. I can share one of my favorites, which is a small-group example. And this one is kind of foundational to think of the practice when we're teaching in our elementary math classrooms. It's not just that interactions between student and teacher, but the interactions between students and students can be very powerful. So, that's why this is one of my favorite examples. I had two students at one point in my practice. And this was Marco and José, and they were in fourth grade. They were having a hard time communicating verbally with one another, and José was trying to convince Marco of his strategy to split the leftovers of an equal-sharing problem into three parts instead of halves.

But his verbal communication of these ideas were not clear to Marco. And José explains to Marco, “You have to cut it into halves.” And Marco would say, “Yes, that is what I did.” Like, frustrated, as if, like, “You have to cut this into halves.” And José would say, and Marco was like, “Yes, that's exactly what I did.” So, this exchange of verbal communication was not really helping both of them showcase how they were trying to communicate. So, then José started to insist, and he said, “No, look.” And then he showed Marco his strategy on his paper. And in his paper, he had split the bar into three parts. And then Marco looked at José and said, “Ah, OK.” Had José not shown this strategy on his paper, then Marco would have never really understood what he meant by “You have to cut it into halves.” And so, I share this example because it really showcases that sometimes what we're trying to say and communicate might come across differently verbally, but we mean something else when we showcase it nonverbally. So, in this instance, José was trying to explain that, but he couldn't figure out how to tell that to Marco. And so, in this instance, I feel like it really showcases the power of the nonverbal communication among students.

Mike: I think what's fascinating about that is, conceptually the strategy was right there. It was kind of like, “I'm going to equally partition into three parts.” The issue at hand was the language choice. I'm essentially referring to this equal partition as a half, this second equal partition as a half, and this third equal partition as a half. That's a question of helping figure out what is the language that we might use to describe those partitions. But if we step back and say, “Mathematically, does the child actually understand the idea of equal partitioning?” Yes. And then it seems as though it becomes a second question about, “How do you work with children to actually say what we call this?” or the way that we name fractions is—that's a different question, as opposed to, “Do you understand equal partitioning, conceptually?”

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

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

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

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

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

Juanita: On that alone? (laughs)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

When you look at the results of your students' work, what types of things are you attending to? 

Many of us were trained to look for the ways that students were not understanding concepts or ideas. But what if we flipped that practice on its head and focused on the things students did understand?

Today on the podcast, we’re talking with Tisha Jones, senior advisor for content development at the MLC about building an asset-based approach to assessment.  

Insπre: Rethinking the Purpose of Math Education | Tisha Jones | TEDxGeorgiaStateU

Mike Wallus: When you look at the results of your students’ work, what types of things are you attending to? Many of us were trained to look for the ways that students were not understanding concepts or ideas. But what if we flipped that practice on its head and focused on the things students did understand? Today on the podcast, we're talking with Tisha Jones, senior advisor for content development at The Math Learning Center, about building an asset-based approach to assessment. 

Mike: Tisha, first of all, thanks for joining us. We're thrilled to have you with us.

Tisha Jones: I'm really excited to be here.

Mike: I have a sense that for a lot of people, the idea of asset-based assessment is something that we might need to unpack to offer, kind of, a basic set of operating principles or a definition. So, my first question is, “How would you describe asset-based assessment? What would that mean for a practitioner?”

Tisha: I think the first part of it is thinking just about assessment. Assessment is a huge part of every school that is in this country. So, there are formative assessments, which are ongoing assessments that teachers are doing while students are considered “in the process of learning”—although we know that students really are never not in the process of learning. And then there are also summative assessments, when we want to see if they have demonstrated proficiency or mastery of the concepts that they've been learning throughout that unit. But when we're thinking about assessments, oftentimes the idea of assessment is that we are looking for what students don't know. And asset-based assessment means that we're taking this idea and we're flipping it, and we're saying, “Let's start by looking at what students are showing us that they do know.” And we're trying to really focus on the things that our students are showing us that they're able to do.

Mike: So, that's a lot. And I think one of many of the things that's going on for me is that that's a pretty profound mind shift, I think, for a lot of folks in the field; not because they necessarily want to look at their students as a set of deficits, but because most of the training that a lot of us got actually was focused on “What are the deficits?”

Tisha: Most of the training when we're talking about kids casually, or with our colleagues or administrators, we're often worried about, “Well, our kids don't know this. Our kids are struggling here.” And that really becomes the way that we see our students, right? And our kids are so much more than that, right? And our kids are coming to us with knowledge, and we can forget that when we're only focused on what they don't know.

Mike: There's a great quote that you're making me think about. It's from the fourteenth century, and the person has said, essentially, “The language that we use becomes the world that we live in.” And I think that's a little bit of where you're going, is that deficit-focused language kind of lives in the DNA of a lot of either the training that we've had or the structures of schools. And so, flipping this is a mind shift, and I think it's really exciting that we're talking about this. I have two things on my mind. I think one is, let's talk about the assessments themselves first. So, if I want to start thinking about using my assessments in an asset-based way, if we just think about the assessments themselves, be they formative or summative, tell me about what you think an educator might do with the assessments that they're using, whether they're coming from a curriculum or whether they're some that they're designing on their own. How should I think about the assessment materials that I have, and are there ways that I should imagine shifting them?

Tisha: That's a great question. I think that when you're looking at your assessments, you may or may not need to change them. They might be fine the way that they are. But the way to know is when you see the opportunities kids have to give their answers, what is that going to tell you about what they understand? So, if you have, for example, a problem that is computation, if you have a problem that has just asked the kids for an answer, or if you have a problem that's multiple choice, what are you learning about their thinking, about their understanding from what they put on the paper? Now, I'm not saying don't ever use those questions. They have their purpose. But that is really what I am asking you to do, is to think about “What is their purpose? What is the intention behind the questions on the assessment?” So, are there ways for you to open up the assessment to give kids more ways of showing what they do understand as opposed to limiting them to saying, “You must show something in this way” or “You're either right or you're wrong”?

Mike: Yeah, that really hits home for me. And I think one of the operating principles that I'm hearing is, regardless of what assessment tools you're using, creating space for kids to show you how they're thinking is really a starting, foundational, kind of, centerpiece for asset-based assessment.

Tisha: Absolutely. And I want to also add that I'm talking a lot about paper and pencil because we think about assessments as paper and pencil. But assessment’s also not just paper and pencil. Assessment, especially formative assessment, it's your conversations that you have with kids in class. As far as I am concerned, there is no better way to know what a kid's thinking than to talk to them. Talk to your kids as much as you absolutely, possibly can. Ask them so many questions.

Mike: Well, you're bringing me to the second piece about the assessments themselves. One piece is, create space, regardless of whether it's a question in a conversation or whether it's a question in a paper-pencil assessment or what have you, for them to show their thinking. The other thing that it makes me think is, part of my work as an educator is to look at the questions and say, “What are the big ideas that I'm really looking for? And what is it that I'm hoping that I can understand about children's thinking with each of these questions that I'm asking?” 

Tisha: Yes.

Mike: Beyond just right and wrong.

Tisha: Yes, this is hard work. But this, to me, is not extra work. When you think about a gap, sometimes that can feel very disheartening. It can feel like, “I can't close it. My kids don't know this. They're never going to get it.” It almost just drains the joy of teaching out. This is the job, and this is the part that I am hoping we can all get excited about. I am excited to know what my kids understand. I feel like that gives me a better entryway to being a better teacher for them. If we can start to shift how we think about assessing our students to looking for what they know, to me, that feels very different. It feels different for your kids, and it feels different for you. It's much more fun to walk into a classroom thinking about what my kids know than what they don't.

Mike: Yeah. And I think you're hinting at the next place that I wanted to go, which is, there's the assessments themselves and both how I use them and how I make space for kids to show their thinking. And then there's “How do I approach the things that kids are showing me in their assessments?” And I think that feels like another one of these mind-shift pieces where, what kept coming to mind for me is, if you and I and a colleague or two were sitting together at a table, and we were teaching third grade and we had a set of student work in front of us, part of what I'm thinking about is what would a conversation sound like if we were really taking an asset-based perspective on looking at our students' work? What questions might we ask? What kind of a process might we use to, kind of, really focus on assets as opposed to focusing on deficits and gaps?

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

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

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

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

Tisha: Right. 

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

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

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

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

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

Tisha: Absolutely. 

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

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

Mike: Uh-hm. 

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

Mike: Oh! Yeah!

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

Mike: Uh-hm.

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

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

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

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

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

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

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

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

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

Kindergarten is a joyful, exciting and challenging grade level to teach. It’s also a time when educators can develop a set of productive norms and routines around discourse that can have a long-lasting effect on students. On today’s podcast, we talk with Dr. Hala Ghousseini, a professor at the University of Wisconsin, about building a solid foundation for math talk in kindergarten and beyond. 

Supporting Mathematics Talk in Kindergarten

Supporting Understanding Using Representations

Exploring Mathematics through Play In the Early Childhood Classroom

Mike Wallus: Kindergarten is a joyful, exciting and challenging grade level to teach. It's also a time when educators can develop a set of productive norms and routines around discourse that can have long-lasting effects on students. On today's podcast, we talk with Dr. Hala Ghousseini, a professor at the University of Wisconsin, about building a solid foundation for math talk in kindergarten and beyond. 

Mike: Welcome, Hala. We're really excited to have you on the podcast today talking about math talk in kindergarten. 

Hala Ghousseini: Thank you very much for having me. This is exciting. I love this topic, and the chance to really talk about this with you is great. 

Mike: Well, I feel the same way. I spent eight of my 17 years teaching kindergarten, so I've been dreaming about a podcast like this for a long time. 

Hala: (laughs) I can imagine the magic of kindergarten just because it's a time where people think that they know what to expect, but literally you don't know what to expect with children in kindergarten. 

Mike: You started to hint at the first thing that I hope to talk about. I would love to talk about norms . This feels so important because the norms and the culture that we set in kindergarten, from my perspective, those might be some of the first messages students receive about what's valued in a mathematics classroom. And I'm wondering if you could talk just a bit about the norms that you think are important. I mean, perhaps what it looks like to support them in kindergarten. 

Hala: Absolutely. And I just want to situate a little bit some of the things that I have been studying and thinking about. When I think of math in kindergarten, it very much exists within the learning altogether that happens in kindergarten; whether it's social-emotional skills, whether they're learning about other subject areas. So, when I think about the norms, I think often of them as embedded within the fabric of what's happening in kindergarten. In the research that we've done, we've seen it happening at two levels. One in relation to what we would call “norms related to what's conceptual,” or what [people might] call more like the disciplinary aspects of norms. So, some of the things that we've seen is, first of all, centered on children's thinking. The idea that first as an individual in class, that I'm a contributor to everyone's understanding. So, the way that is typically continuously communicated by the teacher, in the sense that it's important to share our thinking. And it's important to share it, not just because I'm the teacher and I asked you to do it, but because it's going to contribute to everyone else's learning. 

Hala: My learning as the teacher, others learning in the classroom. And we've seen examples from teachers where often, as they're asking students to get ready to go into their small groups, they would always say, “Remember, it's important to show our thinking and our work because we want to help someone else learn it.” You want to help the class understand this idea better. And even with the use of representations, resources, those were all really in the service of helping someone make their thinking explicit so that someone else is going to understand it or use it or build on it. So, I'll give you another example. The idea of saying, “Remember, we want to listen now to Hala share her thinking because we want to think how we make sense of it, what Hala is helping us think about.” So, those were the typical expressions or things that teachers would say in building these norms in the classroom. 

Hala: The other norm, when it comes to the social aspects of the norm, was really this explicit work on the sense of the collective as an intellectual community. The idea that we are in this together. It's not about me and you as the teacher, but it's about the us. What do we make of it? How do we really flag certain things that may help the group process and think about something? And those were also done constantly across the times we’ve spent in these classrooms, in the way teachers would really point to something that may help us as a group later. “Hey, look at this, this might help us later in the way we’re going to work on certain ideas together.” 

Mike: Well, I do want to ask you about something else that really struck me when I was reading the article. So, you and your co-authors talked a great deal about orienting students to and then encouraging the use of resources to communicate their thinking. That really hit me as a person who used to teach these young kiddos. Can you talk a little bit about what this looks like? 

Hala: Yes. This drew our attention, given where kindergartners are in their language development. They bring a lot of language from home that actually is going to be essential to build on in explaining the reasoning, talking about their thinking, reacting to someone else’s thinking. So, we started thinking about the way students’ think and the way their language that they bring with them becomes a resource that they could use. So, encouraging them that, yes, that is one way you can explain your thinking, so that really they find that language that is going to give them an entry point into the collective as an intellectual community. The second thing in relation to resources, also availing in the classroom. We’ve noticed these teachers that—besides the fact that you have, like, a number line or a hundreds chart displayed on the board or even the physical tools that usually typically students play with—how those become things that the teacher points to and says, “Wow, you know what you're doing.”

Hala: This might help us think about this idea. So, let’s remember that what struck us was that, when students were explaining their thinking, we rarely saw a student asking for permission to go and use something to come and support their thinking. We saw that they were really going to things and bringing them. So that was a norm in that class. That kind of intersects with the idea of normative ways of working. You can just go and reach it. You don’t have to get that teacher’s permission to do it. I think one more thing I’ll say about resources. We’ve noticed the teacher, typically if a student used a particular resource that supported them in their thinking, when they’re sharing, they make sure to actually highlight it, lift it up in what the student is saying so that others see that those resources could be contributions to supporting the reasoning in this class. 

Mike: So, boy, there’s a lot there. I think the first thing that really hits me is this idea that part of the culture that you want to establish, is that the resources are available and it’s contingent on the teacher saying, “Yes, you can go get that right now.”

Hala: Absolutely. And it’s a way of socializing the students to be aware of what’s in their classroom that is actually part of what’s supporting their learning. You know, there is a thing that I always work on when I’m working with teachers, this idea that, you know, children are sensemakers. And we tend to think of children as sensemakers beyond just mathematics. Of course they are, but also they’re sensemakers as learners in general. So, we treat them as sensemakers in the way as teachers. We owe it to them to explain to them why, for example, we’re asking them to do something. And we say, “So, I want you to show your work—not just to please me, because this contributes to the collective work in this way.” And we reinforce this message continuously. Similarly, the idea of what’s in our class, like, when we see, for example, base ten blocks. I have a few things in this corner. The idea that these are there to also support our learning. So, we treat them as sensemakers in the sense—these are all shared tools for our classrooms. So, that's kind of how we think about it in relation to the orienting to resources. 

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

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

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

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

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

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

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

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

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

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

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

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

Hala: So, let me explain what we meant by it in that article. It's literally when—because students may not have that facility to explain their thinking articulately, elaborately—it's when the teacher actually supports them by recapping what they said to the class. And on top of it, building on it and setting it up for further articulation or investigation. So, we try to distinguish here—that's why we're trying to revisit the word “narration” because we don't think of it just as revoicing. We think of it as a way where the teacher is highlighting something the student did and, often, we see it in exchange. It's highlighted not only in terms of the verbatim words that they used or the actions that they took. Highlighting why this is really helping in the task that we are working on together, and then follows it. It positions it in a way where, now this is what Gloria did. 

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

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

Hala: Absolutely. And that sweet spot that I think you are getting at is really knowing when do you do it and when do you hold off. In the sense, I don't think there is a rule, but it all goes to the teacher's ability to know: “Is there a shared language here that the students can access through what a student said?” So, knowing your students in terms of, is this something that I need to further articulate so that now they could engage productively with someone's idea? And if it's not, then actually it's just highlighting, pulling from what a student says, the valuable pieces that you think are going to be important for the continued work of the class, rather than, literally, a student says something, you say verbatim, and then you ask more questions. It's really tracking what seems to be important for the development of everyone's thinking, that collective as an intellectual community that's working together. 

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

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

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

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

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

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

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

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

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

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

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

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

Mike: Oh, thank you so much. 

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

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