Dr. Cathery Yeh, Supporting Neurodiverse Students in Elementary Mathematics Classrooms ROUNDING UP: SEASON 3 | EPISODE 14

What meaning does the term neurodiverse convey and how might it impact a student’s learning experience? 

And how can educators think about the work of designing environments and experiences that support neurodiverse students learning mathematics? 

In this episode, we discuss these questions with Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin. 

BIOGRAPHY

Dr. Cathery Yeh is an assistant professor in STEM education and a core faculty member in the Center for Asian American Studies from the University of Texas at Austin. Her research examines the intersections of race, language, and disability to provide a nuanced analysis of the constructions of ability in mathematics classrooms and education systems.

TRANSCRIPT

Mike Wallus: What meaning does the term neurodiverse convey and how might that language impact a student's learning experience? In this episode, we'll explore those questions. And we'll think about ways that educators can design learning environments that support all of their students. Joining us for this conversation is Dr. Cathery Yeh, a professor in STEM education from the University of Texas at Austin. 

Welcome to the podcast, Cathery. It's really exciting to have you with us today.

Cathery Yeh: Thank you, Mike. Honored to be invited.

Mike: So, I wonder if we can start by offering listeners a common understanding of language that we'll use from time to time throughout the episode. How do you think about the meaning of neurodiversity?

Cathery: Thank you for this thoughtful question. Language matters a lot. For me, neurodiversity refers to the natural variation in our human brains and our neurocognition, challenging this idea that there's a normal brain. I always think of… In Texas, we just had a snow day two days ago. And I think of, just as, there's no two snowflakes that are the same, there's no two brains that are exactly the same, too. I also think of its meaning from a personal perspective. I am not a special educator. I was a bilingual teacher and taught in inclusive settings. And my first exposure to the meaning of neurodiversity came from my own child, who—she openly blogs about it—as a Chinese-American girl, it was actually really hard for her to be diagnosed. Asian Americans, 1 out of 10 are diagnosed—that's the lowest of any ethnic racial group. And I'll often think about when… She's proud of her disabled identity. It is who she is. But what she noticed that when she tells people about her disabled identity, what do you think is the first thing people say when she says, “I'm neurodivergent. I have ADHD. I have autism.” What do you think folks usually say to her? The most common response?

Mike: I'm going to guess that they express some level of surprise, and it might be associated with her ethnic background or racial identity.

Cathery: She doesn't get that as much. The first thing people say is, they apologize to her. They say, “I'm sorry.”

Mike: Wow.

Cathery: And that happens quite a lot. And I say that because–and then I connected back to the term neurodiversity—because I think it's important to know its origins. It came about by Judy Singer. She's a sociologist. And about 30 years ago, she coined the term neurodiversity as an opposition to the medical model of understanding people and human difference as deficits. And her understanding is that difference is beautiful. All of us think and learn and process differently, and that's part of human diversity. So that original definition of neurodiversity was tied to the autism rights movement. But now, when we think about the term, it's expanded to include folks with ADHD, dyslexia, dyscalculia, mental health, conditions like depression, anxiety, and other neuro minorities like Tourette syndrome, and even memory loss. I wanted to name out all these things because sometimes we're looking for a really clean definition, and definitions are messy. There's a personal one. There's a societal one of how we position neurodiversity as something that's deficit, that needs to be fixed. But it's part of who one is. But it's also socially constructed. Because how do you decide when a difference becomes a difference that counts where you qualify as being neurodiverse, right? So, I think there's a lot to consider around that.

Mike: You know, the answer that you shared is really a good segue because the question I was going to ask you involves something that I suspect you hear quite often is people asking you, “What are the best ways that I can support my neurodiverse students?” And it occurs to me that part of the challenge of that question is it assumes that there's this narrow range of things that you do for this narrow range of students who are different. The way that you just talked about the meaning of neurodiversity probably means that you have a different kind of answer to that question when people ask it.

Cathery: I do get this question quite a lot. People email it to me, or they'll ask me. That's usually the first thing people ask. I think my response kind of matches my pink hair question. When they ask me the question, I often ask a question back. And I go, “How would you best educate Chinese children in math?” And they're like, “Why would you ask that?” The underlining assumption is that all Chinese children are the same, and they learn the same ways, they have the same needs, and also that their needs are different than the research-based equity math practices we know and have done 50–60 years of research that we've highlighted our effective teaching practices for all children. We've been part of NCTM for 20 years. We know that tasks that promote reasoning and problem solving have been effectively shown to be good for all. Using a connecting math representation—across math representations in a lesson—is good for all. Multimodal math discourse, not just verbal, written, but embodied in part who we are and, in building on student thinking, and all those things we know. And those are often the recommendations we should ask. But I think an important question is how often are our questions connecting to that instead? How often are we seeing that we assume that certain students cannot engage in these practices? And I think that's something we should prioritize more. I'm not saying that there are not specific struggles or difficulties that the neurodiversity umbrella includes, which includes ADHD, dyslexia, autism, bipolar disorder, on and on, so many things. I'm not saying that they don't experience difficulties in our school environment, but it's also understanding that if you know one neurodiverse student—you know me or my child—you only know one. That's all you know. And by assuming we're all the same, it ignores the other social identities and lived experiences that students have that impact their learning. 

So, I'm going to ask you a question.

Mike: Fire away.

Cathery: OK. What comes to your mind when you hear the term “neurodiverse student”? What does that student look like, sound like, appear like to you?

Mike: I think that's a really great question. There's a version of me not long ago that would have thought of that student as someone who's been categorized as special education, receiving special education services, perhaps a student that has ADHD. I might've used language like “students who have sensory needs or processing.” And I think as I hear myself say some of those things that I would've previously said, what jumps out is two things: One is I'm painting with a really broad brush as opposed to looking at the individual student and the things that they need. And two is the extent to which painting with a broad brush or trying to find a bucket of strategies that's for a particular group of students, that that really limits my thinking around what they can do or all the brilliance that they may have inside them.

Cathery: Thank you for sharing that because that's a reflection I often do. I think about when I learned about my child, I learned about myself. How I automatically went to a deficit lens of like, “Oh, no, how are we going to function in the world? How's she going to function in the world?” But I also do this prompt quite a lot with teachers and others, and I ask them to draw it. When you draw someone, what do you see? And I'll be honest, kind of like drawing a scientist, we often draw Albert Einstein. When I ask folks to draw what a neurodiverse student looks like, they're predominantly white boys, to be honest with you. And I want to name that out. It’s because students of color, especially black, brown, native students—they're disproportionately over- and under-identified as disabled in our schooling. Like we think about this idea that when most of us associate autism or ADHD mainly as part of the neurodiversity branch and as entirely within as white boys, which often happens with many of the teachers that I talk to and parents. We see them as needing services, but in contrast, when we think about, particularly our students of color and our boys—these young men—there's often a contrast of criminalization in being deprived of services for them. And this is not even what I'm saying. It's been 50 years of documented research from the Department of Ed from annual civil rights that repeatedly shows for 50 years now extreme disproportionality for disabled black and Latinx boys, in particular from suspension, expulsion, and in-school arrests. I think one of the most surprising statistics for me that I had learned recently was African-American youth are five times more likely to be misdiagnosed with conduct disorder before receiving the proper diagnosis of autism spectrum disorder. And I appreciate going back to that term of neurodiversity because I think it's really important for us to realize that neurodiversity is an asset-based perspective that makes us shift from looking at it as the student that needs to be fixed, that neurodiversity is the norm, but for us to look at the environment.

And I really believe that we cannot have conversations about disability without fully having conversations about race, language, and the need to question what needs to be fixed, particularly not just our teaching, but our assessment practices. For example, we talk about neurodiversities around what we consider normal or abnormal, which is based on how we make expectations around what society thinks. One of the things that showed up in our own household—when we think about neurodiversity or assessments for autism—is this idea of maintaining eye contact. That's one of the widely considered autistic traits. In the Chinese and in the Asian household, and also in African communities, making eye contact to an adult or somebody with authority? It is considered rude. But we consider that as one of the characteristics when we engage in diagnostic tools. This is where I think there needs to be more deep reflection around how one is diagnosed, how a conversation of disability is not separate from our understanding of students and their language practices, their cultural practices. What do we consider normative? Because normative is highly situated in culture and context.

Mike: I would love to stay on this theme because one of the things that stands out in that last portion of our conversation was this notion that rather than thinking about, “We need to change the child.” Part of what we really want to think about is, “What is the work that we might do to change the learning environment?” And I wonder if you could talk a bit about how educators go about that and what, maybe, some of the tools could be in their toolbox if they were trying to think in that way.

Cathery: I love that question of, “What can we as teachers do? What's some actionable things?” I really appreciate Universal Design for Learning framework, particularly their revised updated version, or 3.0 version, that just came out, I think it was June or July of this year. Let me give you a little bit of background about universal design. And I'm sure you probably already know. I've been reading a lot around its origins. It came about [in the] 1980s, we know from cast.org. But I want to go further back, and it really builds from universal design and the work of architecture. So universal design was coined by a disabled architect. His name was Ronald Mace. And as I was reading his words, it really helped me better understand what UDL is. We know that UDL— Universal Design for Learning and universal design—is about access. Everybody should have access to curriculum. And that sounds great, but I've also seen classrooms where access to curriculum meant doing a different worksheet while everybody else is engaging in small group, whole group problem-based learning. 

Access might mean your desk is in the front of the room where you're self-isolated—where you're really close to the front of the board so you can see it really well—but you can't talk to your peers. Or that access might mean you're in a whole different classroom, doing the same set of worksheets or problems, but you're not with your grade-level peers. 

And when Ronald Mace talks about access, he explained that access in architecture had already been a focus in the late 1900s, around 1998, I think. But he said that universal design is really about the longing. And I think that really shifted the framing. And his argument was that we need to design a place, an environment where folks across a range of bodies and minds feel a sense of belonging there. That we don't need to adapt—the space was already designed for you. And that has been such a transformative perspective: That it shouldn't be going a different route or doing something different, because by doing that, you don't feel like you belong. But if the space is one where you can take part equally and access across the ways you may engage, then you feel a sense of belonging.

Mike: The piece of what you said that I'm really contemplating right now is this notion of belonging. What occurs to me is that approaching design principles for a learning environment or a learning experience with belonging in mind is a really profound shift. Like asking the question, “What would it mean to feel a sense of belonging in this classroom or during this activity that's happening?” That really changes the kinds of things that an educator might consider going through a planning process. I'm wondering if you think you might be able to share an example or two of how you've seen educators apply universal design principles in their classrooms in ways that remove barriers in the environment and support students' mathematical learning.

Cathery: Oh gosh, I feel so blessed. I spend… Tomorrow I'm going to be at a school site all day doing this. UDL is about being responsive to our students and knowing that the best teaching requires us to listen deeply to who they are, honor their mathematical brilliance, and their agency. It's about honoring who they are. I think where UDL ups it to another level, is it asks us to consider who makes the decision. If we are making all the decisions of what is best for that student, that's not fully aligned with UDL. The heart of UDL, it's around multiple ways for me to engage, to represent and express, and then students are given choice. So, one of the things that's an important part of UDL is honoring students' agency, so we do something called “access needs.” At the start of a lesson, we might go, “What do you need to be able to fully participate in math today?” And kids from kindergarten to high school or even my college students will just write out what they need. And usually, it's pretty stereotypical: “I want to talk to someone when I'm learning.” “I would like to see it and not just hear it.” And then you continually go back and you ask, “What are your access needs? What do you need to fully participate?” 

So students are reflecting on their own what they need to be fully present and what they believe is helpful to create a successful learning environment. So that's a very strong UDL principle—that instead of us coming up with a set of norms for our students, we co-develop that. But we're co-developing it based on students reflecting on their experience in their environment. In kindergarten, we have children draw pictures. As they get older, they can draw, they can write. But it's this idea that it's an ongoing process for me to name out what I need to be fully present. And oftentimes, they're going to say things that are pretty critical. It's almost always critical, to be honest with you, but that’s a… I would say that's a core component of UDL. We're allowing students to reflect on what they need so they can name it for themselves, and then we can then design that space together. And along the way, we have kids that name, “You know what? I need the manipulatives to be closer.” That would not come about at the start of me asking about access needs. But if we did a lesson, and it was not close by, they’ll tell me. So it's really around designing an environment where they can fully participate and be their full selves and feel a sense of belonging. So, that's one example. 

Another one that we've been doing is teachers and kids who have traditionally not participated the most in our classrooms or have even engaged in pullout intervention. And we'll have them walk around school, telling us about their day. “Will you walk me through your day and tell me how you feel in each of these spaces, and what are your experiences like?” And again, we're allowing the students to name out what they need. And then they're naming out… Oftentimes, with the students that we're at, where I'm working in mostly multilingual spaces, they'll say, “Oh, I love this teacher because she allows us to speak in Spanish in the room. It’s OK.” So that's going back to ideas of action, expression, engagement, where students are allowed a trans language. That's one of the language principles. 

But we're allowing students and providing spaces and really paying close attention to: “How do we decide how to maximize participation for our students with these set of UDL guidelines? How we are able to listen and make certain decisions on how we can strengthen their participation, their sense of belonging in our classrooms.”

Mike: I think what's lovely about both of those examples—asking them to write or draw what they need or the description of, “Let's walk through the day. Let's walk through the different spaces that you learn in or the humans that you learn with”—is one, it really is listening to them and trying to make meaning of that and using that as your starting point. I think the other piece is that it makes me think that it's something that happens over time. It might shift, you might gain more clarity around the things that students need or they might gain more clarity around the things that they need over time. And those might shift a little bit, or it might come into greater focus. Like, “I thought I needed this” or “I think I needed this, but what I really meant was this.” There's this opportunity for kids to refine their needs and for educators to think about that in the designs that they create.

Cathery: I really appreciate you naming that because it's all of that. It's an ongoing process where we're building a relationship with our students for us to co-design what effective teaching looks like—that it's not a one size fits all. It's disrupting this idea that what works for one works for all. It's around supporting our students to name out what they need. Now, I'm almost 50. I struggle to name out what I need sometimes, so it's not going to happen in, like, one time. It's an ongoing process. And what we need is linked to context, so it has to be ongoing. But there's also in the moments as well. And it's the heart of good teaching in math, when you allow students to solve problems in the ways that make sense to them, that's UDL by design. That's honoring the ideas of multiplicity in action, expression. When you might give a context-based problem and you take the numbers away and you give a set of number choices that students get to choose from. That is also this idea of UDL because there's multiple ways for them to engage. So there are also little things that we do that… note how they're just effective teaching. But we're honoring this idea that children should have agency. All children can engage in doing mathematics. And part of learning mathematics is also supporting our students to see the brilliance in themselves and to leverage that in their own teaching and learning.

Mike: Yeah. Something else that really occurred to me as we've been talking is the difference between the way we've been talking about centering students’ needs and asking them to help us understand them and the process that that kind of kicks off. I think what strikes me is that it's actually opening up the possibilities of what might happen or the ways that a student could be successful as opposed to this notion that “You're neurodiverse, you fit in this bucket. There's a set of strategies that I’m going to do just for you,” and those strategies might actually limit or constrict the options you have. For example, in terms of mathematics, what I remember happening very often when I was teaching is, I would create an open space for students to think about ways that they could solve problems. And at the time, often what would happen is kids who were characterized as neurodiverse wouldn't get access to those same strategies. It would be kind of the idea that “This is the way we should show them how to do it.” It just strikes me how different that experience is. I suspect that that was done with the best of intentions, but I think the impact unfortunately probably really didn't match the intent.

Cathery: I love how you're being honest. I did the same thing when I was teaching, too, because we were often instructed to engage in whole-group instruction and probably do a small-group pullout. That was how I was taught. And when the same kids are repeatedly pulled out because we're saying that they're not able to engage in the instruction. I think that part of UDL is UDL is a process, realizing that if students are not engaging fully in the ways that we had hoped, instead of trying to fix the child, we look at the environment and think about what changes we need to make in tier one. So whole-group instruction, whole-group participation first to see how we can maximize their participation. And it's not one strategy, because it depends; it really depends. I think of, for example, with a group of teachers in California and Texas now, we've been looking at how we can track participation in whole-group settings. And we look at them across social demographics, and then we started to notice that when we promote multimodal whole-group participation, like kids have access to manipulatives even during whole-group share out. Or they have visuals that they can point to, their participation and who gets to participate drastically increase.

So there's many ways in which, by nature, we engage in some narrow practices because, too, oftentimes whole group discussion is almost completely verbal and, at times, written, and usually the teacher's writing. So it's going back to the idea of, “Can we look at what we want our students to do at that moment? So starting on the math concept and practices, but then looking at our students and when they're not participating fully, it's not them. What are the UDL principles and things that I know and strategies that I have with my colleagues that I can make some small shifts?”

Mike: You know, one of the things that I enjoy most about the podcast is that we really can take a deep dive into some big ideas, and the limitation is we have 20 minutes to perhaps a half hour. And I suspect there are a lot of people who are trying to make meaning of what we're talking about and thinking about, “How might I follow up? How might I take action on some of the ideas?” So I want to turn just for a little while to resources, and I'm wondering if there are resources that you would suggest for a listener who wants to continue learning about universal design in a mathematics classroom?

Cathery: Oh, my goodness, that's such a hard question because there’s so many. Some good ones overall: I would definitely encourage folks to dive into the UDL guidelines—the 3.0 updates. They're amazing. They're so joyful and transformative that they even have, one of the principles is centering joy in play, and for us to imagine that, right?

Mike: Yes!

Cathery: What does that mean to do that in a math classroom? We can name out 50 different ways. So how often do we get to see that? So, I would highly encourage folks to download that, engage in deep discussion because it was a 2.2 version for, I think, quite a few years. I would also lean into a resource that I'm glad to email later on so it's more easily accessible. I talked about access needs, this idea of asking students, asking community members, asking folks to give this opportunity to name out what they need. It's written by a colleague, Dr. Daniel Reinholz and Dr. Samantha Ridgway. It's a lovely reading, and it focuses specifically in STEM but I think it's a great place to read. I would say that Dr. Rachel Lambert's new book on UDL math is an excellent read. It's a great joyful read to think about. I'm going to give one shout out to the book called the Year of the Tiger: An Activist’s Life. It's by Alice Wong. I encourage that because how often do we put the word activism next to disability? And Alice Wong is one of the most amazing humans in the world, and it's a graphic novel. So it's just joyful. It's words with poetry and graphic novel mixed together to see the life of what it means to be a disabled activist and how activism and disability goes hand in hand. Because when you are disabled and multi-marginalized, you are often advocating for yourself and others. It's amazing. So I'll stop there. There's endless amounts.

Mike: So for listeners, we'll link the resources that Cathery was talking about in our show notes. I could keep going, but I think this is probably a great place to stop. I want to thank you so much for joining us. It's really been a pleasure talking with you.

Cathery: Thank you. Thank you.

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

© 2025 The Math Learning Center | www.mathlearningcenter.org

Assessment in the Early Years Guest: Shelly Scheafer

ROUNDING UP: SEASON 3 | EPISODE 13

Mike (00:09.127) Welcome to the podcast Shelley. Thank you so much for joining us today.

Shelly (00:12.956) Thank you, Mike, for having me.

Mike (00:16.078) So I'd like to start with this question. What makes the work of assessing younger children, particularly students in grades K through two, different from assessing students in upper elementary grades or even beyond?

Shelly (00:30.3) There's a lot to that question, Mike. I think there's some obvious things. So effective assessment of our youngest learners is different because obviously our pre-K, first, even our second grade students are developmentally different from fourth and fifth graders. So when we think about assessing these early primary students,

we need to use appropriate assessment methods that match their stage of development. For example, when we think of typical paper pencil assessments and how we often ask students to show their thinking with pictures, numbers and words, our youngest learners are just starting to connect symbolic representations to mathematical ideas, let alone, you know, put letters together to make words. So

When we think of these assessments, we need to take into consideration that primary students are in the early stages of development with respect to their language, their reading, and their writing skills. And this in itself makes it challenging for them to fully articulate, write, sketch any of their mathematical thinking. So we often find that with young children in reviews,

you know, individual interviews can be really helpful. But even then, there's some drawbacks. Some children find it challenging, you know, to be put on the spot, to show in the moment, you know, on demand, you know, what they know. Others, you know, just aren't fully engaged or interested because you've called them over from something that they're busy doing. Or maybe, you know,

they're not yet comfortable with the setting or even the person doing the interview. So when we work with young children, we need to recognize all of these little peculiarities that come with working with that age. We also need to understand that their mathematical development is fluid, it's continually evolving. And this is why

Shelly (02:47.42) they often or some may respond differently to the same proper question, especially if the setting or the context is changed. We may find that a kindergarten student who counts to 29 on Monday may count to 69 or even 100 later in the week, kind of depending on what's going on in their mind at the time. So this means that assessment with young children needs to be frequent.

informative and ongoing. So we're not necessarily waiting for the end of the unit to see, aha, did they get this? You know, what do we do? You know, we're looking at their work all of the time. And fortunately, some of the best assessments on young children are the observations in their natural setting, like times when maybe they're playing a math game or working with a center activity or even during just your classroom routines.

And it's these authentic situations that we can look at as assessments to help us capture a more accurate picture of their abilities because we not only get to hear what they say or see what they write on paper, we get to watch them in action. We get to see what they do when they're engaged in small group activities or playing games with friends.

Mike (04:11.832) So I wanna go back to something you said and even in particular the way that you said it. You were talking about watching or noticing what students can do and you really emphasize the words do. Talk a little bit about what you were trying to convey with that, Shelley.

Shelly (04:27.548) So young children are doers. When they work on a math task, they show their thinking and their actions with finger formations and objects. And we can see if a student has one-to-one correspondence when they're counting, if they group their objects, how they line them up, do they tag them, do they move them as they count them. They may not always have the verbal skills to articulate their thinking, but we can also attend to things like head nodding,

finger counting, and even how they cluster or match objects. So I'm going to give you an example. So let's say that I'm watching some early first graders, and they're solving the expression 6 plus 7. And the first student picks up a number rack or a rec and rec. And if you're not familiar with a number rack, it's a tool with two rows of beads.

And on the first row, there are five red beads and five white beads. And on the second row, there's five red beads and five white beads. And the student solving six plus seven begins by pushing over five red beads in one push and then one more bead on the top row. And then they do the same thing for the seven. They push over five red beads and two white beads. And they haven't said a word to me. I'm just watching their actions.

And I'm already able to tell, hmm, that student could subitize a group of five, because I saw him push over all five beads in one push. And that they know that six is composed of five and one, and seven is composed of five and two. And they haven't said a word. I'm just watching what they're doing. And then I might watch the student, and they see it.

I see him pause, know, nothing's being said, but I start to notice this slight little head nodding.

Shelly (06:26.748) And then they say 13 and they give me the answer and they're really pleased. I didn't get a lot of language from them, but boy, did I get a lot from watching how they solve that problem. And I want to contrast that observation with a student who might be solving the same expression six plus seven and they might go six and then they start popping up one finger at a time while counting seven, eight.

9, 10, 11, 12, 13. And when they get seven fingers held up, they say 13 again. They've approached that problem quite differently. But again, I get that information that they understood the equation. They were able to count on starting with six. And they kept track of their count with their fingers. And they knew to stop when seven fingers were raised.

And I might even have a different student that solves the problem by thinking, hmm, and they talk to themselves or they know I'm watching and they might start talking to me. And they say, well, 6 plus 6 is 12 and 7 is 1 more than 6. So the answer is 16 or 13.

And if this were being done on a paper pencil as an assessment item or they were answering on some kind of a device, all I would know about my students is that they were able to get the correct answer. I wouldn't really know a lot about how they got the answer. What skills do they have? What was their thinking? And there's not a lot that I can work with to plan my instruction. Does that kind of make sense?

Mike (08:20.84) Absolutely. I think the, the way that you described this really attending to behaviors, to gestures, to the way that kids are interacting with manipulatives, the self-talk that's happening. It makes a ton of sense. And I think for me, when I think back to my own practice, I wish I could wind the clock back because I think I was attending a lot to what kids were saying.

and sometimes they're written communication, and there was a lot that I could have also taken in if I was attending to those things in a little bit more depth. It also strikes me that this might feel a little bit overwhelming for an educator. How do you think about what an educator, let me back that up. How can an educator know what they're looking for?

Shelly (09:17.5) to start, Mike, by honoring your feelings, because I do think it can feel overwhelming at first. But as teachers begin to make informal observations, really listening to you and watching students' actions as part of just their daily practice, something that they're doing, you know, just on a normal basis, they start to develop these kind of intuitive understandings of how children learn, what to expect

them to do, what they might say next if they see a certain actions. And after several years, let's say teaching kindergarten, if you've been a kindergarten teacher for four, five, six, 20, you know, plus years, you start to notice these patterns of behavior, things that five and six year olds seem to say and think and do on a fairly consistent basis. And that kind of helps you know, you know, what you're looking at. But before you say anything,

I know that isn't especially helpful for teachers new to the profession or new to a grade level. And fortunately, we have several researchers that have been, let's say, kid watching for 40, I don't know, 50 years, and they have identified stages through which most children pass as they develop their counting skills or maybe strategies for solving addition and subtraction problems. And these stages are

laid out as progressions of thinking or actions that students exhibit as they develop understanding over periods of time. listeners might, you know, know these as learning progressions or learning trajectories. And these are ways to convey an idea of concept in little bits of understanding. So.

When I was sharing the thinking and actions of three students solving six plus seven, listeners familiar with cognitively guided instruction, CGI, they might have recognized the sequence of strategies that children go through when they're solving addition and subtraction problems. So in my first student, they didn't say anything but gave me an answer.

Shelly (11:40.068) was using direct modeling. We saw them push over five and one beads for six and then five and two beads for seven and then kind of pause at their model. And I could tell, you know, with their head nodding that they were counting quietly in their head, counting all the beads to get the answer. And, you know, that's kind of one of those first stages that we see and recognize with direct modeling. And that gives me information on what I might do with a student.

coming next time, I might work on the second strategy that I conveyed with my second student where they were able to count on. They started with that six and then they counted seven more using their fingers to keep track of their count and got the answer. And then that third kind of level in that progression as we're moving of understanding.

was shown with my third student when they were able to use a derived fact strategy. The student said, well, I know that 6 plus 6 is 12. I knew my double fact. And then I used that relationship of knowing that 7 is 1 more than 6. And so that's kind of how we move kids through. And so when I'm watching them, I can kind of pinpoint where they are and where they might go next. And I can also think about what I might do.

And so it's this knowledge of development and progressions and how children learn number concepts that can help teachers recognize the skills as they emerge, as they begin to see them with their students. And they can use those, you know, to guide their instruction for that student or, you know, look at the class overall and plan their instruction or think about more open-ended kinds of questions that they can ask that recognize these different levels that students are working with.

Mike (13:39.17) You know, as a K-1 teacher, I remember that I spent a lot of my time tracking students with things like checklists. You know, so I'd note if students quote unquote had or didn't have a skill. And I think as I hear you talk, that feels fairly oversimplified when we think about this idea of developmental progressions. How do you suggest that teachers approach capturing evidence of student learning,

Shelly (14:09.604) well, I think it's important to know that if, you know, it takes us belief. We have to really think about assessment and children's learning is something that is ongoing and evolving. And if we do, it just kind of becomes part of what we can do every day. We can look for opportunities to observe students skills in authentic settings. Many in the moment.

types of assessment opportunities happen when we pose a question to the class and then we kind of scan looking for a response. Maybe it's something that we're having them write down on their whiteboard or maybe it's something where they're showing the answer with finger formations or we're giving a thumbs up or a thumbs down, know, kind of to check in on their understanding. We might not be checking on every student, but we're capturing the one, you know, a few.

And we can take note because we're doing this on a daily basis of who we want to check in with. What do we want to see? We can also do a little more formal planning when we draw from what we're going to do already in our lesson. Let's say, for example, that our lesson today includes a dot talk or a number talk, something that we're going to write down. We're going to record student thinking.

And so during the lesson, the teacher is going to be busy facilitating the discussion, recording the students thinking, you know, and making all of those notes. But if we write the child's name, kind of honor their thinking and give it that caption on that public record, at the end of the lesson, you know, we can capture a picture, just, you use our phone, use an iPad, quickly take a picture of that student's thinking, and then we can record that.

you know, where we're keeping track of our students. So we have, OK, another moment in time. And it's this collection of evidence that we keep kind of growing. We can also, you by capturing these public records, note whose voice and thinking were elevating in the classroom. So it kind of gives us how are they thinking and who are we listening to and making sure that we're kind of spreading that out and hearing everyone.

Shelly (16:31.728) I think, Mikey, you checklists that you used. Yeah, and even checklists can play a role in observation and assessments when they have a focus and a way to capture students' thinking. So one of the things we did in Bridges 3rd edition is we designed additional tools for gathering and recording information during workplaces.

Mike (16:35.501) I did.

Shelly (16:56.208) That's a routine where students are playing games and or engaged with partners doing some sort of a math activity. And we designed these based on what we might see students do at these different games and activities. And we didn't necessarily think about this is something you're going to do with every student. You know, or even, you know,

in one day because these are spanned out over a period of four to six weeks where they can go to these games. And we might even see the students go to these activities multiple times. And so let's say that kindergarten students are playing something like the game Beat You to 10, where they're spinning a spinner, they're counting cubes, and they're trying to race their partner to collect 10 cubes. And with an activity like that,

I might just want to focus on students who I still want to see, do they have one-to-one correspondence? Are they developing cardinality? Are they able to count out a set? And so those might, you know, of objects, you know, based on the number, they spin a four, can they count out four? And those might be kinds of skills that you might have had typically on a checklist, right, Mike, for kindergarten? But I could use this activity to kind of gap.

gather that note and make any comments. So just for those kids I'm looking at or maybe first graders are playing a game like sort the sum where they're drawing two different dominoes and they're supposed to find how many they have in all. And so with a game like that, I might focus on what are their strategies? Are they counting all the dots? Are they counting on from the dot? And if one set of the dots on one side and then counting on the other.

Are they starting with the greater number or the most dots? Are they starting with the one always on the left? Or I might even see they might instantly recognize some of those. So I might know the skills that I want to look for with those games and be making notes, which kind of feels checklist-like. But I can target that time to do it on students I want that information by thinking ahead of time.

Shelly (19:18.684) What can I get by watching, observing these students at these games? trust, I mean, as you know, young children love it. Older children love it. When the teacher goes over and wants to watch them play, or even better, wants to engage in the game play with them, but I can still use that as an assessment.

Mike (19:39.32) think that's really helpful, Shelly, for a couple reasons. First, I think it helps me rethink, like you said, one, getting really a lot clearer on like, love the, I'm gonna back that up. I think one of the things that you said was really powerful is thinking about not just the assessment tools that might be within your curriculum, but looking at the task itself that you're gonna have students engage with, be it a game or a,

Shelly (19:39.356) and

Mike (20:07.96) project or some kind of activity and really thinking like, what can I get from this as a person who's trying to make sense of students thinking? And I think my checklist suddenly feels really different when I've got a clear vision of like, what can I get from this task or this game that students are playing and looking for evidence of that versus feeling like I was pulling kids over one-on-one, which I think I would still do because there's some depth that I might want to capture.

But it it changes the way that I think about what I might do and also what I might get out of a task So that that really resonates for me

Shelly (20:47.066) Yeah, and I think absolutely, you know, I didn't want to make individual interviews or anything sound bad because we can't do them. just, you there's the downfall of, you know, kids comfort level with that and ask them to do something on demand. But we do want more depth and it's that depth that, you know, we know who we want more depth on because of these informal types of observations that we're gathering on a daily basis in our class. You know, might, says something and

we take note I want to touch bases with that thinking or I think I'm going to go observe that child during that workplace or maybe we're seeing some things happening during a game and instead of you know like stopping the game and really doing some in-depth interview with the student at that moment because you need more information I can might I might want to call them over and do that more privately at a different time so you're absolutely on there's a place there's a place for you know both

Mike (21:42.466) The other thing that you made me think about is the extent to which, like one of the things that I remember thinking is like, I need to make sure if a student has got it or not got it. And I think what you're making me think can really come out of this experience of observing students in the wild, so to speak, when they're working on a task or with a partner is that I can gather a lot more evidence about the application of that idea. I can see the extent to which students are.

doing something like counting on in the context of a game or a task. And maybe that adds to the evidence that I gather in a one-on-one interview with them. But it gives me a chance to kind of see, is this way of thinking something that students are applying in different contexts, or did it just happen at that one particular moment in time when I was with them? So that really helps me think about, I think, how those two...

maybe different ways of assessing students, be it one-on-one or observing them and seeing what's happening, kind of support one another.

Shelly (22:46.268) think you also made me think, you know, really hit on this idea that students, like I said, you their learning is evolving over time. And it might change with the context so that they, you know, they show us that they know something in one context with these numbers or this, you know, scenario. But they don't necessarily always see that it applies across the board. I mean, they don't, you know, make these.

generalizations. That's something that we really have to work with students to develop. they're also, they're young children. Think about how quickly a three-year-old and a four-year-old change, you know, the same five to six, six to seven. I mean, they're evolving all the time. And so we want to get this information for them on a regular basis. You know, a unit of instruction may be a month or more long.

And a lot can happen in that time. So we want to make sure that we continue to check in with them and help them to develop if needed or that we advance them. know, we nudge them along. We challenge them with maybe a question. Will that apply to every number? So a student discovers, when we add one to every number, it's like saying the next number. So six and one more is seven and eight and one more is nine.

And you can challenge them, ooh, does that always work? What if the number was 22? What if it was 132? Would it always work? you know, when you're checking in with kids, you have those opportunities to keep them thinking, to help them grow.

Mike (24:23.426) I want to pick up on something that we haven't necessarily said aloud, but I'd like to explore it. You know, looking at young students work from an asset-based perspective, particularly with younger students, I think I often had points in time where there felt like so much that I needed to teach them. And sometimes I felt myself focusing on what they couldn't do. Looking back, I wish I had thought about my work as noticing the assets, the strategies, the ways of thinking.

that they were accumulating.

Are there practices you think support an asset-based approach to assessment with young learners?

Shelly (25:06.278) think probably the biggest thing we can do is broaden our thinking about assessment. The National Council of Teachers of Mathematics wrote in Catalyzing Change in Early Childhood and Elementary Mathematics that the primary purpose of assessment is to gather evidence of children's thinking, understanding, and reasoning to inform both instructional decisions and student and teaching learning.

If we consider assessments and observations as tools to inform our instruction, we need to pay attention to the details of the child's thinking. And when we're paying attention to the details, what the child is bringing to the table, what they can do, that's where our focus goes. So the question becomes, what is the student understanding?

What assets do they bring to the task? It's no longer, can they do it or can they not do it? And when we know, when we're focusing on just what that student can do, and we have some understanding of learning progressions, how students learn, then we can place what they're doing kind of on that trajectory, in that progression, and that becomes knowledge.

And with that knowledge, then we can help students move along the progression to develop more developed understanding. For example, again, if I go back to my six plus seven and we notice that a student is direct modeling, they're counting out each of the sets and counting all, we can start to nudge them toward counting on.

We might cover, you know, they were using that number rec, we might cover the first row and say, you just really showed me a good physical representation of six plus seven. And I kind of noticed that you were counting the beads to see how many were there. I'm wondering if I cover this first row. How many beads am I covering? Hmm. I wonder, could you start your counting at six? You know, we can kind of work with what they know. And I can do that because of

Shelly (27:31.928) I haven't, I've focused on where they are in that progression and where that development is going. And I kind of have a goal of where I want students to go, you know, to further their thinking. Not that being in one place is right or wrong, or yes they can do it, no they can't. It's my understanding of what assets they bring that I can build on. Is that kind of what you're after?

Mike (27:58.51) It is, and I think you also addressed something that again has gone unsaid, but I think you, you, you unpacked it there, which is assessment is really designed to inform my instruction. And I think the example you offered us a really lovely one where, we have a student who's direct modeling and they're making sense of number in a certain way and their strategy reflects that. And that helps us think about the kinds of nudges we can offer.

that might shift that thinking or press them to make sense of numbers in a different way. That really the assessment is, it is a moment in time, but it also informs the way that you think about what you're gonna do next to keep nudging that student's thinking.

Shelly (28:44.348) Exactly, and we have to know that if we have 20 students, they all might be, you know, have 20 little plans that they're on, 20 little pathways of their learning. And so we need to think about everybody, you know. So we're going to ask questions that help them do them, and we're going to honor their thinking. And then we can, you know, like so again, I'm going go back to like doing that dot talk with those students.

And so I'm honoring all these different ways that students are finding the total number of dots. And then I'm asking them to look for what's the same within their thinking so that other students also can serve to nudge kids, to have them let them try and explore a different idea or, ooh, can we try that Mike's way and see if we can do that? hmm, what do you notice about?

how Mike solved the problem and how Shelly solved the problem. Where is their thinking the same? Where is it different? And so we're honoring everybody's place of where they're at, but they're still learning from each other.

Mike (29:51.224) You know, you have made multiple mentions to this idea of progressions or trajectories, and I'm wondering if there are resources that have informed your thinking about assessment at the early ages. Is there anything you would invite listeners to engage with if they wanted to continue learning, Shelley?

Shelly (30:13.008) I think Mike, had that question earlier, so just pause this for a second. Okay. I know you will. I just know it's right here.

Mike (30:16.558) That's okay, no worries. We'll cut every single bit of this out and it will sound supernatural. Yeah, yeah.

Mike (30:39.854) Brent's over here multitasking.

Shelly (30:41.85) OK. OK, I'm just making sure that I'm not going to blow it. I think you're spot on. I think I thought we skipped something. No, it's up here.

Mike (30:53.132) Okay, just pick up whenever you're ready.

Shelly (30:55.108) Yeah, I just have too many notes here.

Shelly (31:10.084) OK, I've got it. Do you want to ask the question again?

Mike (31:12.258) Go for it. Absolutely. Yep. Are there resources you'd invite listeners to engage with if they wanted to keep learning, Shelly?

Shelly (31:28.368) You phrased that a little bit different. What I answered was, what are some of the resources that helped you build an understanding of children's developmental progressions? Do you have that question? Or I can jump on from what you asked, too.

Mike (31:35.5) Okay. Yeah, let let. No, no, no, let me let me ask the question that way.

Shelly (31:42.202) Okay.

Mike (31:45.774) Okay, how did we have it in the thing? Can you say it one more time and I'll say it back in the question?

Shelly (31:50.768) What are some of the resources that help to build an understanding of children's developmental progressions?

Mike (31:56.504) Perfect. What are some of the resources that helped you build an understanding of children's developmental progression, Shelley?

Shelly (32:05.34) Honestly, I can say that I learned a lot from the students I taught in my classroom. My roots run deep in early childhood. And I can also proudly say that I have a career-long relationship with the Math Learning Center and Bridges Curriculum, which has always been developmentally appropriate curriculum for young learners. And with that said, I think I stand on the back of giants.

practitioner researchers for early childhood who have spent decades observing children and recording their thinking. I briefly mentioned Cognitively Guided Instruction, which features the research of Thomas Carpenter and his team. And their book, Children's Mathematics, is a great guide for K-5 teachers. I love it because

I mean, the recent edition has QR codes where you can watch teachers and students in action. You can see some interviews. You can see some classroom lessons. And they also wrote young children's mathematics on cognitive-guided instruction in early childhood education. So I mean, they're just a great resource. Another teacher researcher.

is Kathy Richardson, and some listeners may know her from her books, the developing number concept series or number talks in the primary classroom. And she also wrote a book called How Children Learn Number Concepts, Guide to the Critical Learning Phases, which targets pre-kindergarten through grade four. And I love that Kathy writes.

in her acknowledgments that this work is the culmination of more than 40 years working with children and teachers observing, wondering, discussing, reading, and thinking.

Shelly (34:11.692) It is. So spot on to the observations and the things that I noticed in my own teaching, but it's also still one of the most referenced resources that I use. And if podcasts had a video, I would be able to hold up and show you my dog eared book with sticky notes coming out the all the sides because it is just something that.

just resonates with me again. And then I think also maybe less familiar.

Mike (36:38.958) I think you mentioned giants and those are some gigantic folks in the world of mathematics education. The other piece that I think really resonates for me is I had a really similar experience with both CGI and Kathy Richardson in that a lot of what they're describing are the things that I was seeing in classrooms. What it really helped me do is understand

how to place that behavior and what the meaning of it was in terms of students understanding of mathematics. And it also helped me think about that as an asset that then I could build on. Shelley, I think this is probably a great place to stop, but I wanna thank you so much for joining us. It has really been a pleasure talking with you.

Mike (37:28.95) Say thank you again, but definitively.

Mike (37:35.48) Brent, how do you feel about that?

Mike (37:41.966) do you want to jump in? Yeah, feel free.

Mike (38:04.194) is

Is there a question I could ask that would set you up?

Mike (38:14.574) can work that in to a new ending.

Mike (38:42.872) Do you, there something that you want to add though, Shelley? Cause we can, we can edit it, edit content in, and we can sequence content in too. So if there's something that mattered to you, we can absolutely add it.

Mike (39:14.542) Let's do a question like that then.

Mike (39:35.02) What if we see...

Mike (39:40.578) Why don't you, why don't,

Yeah, why don't you say it? Go ahead and say it the way that you you it was going to flow out and then we'll we can edit this in definitely.

Mike (39:59.191) Okay.

Okay, go for it. Yeah, yeah.

Mike (40:06.676) can I tell you this is one of the smoothest podcast recordings we have had? There's nothing to be sorry about.

Mike (40:18.562) There, okay, I was, can you ask the question again, Mike? So that it's clean.

Mike (40:29.752) Are there resources you would invite our listeners to engage with if they want to continue learning?

Mike (42:26.328) I think that's a great place to stop. Shelly Schaefer, thank you so much for joining us.

Mike (42:38.602) That was perfect. Yeah, fantastic. I'm gonna cut roll. No, there's nothing to be sorry about

© 2025 The Math Learning Center | www.mathlearningcenter.org

Dr. Victoria Jacobs, Examining the Meaning and Purpose of our Questions ROUNDING UP: SEASON 3 | EPISODE 12

Mike (00:03): The questions educators ask their students matter. They can have a profound impact on students' thinking and the shape of their mathematical identities. Today we're examining different types of questions, their purpose and the meaning students make of them. Joining us for this conversation is Dr. Vicki Jacobs from the University of North Carolina Greensboro. Welcome to the podcast, Vicki. I'm really excited to talk with you today.

Vicki (00:33): Thanks so much for having me. I'm excited to be here.

Mike (00:36): So you've been examining the ways that educators use questioning to explore the details of students' thinking. And I wonder if we could start by having you share what drew you to the topic.

Vicki (00:47): For me, it all starts with children's thinking because it's absolutely fascinating, but it's also mathematically rich. And so a core part of good math instruction is when teachers elicit children's ideas and then build instruction based on that. And so questioning obviously plays a big role in that, but it's hard. It's hard to do that well in the moment. So I found questioning to explore children's thinking to be a worthwhile thing to spend time thinking about and working on.

Mike (01:17): Well, let's dig into the ideas that have emerged from that work. How can teachers think about the types of questions that they might ask their students?

Vicki (01:24): Happy to share. But before I talk about what I've learned about questioning, I really need to acknowledge some of the many people that have helped me learn about questioning over the years. And I want to give a particular shout out to the teachers and researchers in the wonderful cognitively guided instruction or CGI community as well as my long-term research collaborators at San Diego State University. And more recently, Susan Sen. This work isn't done alone, but what have we learned about teacher questioning across a variety of projects? I'll share two big ideas and the first relates to the goals of questioning and the second addresses more directly the types of questions teachers might ask. So let's start with the goals of questioning because there are lots of reasons teachers might ask questions in math classrooms. And one common way to think about the goal of questioning is that we need to direct children to particular strategies during problem solving.

(02:23): So if children are stuck or they're headed down a wrong path, we can use questions to redirect them so that they can get to correct answers with particular strategies. Sometimes that may be okay, but when we only do that, we're missing a big opportunity to tap into children's sense-making. Another way to think about the goal of questioning is that we're trying to explore children's thinking during problem solving. So think about a math task where multiple strategies are encouraged and children can approach problem solving in any way that makes sense to. So we can then ask questions that are designed to reveal how children are thinking about the problem solving, not just how well they're executing our strategies. And we can ask these questions when children are stuck, but also when they solve problems correctly. So this shift in the purpose of questioning is huge. And I want to share a quote from a teacher that I think captures the enormity of this shift.

(03:26): She's a fifth grade teacher, and what she said was the biggest thing I learned from the professional development was not asking questions to get them to the answers so that I could move them up a strategy, but to understand their thinking. That literally changed my world. It changed everything. So I love this quote because it shows how transformative this shift can be because when teachers become curious about how children are thinking about problem solving, they give children more space to problem solve in multiple ways, and then they can question to understand and support children's ideas. And these types of questions are great because they increase learning opportunities for both children and teachers. So children get more opportunities to learn how to talk math in a way that's meaningful to them because they're talking about their own ideas and they also get to clarify what they did think more about important math that's embedded in their strategies and sometimes to even self-correct. And then as teachers, these types of questions give us a window into children's understandings, and that helps us determine our next steps. Questioning can have a different and powerful purpose when we shift from directing children toward particular strategies to exploring their mathematical thinking.

Mike (04:54): I keep going back to the quote that you shared, and I think the details of the why and kind of the difference in the experience for students really jump out. But I'm really compelled by what that teacher said to you about how it changes everything. And I wonder if we could just linger there for a moment and you could talk about some of the things that you've seen happen for educators who have that kind of aha moment in the same way that that teacher did, how that impacts the work that they're doing with children or how they see themselves as an educator.

Vicki (05:28): That's a great question. I think it's freeing in some way because it changes how educators think about what their next steps are. Every teacher has lots of pressures from standards and sometimes pacing guides and grade level teams that are working on the same page, all sorts of things that are a big part of teaching. But it puts the focus back on children and children's thinking and that my next steps should then come from there. And so in some ways, I think it gives a clearer direction for how to navigate all those various pressures that teachers have.

Mike (06:14): I love that. Let's talk about part two.

Vicki (06:17): Sure. So if we have the goal of questioning to explore children's thinking, how do we decide what questions to ask? So first of all, there's never a best question. There are many questioning frameworks out there that can provide lots of ideas, but what we've found is that the most productive questions always start with what children say and do. So that means I can't plan all my questions in advance, and instead I have to pay close attention to what children are saying and doing during problem solving. And to help us with that, we found a distinction between inside questions and outside questions. And that distinction has been really useful to us and also usable even during instruction. So inside questions are questions that explore details that are part of inside children's current strategies. And outside questions are questions that focus on strategies or representations that are not what children have done and may even be linked to how we as teachers are thinking about problem solving.

(07:26): So I promised an example, and this is from our recent research project on teaching and learning about fractions. And we asked teachers to think about a child's written strategy for a fraction story problem. And the problem was that there are six children equally sharing four pancakes, and they need to figure out how much pancake each child can get. So we're going to talk about Joy's strategy for solving this problem. She is a fourth grader who solved the problem successfully, but in a complex and rather unconventional way. So I'm going to describe her strategy as a reminder. We have six children sharing four pancakes. So she drew the four pancakes. She split the first three pancakes into fourths and distributed the pieces to the six children, and that works out to two fourths for each child. But now she has a problem because she has one pancake left and fourths aren't going to work anymore because that's not enough pieces for her six children.

(08:23): So she split the pancake first into eighths and then into 20 fourths and distributed those pieces. So each child ends up receiving two fourths, one eighth and one 24th. And when you put all those amounts together, they equal the correct amount of two thirds pancake per child. But Joy left her answer in pieces as two fourths, one eighth and one 24th, and she wrote those fractions in words rather than using symbols. Okay, so there's a lot going on in this strategy. And the specific strategy doesn't matter so much for our conversation, but the situation does. Here we have a child who has successfully solved the problem, but how she solved it and how she represented her answer are different than what we as adults typically do. So we ask teachers to think about what kind of follow-up conversation would you want to have with joy?

(09:23): What types of questions would you want to ask her? And there were these two main questioning approaches, what we call inside questioning and outside questioning. So let's start with outside questioning. These teachers focused on improving Joy's strategy. So they ask follow-up questions like, is there another way you can share the four pancakes with six children? Or is your strategy the most efficient way you could share the pancakes? Or is there a way to cut bigger servings that would be more efficient? So given the complexity of Joy's strategy, we can appreciate these teachers' goals of helping joy move to a more efficient strategy. But all of these questions are pushing her to use a different strategy. So we considered them outside questions because they were outside of her current strategy. And outside questions can sometimes be productive, but they tend to get overused. And when we use them a lot, they can communicate to kids that what they're actually doing was wrong and that it needs fixing.

(10:29): So let's think about the other approach of inside questioning. These teachers started by exploring what Joy had done in all of its complexity. And they ask a variety of questions. Usually it started with a general question, can you tell me what you did? But then they zoomed in on some of the many details. So for examples, they've asked how she split the pancakes. They offered questions like, why did you split the first three pancakes into four pieces? Or Tell me about the last pancake. That was the one that she split into eights and 20 fourths. Or they might ask about how she knew how to name each of the fractional amounts, especially the one 24th, because that's something that many children might've struggled with. And then there were questions about a variety of other details. Some of them are hard to explain without showing you a picture of the strategy, but the point is that the teachers took seriously what Joy had done and elevated it to the focus of the conversation. So Joy had a chance to share her reasoning and reflect on it, and the teachers could better understand Joy's approach to problem solving. So we found this distinction between inside and outside questioning to be useful to teachers and even in the midst of instruction because teachers can quickly check in with themselves. Am I asking an inside question or an outside question?

Mike (11:49): Well, I have so many questions about inside and outside questions, but I want to linger on inside questions. What I found myself thinking is that for the learner, there are benefits for building number sense or conceptual understanding. The other thing that strikes me is that inside questions are also an opportunity to support students' math identity. And I wonder if that's something that you've seen in your work with teachers and with students.

Vicki (12:14): Absolutely. I love your question. One of my favorite things about inside questions is that children see that their ideas are being taken seriously. And that's so empowering. It helps children believe that they can do math and that they are in charge of their mathematical thinking. I'll share a short story that was memorable for me, and this was from a while ago when I was in graduate school. So I was working on a research project and we were conducting problem solving interviews with young children. And our job was to document their strategies. So if we could see exactly what they did, we were told to write down the strategy and move on. But if we needed to clarify something, we could ask follow up questions. I was working with a first grader who had just spent a really long time solving a story problem. He had solved it successfully, and he had done that by joining many, many, many unifix cubes into a very long train.

(13:10): And then he had counted them by ones multiple times. So he had been successful. I could tell exactly what he had done. So I started to move on to the next problem. So this young child looked at me a little incredulous and simply asked, don't you want to know how I did it? And he had come from a class where his math thinking was valued, and talking about children's thinking was a regular part of what they did. So he couldn't quite understand why this adult was not interested in how he had thought about the problem. Well, I was a little embarrassed and of course backtracked and listened to his full explanation. But the interaction stuck with me because it showed me how empowering it was for children to truly be listened to as math thinkers. And I think that's something we want for all children.

Mike (14:00): The other thing that's hitting me in that story and in the story of joy is mea culpa. I am a person who has lived in the cult of efficiency where I looked at a student's work and my initial thought was, how do I nip the edges of this to get to more efficiency? But I really am struck by it how different the idea of asking the student to explain their thinking or the why behind it. I find myself thinking about joy, and it appears that she was intent on making sure that there were equal shares for each person. So there's ways that she could build to a different level of efficiency. But I think recognizing that there's something here that is really important to note about how and why she chose that, that would feel really meaningful as a learner.

Vicki (14:44): I agree. I think what I like about inside questions is that they encourage us to, that children's thinking makes sense, even if it's different than how we think about it. It's our job to figure out how it makes sense. And then to build from there.

Mike (15:03): Can you just say more about that? That feels like kind of a revelation.

Vicki (15:08): Well, if we start with how kids are thinking and we take that seriously and we make that the center of the conversation, then we're acknowledging to the student and to ourselves that the child has something meaningful to bring to this conversation. And so we need to figure out how the child is thinking all the kind of kernels of mathematical strength in that thinking. And then yes, we can build from there, but we start with where they are as opposed to how we might solve the problem.

Mike (15:49): If you were to offer educators a universal inside question or a few sentence frames for inside questions, is it possible to construct something like that that's generic or do you have other advice for us?

Vicki (16:02): So that's a nice trick question. I wish it were that easy. I don't really think there are any universal inside questions. Perhaps the only universal one I can think of is something like, how did you solve this problem? It's a great general open-ended question. That's a good starter question in most situations. But the really powerful questions generally come from noticing mathematically important details in children's strategies. So a sentence stem that has been helpful in our work is, I noticed blank, so I wonder blank. Obviously questions don't have to be phrased exactly like this, but the idea is that we pick something that the child has done in their strategy and ask a question about the child's thinking behind that strategy detail. And that keeps us honest because the question absolutely has to begin with something in the child's strategy rather than inadvertently kind of slipping into our strategy.

Mike (17:04): Vicki, what do you think about the purpose of outside questions? Are there circumstances where we would want to ask our students an outside question?

Vicki (17:12): Absolutely. Sometimes we need to push children's thinking or share particular ideas, and that's okay. It's not that all outside questions are bad, it's just that we tend to overuse them and we could use them at more productive times. And by that I mean that we generally want to understand children's thinking before nudging their thinking forward with outside questions. So let's go back to the earlier example of Joy. Who was solving that problem about six children sharing four pancakes. And we had the two groups of teachers that had the different approaches to follow up questioning. There was the outside questioning that immediately zeroed in on improving Joy's strategy and the inside questioning that spent time exploring Joy's reasoning behind her strategy. So I'm thinking of two specific teachers right now. One generally took the outside questioning approach and the other inside questioning approach. And what was interesting about this pair was that they both asked the same outside question, could Joy partition the pancakes in a different way?

(18:19): But they asked this question at different times and the timing really matters. So the teacher who took an outside questioning approach wanted to begin her conversation that way. She wanted to ask Joy, could she partition in a different way? But in contrast, the teacher who took an inside questioning approach wanted to ask Joy lots of questions about the details of her existing strategy, and then posed this very same question at the end to see if Joy had some new ideas for partitioning after their conversation about her existing strategy. And that feels really different to children. So the exact same question can send children different messages when outside questions are posed. First they communicate to children that what they did was wrong and needs fixing. But when outside questions are posed after a conversation about their thinking, it communicates a puzzle or a problem to be solved.

(19:17): And children often are better equipped to consider this new problem having thoroughly discussed their own strategy. So I guess when I think about outside questions, I think of timing and amount. We generally want to start with inside questions, and we want most of our questions to be inside questions, but some outside questions can be productive. It's just that we overuse them. I want to mention one other thing about outside questions, and I think we often need fewer outside questions than we think we do, as long as we have space for children to learn from other children's thinking. So think about a typical lesson structure like launch, explore, discuss where children solve problems independently. And then the lesson concludes with a whole class discussion where children share their strategies and reflect on their problem solving. Will these sharing sessions serve as natural outside questions? Because children get to think about strategies that are outside of their own, but in a way that doesn't point to their own strategy as lacking in some way. So outside questions definitely have a place we just need to think about when we ask them and how many of them are really necessary.

Mike (20:34): That is really helpful. I find myself thinking about my own process when I'm working on a problem, be it mathematical or organizational or what have you. When someone asks me to talk about how I've thought about it, engaging in that process in some ways primes me, right? Because I've gotten clearer on my own thinking. I suspect that the person who's asking me the question is also clearer on that, which allows them to ask a different kind of outside question if and when they get to the point. So there's the benefit for the learner in that their clarifying their own thinking. There's the benefit in the educator who's engaging with the learner and getting just a much clearer sense of how that thinking was happening. And I suspect that leads to an outside question that's much more productive.

Vicki (21:16): It's a win-win situation.

Mike (21:18): Absolutely. This conversation has been wonderful. The challenge of having a podcast, of course, is that we've got about 20 to 25 minutes to talk about a really big idea that has profound implications for teachers. If someone wanted to pick up on the things we've been talking about today, where would you start, Vicki?

Vicki (21:38): I would encourage them to go talk to children. Children's thinking is so mathematically rich and it's so fascinating. So be curious about their thinking. Ask questions, ask those inside questions. Don't worry about asking the best question. It doesn't exist, but ask questions and then children will be your guides. They'll help you know where to go next. The other thing I would suggest is these journeys are always best done with your colleagues. And so get a colleague together and think about questioning together what we were talking about earlier with joy strategy teachers. We're looking at students' written work. That's a great place to practice. You can look at children's written work and talk together to figure out what types of conversations do you want to have with this child afterwards.

Mike (22:28): I think that's a great place to stop. I want to thank you so much for joining us today, Vicki, it has really been a pleasure talking with you.

Vicki (22:34): That was fun. Thanks for having me.

Mike (22:39): This podcast is brought to you by the Math Learning Center and the Meyer Math Foundation dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.

© 2025 The Math Learning Center | www.mathlearningcenter.org

 

Dr. Karisma Morton, Understanding and Supporting Math Identity   ROUNDING UP: SEASON 3 | EPISODE 11

In this episode, we will explore the connection between identity and mathematics learning. We’ll examine the factors that may have shaped our own identities and those of our students. We’ll also discuss ways to practice affirming students' identities in mathematics instruction.

BIOGRAPHIES

Dr. Karisma Morton is an assistant professor of mathematics education at the University of North Texas. Her research explores elementary preservice teachers’ ability to teach mathematics in equitable ways, particularly through the development of their critical racial consciousness. Findings from her research have been published in the Journal for Research in Mathematics Education and Educational Researcher. ​

RESOURCES

The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices by Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin

Rough Draft Math: Revising to Learn by Amanda Jansen

Olga Torres' “Rights of the Learner” framework

Cultivating Mathematical Hearts: Culturally Responsive Mathematics Teaching in Elementary Classrooms by Maria del Rosario Zavala and Julia Maria Aguirre

TRANSCRIPT

Mike Wallus: If someone asked you if you were good at math, what would you say, and what justification would you provide for your answer? Regardless of whether you said yes or no, there are some big assumptions baked into this question. In this episode, we're talking with Dr. Karisma Morton about the ways the mathematics identities we formed in childhood impact our instructional practices as adults and how we can support students' mathematical identity formation in the here and now. 

Welcome to the podcast, Karisma. I am really excited to be talking with you about affirming our students’ mathematics identities.

Karisma: Oh, I am really, really excited to be here, Mike. Thank you so much for the invitation to come speak to your audience about this.

Mike: As we were preparing for this podcast, one of the things that you mentioned was the need to move away from this idea that there are math people and nonmath people. While it may seem obvious to some folks, I'm wondering if you can talk about why is this such an important thing and what type of stance educators might adopt in its place?

Karisma: So, the thing is, there is no such thing as a math person, right? We are all math people. And so, if we want to move away from this idea, it means moving away from the belief that people are inherently good or bad at math. The truth is, we all engage in mathematical activity every single day, whether we realize it or not. We are all mathematicians. And so, the key is, as math teachers, we want to remove that barrier in our classrooms that says that only some students are math capable. 

In the math classroom, we can begin doing that by leveraging what students know mathematically, how they experience mathematics in their daily life. And then we as educators can then incorporate some of those types of activities into the everyday learning of math in our classrooms. So, the idea is to get students to realize they are capable math doers, that they are math people. And you're showing them the evidence that they are by bringing in what they're already doing. And not just that they are math doers, but that those peers that are also engaged in the classroom with them are capable math doers. And so, breaking down those barriers that say that some students are and some students aren't is really key. So, we are all math people.

Mike: I love that sentiment. You know, I've seen you facilitate an activity with educators that I'm hoping that we could replicate on the podcast. You asked educators to sort themselves into one of four groups that best describe their experience when they were a learner of mathematics. And I'm wondering if you could read the categories aloud and then I'm going to ask our listeners to think about the description that best describes their own experiences.

Karisma: OK, great. So, there are four groups. And so, if you believe that your experience is one where you dreaded math and you had an overall bad experience with it, then you would choose group 1. If you believe that math was difficult but you could solve problems with tutoring or help, then you would select group 2. If you found that math was easy because you were able to memorize and follow procedures but you had to practice a lot, then you'd be in group 3. And finally, if you had very few difficulties with math or you were kind of considered a math whiz, then you would select group 4.

Mike: I had such a strong reaction when I participated in this activity for the first time. So, I have had my own reckoning with this experience, but I wonder what impact you've seen this have on educators. Why do it? What's the impact that you hope it has for someone who's participating?

Karisma: Yeah. So, I would say that a key part of promoting that message that we started off talking about is for teachers to go back, to reflect. We have to have that experience of thinking about what it was like for us as math learners. Because oftentimes we go into the classroom and we're like, “All right, I got to do this thing.” But we don't take a minute to reflect: “What was it like for me as a math learner?” And I wanted to first also say that I did not develop this activity. This is not a Karisma original. I did see this presented at a math teacher-educator conference about five years ago by Jennifer Ward. I think she's at Kennesaw State [University] right now. But the premise is the same: We want to give teachers an opportunity to reflect over their own experiences as math learners as a good starting place for helping them to identify with each other and also with the students that they're teaching.

And so, whenever I have this activity done, I have each of the participants reflect. And then they have conversations around why they chose what they chose. And this is the opportunity for them to have what we call “windows,” “mirrors,” and “sliding glass doors,” right? So, you either can see yourself in another person's experience and feel like, “Oh, I'm not alone here,” especially if it were a negative experience. Or you may get to see or take a glimpse into what someone else has experienced that was very different from your own and really get a chance to understand what it was like for them. They may have been the math whiz, and you're looking at them like they're an alien that fell from the sky because you're like, “How did that happen,” right? But you can begin to have those kinds of conversations: “Why was it like this for you?” and “It wasn't like that for me.” Or “It was the same for me, but what did it look like in your instance versus my instance?” 

I honestly feel like sometimes people don't realize that their experience is not necessarily unique, especially if it's coming from a math trauma perspective. Some people don't want to talk about their experience because they feel like it was just theirs. But they sometimes can begin to realize that, “Hey, you had that experience too, and let's kind of break down what that means.” Do you want to be that type of teacher? Do you want to create the type of environment where you felt like you weren't a capable math doer? So powerful, powerful exercise. I encourage your listeners to try it with a group of friends or colleagues at work and really have that conversation.

Mike: Gosh, I'm just processing this. One of the things that I keep going back to is you challenging us to discard the idea that some people are inherently good at math and other people are not. And I'm making a connection that if I'm a person who identified with group 1, where I dreaded math and it was really a rough experience, what does it mean for me to discard the idea that some people are inherently good or inherently not good at math versus if I identified as a person who was treated as the math whiz and it came easy for me, again, what's required for me? 

It feels like there's things that we can agree with on the surface. We can agree that people are not good inherently at mathematics. But I find myself really thinking about how my own experience actually colors my beliefs and my actions, how agreeing to that on the surface and then really digging into how your own experience plays out in your practice or the ways that you interact with kids. There's some work to be done there, it seems like.

Karisma: Absolutely. You hit the nail on the head there. It's important to do that work. It's really important for us to take that moment to reflect and think about how our own experience may be impacting how we're teaching mathematics to children.

Mike: I think that's a great place to make a shift and talk about areas where teachers could take action to cultivate a positive mathematics identity for kids. I wonder if we can begin by talking about expectations and norms when it comes to problem solving.

Karisma: Yes. So, Julia Aguirre, Karen Mayfield-Ingram, and Danny Martin wrote this amazing book, called The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices. And one of those equity-based practices is affirming math learners’ identities. And so, one of the ways we can do this in the math classroom is when having students engaged in problem solving. And so, one of the things that we want to be thinking about when we are having students engaged in math problem solving is we want to be promoting students' persistence and reasoning during problem solving. And you might wonder, “Well, what does that actually look like?” 

Well, it might be helpful to see what it doesn't look like, right? So, in the typical math classroom, we often see an emphasis on speed: who got it done quickly, who got it done first, who even got it done within the time allotted. And then also this idea of competition. So, that is really hard for kids because we all need time to process and think through our problem-solving strategies. And if we're putting value on speed, and we're putting value on competition, are we in fact putting value on a problem-solving strategy or the process of problem-solving? So, one way to affirm math learners' identities is to move away from this idea of speed and competition and foster the type of environment where we're valuing students' persistence with the problem. We're valuing students' processes in solving a problem, how they're reasoning, how they're justifying their steps or their solutions’ strategies, as opposed to who's getting done quickly. 

Another thing to be thinking about is reframing making mistakes. There's so many great resources about this. What comes to mind immediately is Rough Draft Math by Amanda Jansen, which is really helping us to reframe the idea that we can make some mistakes, and we can revise our thinking. We can revise our reasoning, and that's perfectly OK. 

Olga Torres' “Rights of the Learner” framework talks a lot about the right to make a mistake is one of the four rights of the learner in the mathematics classroom. And so, when having kids engaged in problem-solving and mathematics, mistakes should be seen more like what Olga Torres calls “celebrations,” because there are opportunities for learning to occur. We can focus on this mistake and think about and problem-solve through the mistake. “Well, how did we get here?” Use it as a moment that all students can benefit from. And so, kids then become less afraid to make mistakes because they're not ridiculed or made to feel less than because they've done so. Instead, it empowers them to know that “Hey, I made this mistake, but in actuality, this is going to help me learn. And it's also going to help my classmates.”

Mike: I suspect a lot of those moments, people really appreciate when there's the “aha!” or the “oh!” What was happening before that might've been some struggle or some misconceptions or a mistake. You're making me think that we kind of have to leave space for those mistakes or those misconceptions to emerge if we really want to have those “aha!”s or those “oh!”s in our classroom.

Karisma: That's exactly right. And imagine if you are the one who's like, “Oh!”—what that does for your self-confidence. And even having your peers recognize that you've come to this answer or this understanding. It almost becomes like a collective win if you have fostered a type of environment where it's less about me against you and more about all of us learning together.

Mike: The other thing that came to me is that I'm thinking back to the four groups. I would've identified as a person who would fit into group 2, meaning that there were definitely points where math was difficult for me, but I could figure it out with tutoring or with help from a teacher. I start to wonder now how much of my perception was about the fact that it just took me a little bit longer to process and think about it. So, it wasn't that math was difficult. It was that I was measuring my sense of myself in mathematics around whether I was the first person, or I was fast, or I got it right away, or I got it right the first time, as opposed to really thinking about, “Do I understand this?” And to me, that really feels connected to what you're saying, which is the way that we as teachers value students' actions, their rough-draft attempts, their mistakes, and position those as part of the process—that can have a really concrete impact on how I think about myself and also how I think about what it is to do math. 

Well, let's shift again and talk about another area where educators could support positive identity. I’m thinking about the ways that they can engage with students' background knowledge and their life experiences.

Karisma: Hmm, yeah. This is a huge one. And this really, again, comes back to recognizing that our students are whole human beings. They have experiences that we should want to leverage in the math classroom, that they don't need to keep certain parts of themselves at the door when they come in. And so, how do we take advantage of what our students are bringing to the table? And so, we want to be thinking a lot about, “Well, who is the student?” “What do they know?” “What other identities do they hold?” “What's important to them?” “What kinds of experiences do they have in their everyday life that I can bring into the math classroom?” “What are their strengths?” “What do they enjoy doing?” The truth of the matter is really great teachers do this all the time, you know? You know who your students are for the most part, right? And students come to us with a whole host of experiences that we want to leverage and come with all sorts of experiences that we could use in the math classroom.

I think oftentimes we don't think about making connections between those things and how to connect them to the mathematics that's happening in the classroom. So, oftentimes we don't necessarily see a reason to connect what we know about our students to mathematics. And so, it's really just a simple extra step because really amazing teachers—which I know they're amazing teachers that are listening right now—you know who your students are. So how do we take what we know about them and bring that into the mathematics learning? Again, as with problem solving, what is it that we want to stay away from? We want to be staying away from connecting math identity only with correct answers and how fast a kid is at solving a problem. Their math identity shouldn't be dependent on how many items they got correct on an assessment. It should be more about, “Well, what is it that they know? And how are we able to use this in the math classroom?”

Mike: You're making me think about how oftentimes there's this distinction that happens in people's minds between school math and math that happens everywhere in the real world. Part of what I hear you suggesting is that when you help kids connect to their real world, you're actually doing them another service and that you're helping them see, like, “Oh, these lived experiences that I might not have called mathematics, they are,” right? “I do mathematics. I'm a doer.” And part of our work in bringing that in is helping them see what's already there.

Karisma: I love that. Helping them see what's already there. That's exactly right.

Mike: Well, before we go, I'm wondering if you could talk about some of the resources that have informed your thinking about this and that you think might also help a person who's listening who wants to keep learning.

Karisma: Yeah. There's a lot of great resources out there. The one that I rely on heavily is The Impact of Identity in K–8 Mathematics: Rethinking Equity-Based Practices. I really like this book because it's very accessible. It does a really great job of setting the stage for why we need to be thinking about equity-based practices. And I really enjoy how practical things are. So, the book goes through describing what a representative lesson would look like. And so, it's a really nice blueprint for teachers as they're thinking about students' identities and how to promote positive math identity amongst their students. And then I think we also mentioned Rough Draft Math by Amanda Jansen, which is a good read. And then there's also a new book that came out recently, Cultivating Mathematical Hearts: Culturally Responsive [Mathematics] Teaching in Elementary Classrooms. And this book goes even deeper by having vignettes and having specific classroom examples of what teaching in this kind of way can look like. So those are three resources off the top of my head that you could dig into and have book clubs at your schools and engage with your fellow educators and grow together.

Mike: I think that's a great place to stop. Thank you so much for joining us today. This has really been a pleasure.

Karisma: Oh, it's been a pleasure talking to you too. Thank you so much for this opportunity.

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

© 2025 The Math Learning Center | www.mathlearningcenter.org

 

Sue Kim and Myuriel Von Aspen, Building Productive Partnerships   ROUNDING UP: SEASON 3 | EPISODE 10

In this episode, we examine the practice of building productive student partnerships. We’ll talk about ways  educators can cultivate joyful and productive partnerships and the role the educator plays once students are engaged with their partner. 

BIOGRAPHIES

Sue Kim is an advocate for children’s thinking and providing them a voice in learning mathematics. She received her teaching credential and master of education from Biola University in Southern California. She has been an educator for 15 years and has taught and coached across TK–5th grade classrooms including Los Angeles Unified School District and El Segundo Unified School District as well as several other Orange County, California, school districts. 

Myuriel von Aspen believes in fostering collaborative partnerships with teachers with the goal of advancing equitable, high-quality learning opportunities for all children. Myuriel earned a master of arts in teaching and a master of business administration from the University of California, Irvine and a bachelor of science in computer science from Florida International University. She currently serves as a math coordinator of the Teaching, Learning, and Instructional Leadership Collaborative. ​

RESOURCES

Catalyzing Change in Early Childhood and Elementary Mathematics by National Council of Teachers of Mathematics

Purposeful Play by Kristine Mraz, Alison Porcelli, and Cheryl Tyler 

Hands Down, Speak Out: Listening and Talking Across Literacy and Math K–5 by Kassia Omohundro Wedekind and Christy Hermann Thompson

TRANSCRIPT

Mike Wallus: What are the keys to establishing productive student partnerships in an elementary classroom? And how can educators leverage the learning that happens in partnerships for the benefit of the entire class? We'll explore these and other questions with Sue Kim and Myuriel von Aspen from the Orange County Office of Education on this episode of Rounding Up. 

Well, hi, Sue and Myuriel. Welcome to the podcast.

Myuriel von Aspen: Hi, Mike. 

Sue Kim: Thanks for having us.

Mike: Thrilled to have you both. 

So, I first heard you two talk about the power of student partnerships in a context that involved counting collections. And during that presentation, you all said a few things that I have been thinking about ever since. The first thing that you said was that neuroscience shows that you can't really separate emotions from the way that we learn. And I wonder what do you mean when you say that and why do you think it's important when we're thinking about student partnerships?

Myuriel: Yes, absolutely. So, this idea comes directly from neuroscience research, the idea that we cannot build memories without emotions. I'm going to read to you a short quote from the NCTM [National Council of Teachers of Mathematics] publication Catalyzing Change in Early Childhood and Elementary Mathematics that says, “Emerging evidence from neuroscience strongly shows that one cannot separate the learning of mathematics content from children's views and feelings toward mathematics.” 

So, to me, what that says is that how children feel has a huge influence on their ability to learn math and also on how they feel about themselves as learners of math. So, depending on how they feel, they might be willing to engage in the content or not. And so, as they're engaging in counting collections and they're enjoying counting and they feel joyful and they're doing this with friends, they will learn better because they enjoy it, and they care about what they're doing and what they're learning.

Mike: You know, this is a nice segue to the other thing that has been on my mind since I heard you all talk about this because I remember you said that students don't think about a task like counting collections as work, that they see it as play. And I wonder what you think the ramifications of that are for how we approach student partnership?

Sue: Yeah, you know, I've been in so many classrooms across TK through fifth [grade], and when I watch kids count collections, we see joy, we see engagement in these ways. But I've also been thinking about this idea of how play is even defined, in a way, since you asked that question that they think of it as play. 

Kristine Mraz, teacher, author, and a consultant, has [coauthored] a book called Purposeful Play. And I remember this was the first time I hear about this reference about Vivian Paley, an American early childhood educator and researcher, stress through her career, the importance of play for children when she discovered in her work that play’s actually a very complex activity and that it is indeed hard work. It's the work of kids. It's the work of what children do. That's their life, in a sense. And so, something I've been thinking about is how kids perceive play is different than how adults perceive play. And so, they take it with seriousness. There is a complex, very intentionality behind things that they do and say. And so, when we are in our session, and we reference Megan Franke, she says that when young people are engaging with each other's ideas, what they're able to do is mathematically important. But it's also important because they're learning to learn together. They're learning to hear each other. They're developing social and emotional skills as they try and navigate and negotiate each other's ideas. And I think for kids that this could be considered play, and I think that's so fascinating because it's so meaningful to them. And even in a task like counting, they're doing all these complex things. But as adults we see them, and we’re like, “Oh, they're playing.” But they are really thinking deeply about some of these ideas while they're developing these very critical skills that we need to give opportunities for them to develop.

Myuriel: I like that idea of leaning into the play that you consider maybe not as serious, but they are. Whether they're playing seriously or not, that you might take that opportunity to make it into a mathematical question or a mathematical reflection.

Sue: I totally agree with you. And taking it back to that question that you asked, Mike, about, “How do we approach student partnerships then?” And I think that we need to approach it with this lens of curiosity while we let kids engage in these ways and opportunities of learning to hear each other and develop these social-emotional skills, like we said. And so, when you see kids that we think are “playing” or they're building a tower: How might we enter that space with a lens of curiosity? Because to them, I think it's serious work. We can't just think, “Oh, they're not really in the task” or “They're not doing what they were supposed to do.” But how do we lean into that space with a lens of curiosity as Megan reminded us to do, to see what mathematical things we can tap into? And I think that kids always rise to the occasion.

Mike: I love that. So, let's talk about how educators can cultivate joyful and productive student partnerships. I'm going to guess that as is often the case, this starts by examining existing beliefs that I might have and some of my expectations.

Sue: Yeah, I think it really begins with your outlook and your identity as a teacher. What's your outlook on what's actually possible for kids in your class? Do you believe that kids as young as 4-year-olds can take on this responsibility of engaging with each other in these intelligent ways? Unless we begin there and we really think and reflect and examine what our beliefs are about that, I think it's hard to go and move beyond that, if that makes sense. 

And like what we just talked about, it's being open to the curiosity of what could be the capacity of how kids learn. I've seen enough 4-year-olds in TK classrooms doing these big things. They always blow my mind, blow my expectations, when opportunities are given to them and consistently given to them. And it's a process, right? They're not going to start on day one doing some of these more complex things. But they can learn from one another, and they also learn from you as a teacher because they are really paying attention. They are attending to some of these complex ideas that we put in front of them.

Mike: Well, you hit on the question that I was thinking about. Because I remember you saying that part of nurturing partnerships starts with a teacher and perhaps a pair of children at a table. Can you all paint a picture of what that might look like for educators who are listening?

Sue: Yeah, so actually in one of the most recent classrooms, I went in, and this teacher allowed me to partner with her in this work. She wanted to be able to observe and do it in a structured way so that she could pick up on some details of noticing the things that kids were doing. And so, she would have a collection out, or they got to choose. She was really good about offering choice to kids, another way to really engage them. And so, they would choose. They would come together. And then she started just taking some anecdotal notes on what she heard kids saying, what she saw them doing, what they had to actually navigate through some of the things, the stuck moments that came up. 

From that, we were able to develop, “OK, what are some goals? We noticed Students A and B doing this and speaking in these ways. What might be the next step that we might want to put into a mini lesson or model out or have them actually share with the class what they were working on mathematically?” Whether it was organization, or how they decided they wanted to represent their count, how they counted and things like that. 

And so, it was just this really natural process that took place that we were able to really lean into and leverage that kids really responded to because it wasn't someone else's work or a page from a textbook. It was their work, their collection that was meaningful to them and they had a true voice and a stake in that work.

Mike: I feel like there have been points in time where my understanding of building groups was almost like an engineering problem, where you needed to model what you wanted kids to do and have them rehearse it so specifically. But I think what sits at the bottom of that approach is more about compliance. And what I loved about what you described, Sue, is a process where you're building on the mathematical assets that kids are showing you during their time together—but also on the social assets that they're showing you. So, in that time when you might be observing a pair or a partnership playing together, working together with something like counting collections, you have a chance to observe the mathematics that's happening. You also have a chance to observe the social assets that you see happening. And you can use that as a way to build for that group, but also to build for the larger group of children. And that just feels really profoundly different than, I think, how I used to think about what it was to build partnerships that were “effective.”

Myuriel: You know, Mike, I think it's not only compliance. It's also that control. And what it makes me think about is, when we want to model ourselves what we want students to do, instead of—exactly what you said, looking at what they're doing and bringing that knowledge, those skills, that wisdom that's in the room from the students to show to others so that they feel like their knowledge counts. The teacher is not only the only authority or the only source of knowledge in the room—we bring so much, and we can learn from each other. So, I think it's so much more productive and so effective in developing the identity of students when you are showing something that they're doing to their peers versus you as an adult telling them what to do.

Mike: Yeah. Are there any particular resources that you all have found helpful for crafting mini lessons as students are learning about how to become a partnership or to be productive in a partnership?

Myuriel: Yes. One book that I love, it's not specific to counting collections, but it does provide opportunities for teachers to create micro-lessons when students are listening and talking to each other. It's Hands Down, Speak Out: Listening and Talking Across Literacy and Math K–5 by Kassia [Omohundro] Wedekind and Christy [Hermann] Thompson. And the reason why I love this book is because it provides, again, these micro-lessons depending on what the teacher is noticing, whether it is that the teacher is noticing that students need support listening to each other or maybe making their ideas clear. Or maybe students need to learn how to ask questions more effectively or even reflect on setting and reflecting on the goals that they have as partners. It does provide ideas for teachers to create those micro-lessons based on what the teacher is noticing.

Sue: Yeah, I guess I want to add to that, Mike, as well, the resources that Myuriel said. But also, I think this is something I really learned along the process of walking alongside this teacher, was looking at partnerships through a mathematical lens and then a social lens. And so, the mini lesson could be birthed out of watching kids in one day. It might be a social lens thinking about, “They were kind of stuck because they wanted to choose different collections. What might we do about that?” And that kind of is tied to this problem-solving type of skill and goal that we would want kids to work on. That’s definitely something that's going to come up as kids are working in partnerships. These partnerships are not perfect and pristine all the time. I think that's the nature of the job. And just as humans, they're learning how to get along, they're learning how to communicate and navigate and negotiate these things.

And I think those are beautiful opportunities for kids and for teachers, then, to really lean into as goals, as mini lessons that can be out of this. And these mini lessons don't have to be long and drawn out. They can be a quick 5-, 10-minute thing. Or you can pause in the middle of counting and kind of spotlight the fact that “Mike and Brent had this problem, but we want to learn from them because they figured out how to solve it. And this is how. Let's listen to what happened.” So, these natural, not only places in a lesson that these opportunities for teaching can pop up, but that these mini lessons come straight from kids and how they are interacting and how they are taking up partnerships, whether it be mathematical or social.

Mike: I think you're helping me address something that if I'm transparent about was challenging for me when I was a classroom teacher. I got a little bit nervous about what was happening and sometimes I would shut things down if I perceived partnerships to be, I don't know, overwhelming or maybe even messy. But you're making me think now that part of this work is actually noticing what are the assets that kids have in their social interactions in the way that they're playing together, collaborating together, the mathematics? And I think that's a big shift in my mind from the way that I was thinking about this work before. And I wonder, first of all, is this something that you all notice that teachers sometimes are challenged by? And two, how you talk to someone who's struggling with that question of like, “Oh my gosh, what's happening in my classroom?”

Myuriel: Yes, I can totally understand how teachers might get overwhelmed. We hear this from, not only from teachers trying to do the work of counting collections, but even just using tools for students to problem-solve because it does get messy. I like the way Sue keeps emphasizing how it will be messy. When you have rich mathematical learning happening, and you're using tools and collections and you have 30 students having conversations, it definitely will get messy. But I would say that something that teachers can do to mitigate some of that messiness is to think about the logistics ahead of time and be intentional about what you are planning to do. So, some of the things that they may want to think about is: How are students going to access the counting collections? Where are you going to [put] the tools that they're going to be using? Where physically in the classrooms will students get together to have collections so that they have enough room to spread out and record and talk to each other? And just like Sue was mentioning: How do I partner students so that they do have a good experience, and they support each other? So, all of these things that might cost a bit of chaos if you don't think about them, you can actually think about each one of those ahead of time so that you do have a plan for each one of those. 

Another thing that teachers may want to consider thinking about is, what do they want to pay attention to when they are facilitating or walking around? There's a lot that they need to pay attention to. Just like Sue mentioned, it is important for them to pay attention to something because you want to bring what's in the room to connect it and have these mini lessons of what students actually need. And also, thinking about after the counting collections: What worked and what didn’t? And what changes do I want to make next time when I do this again? Just so that there is a process of improvement every time. Because as Sue had mentioned, it's not going to happen on day one. You are learning as a teacher, and the students are learning. So, everybody in that room is learning to make this a productive and joyful experience.

Sue: Yeah, and another thing that I would definitely remind teachers about is that there's actually research out there about how important it is for kids to engage with one another's mathematical ideas. I'm so thankful that people are researching out there doing this work for us. And this goes along with what Myuriel was saying, but the expectations that we put on ourselves as teachers sometimes are too far. We're our biggest critique-ers of the work that we do. And of course we want things to go well, but to make it more low-risk for yourself. I think that when we lower those stakes, we're more prone to let kids take ownership of working together in these ways, to use language and communication that makes sense while doing math and using these cognitive abilities that are still in the process of developing. And I think they need to remember that it takes time to develop, and it's going to get there. And kids are going to learn. Kids are going to do some really big things with their understanding. But giving [yourself] space, the time to learn along with your students, I think is very critical so that you feel like it's manageable. You feel like you can do it again the next day.

Mike: Tell me a little bit about how you have seen educators use things like authentic images or even video to help their students make sense of what it means to work in a partnership. What have you seen teachers do?

Sue: Yeah. Not to mention how that is one sure way to get kids engaged. I don't know if you've been in a room full of first graders or kindergartners, but if you put a video image up that's them counting and showing how they are thinking about things, they are one-hundred-percent there with you. They love being acknowledged and recognized as being the doers and the sensemakers of mathematics. And it goes into this idea of how we position kids competently, and this is another way that we can do that. But capturing student thinking in photos or a short clip has really been a powerful tool to get kids to engage in each other's ideas in a deeper way. I think it allows teachers and students to pause and slow down and really focus in on the skill of noticing. I think people forget that noticing is a skill you have to teach. And you have to give opportunities for kids to actually do these things so they can see mathematically what's happening within the freeze-frame of this image, of this collection, and how we might ask questions to help facilitate and guide their thinking to think deeply about these ideas.

And so, I've seen teachers use them with partners, and they may say, “Hey, here's one way that they were counting. How do you think they counted within the frame of this picture or this photo that we took?” And then kids will have these conversations. They'll engage mathematically what they think, and then they might show the video clip of the students actually counting. And they get to make predictions. They get to navigate the language around what they think. And it's just, again, been a really nice tool that has then branched out into whole-group discussions. So, you can use it with partnerships and engage certain kids in specific ways, but then being able to utilize that and leverage that in whole-group settings has really been powerful to see.

Myuriel: I also recently observed a teacher with pictures, showing students different tools that different partners were using and having those discussions about, “Why did this tool work and why didn't this one?” or “What will you have to do if your collection gets bigger?” So, it is a great opportunity to really show from what they're using and having those discussions about what works and what doesn’t, and “Why would I use this versus this?” from their own work.

Mike: Myuriel, what you made me wonder is if you could apply this same idea of using video or images to help support some of those social goals that we were talking about for students as well.

Myuriel: I think that you could. I can just imagine that if you see two students working together and supporting each other or asking some good questions and being curious, you could record them and then show that to the others to ask them what they're noticing. “How are these two students supporting each other in their learning?” Even “How are they being kind to each other when they make a mistake?” So, there is so much power in using video for not just the mathematical skills, but also for the social skills.

Sue: Myuriel, when you're talking, you're reminding me about two particular students that we have watched, and we have recorded video around, actually, when they came to a disagreement. 

There was this one instance when a couple of students came to a disagreement about what to call the next number of the sequence. And that was a really cool moment because we actually discovered, “Wow, these two peers had enough trust in each other to pause, to listen to both sides.” And then when it came time to actually call the number and the sequence, the other student actually trusted enough and listened to the reasoning of the other student to say, “OK, I'm going to go along with you, and I think that should be what the sequence is.” And it was just a really neat opportunity and—that this teacher actually showed in front of kids just to see what kids would say in response to that particular moment.

Myuriel: It was actually one very cute, but very interesting moment when you see that second student who's listening to the other one. And actually at first she kind of argued with him a little bit about, “No, it's not this number.” But the second time around, when she counted, she paused right at that same spot where she had trouble before, and she set the number that he had suggested the earlier time so that you see that she's listening, she's considering someone else's ideas, and she's learning the correct sequence. Yes, that was really amazing to see.

Sue: So, it's the sequence of numbers that they're working on, but think about all the social aspects of what is happening and developing, and I think that they're addressing it and that they're having to engage with [it]. It's [a] very complex situation that they're learning a lot of skills around in that very moment.

Mike: You know, I wonder how an educator might think about their role once students are actually engaged with a partner. How do you all think about goals, or the role of the teacher, once students are working with a partner?

Sue: I think that one of the things we're really thinking about and being more intentional about is: When do we actually interject, or when do we as teachers actually say something? When and how do we make those decisions? And for several years now, I've really taken on this notion that we are facilitators. Yes, we're teachers. But more than anything, we are facilitators of the students in our class, and we want to really give them the opportunity to work through some of these ideas. And we will have set up partnerships based on what we've seen and notes that we took as kids have been working. But it's an ever-innovated process, I think. And I think something that's always going to be on the forefront is that idea: How are we facilitating? How are we deciding when we want to say something or interject, and why? And what is it that we are trying to get kids to think about?

Because I think we need to help students realize that they are always in the driver's seat of what they're doing, especially if they're in a partnership. And there are targeted things that we can have them maybe think about when we drop a question based on what we're noticing. Or maybe when they're stuck, and they're in the middle of negotiating something. But I really think that it starts there with us kind of thinking about: What is our role? Is it OK that we step back and we just watch even if they have to problem-solve through something that feels like, “Oh, I don't know if they're going to get through that moment.” But we’ve got to let them. We’ve got to give them opportunities to do that without having to rescue them every single time.

Myuriel: And you're right, Sue, we've seen it so many times when if you just bite your tongue, 10 seconds later, it's happening, right? They're helping each other, and they get to the idea that you thought you had to bring up to them. But they were able to resolve it. So, if we only allow that time for them to process the idea or to revise their thinking or to allow the other partner to support their partner, it will happen.

Sue: Yeah, and I think that doesn't mean that we can't set kids up. I've seen teachers launch the lesson with something a partner did before yesterday, and they will have referred to a protocol or something they're working on. And then as facilitators, we can then go out, and we might already be thinking about, “Oh, I want to be watching these two partnerships today”—having in mind, “OK, this is my target idea for them, my target goal for them.” So, there are definite ways that we can frame and decide who we want to watch and observe, but while in the balance of letting kids do what they're going to do and what the expectation of being surprised. Because kids always surprise us with their brilliance.

Mike: Yeah, there's multiple things that came to mind as I was listening to you all talk about this. The first one is how it's possible to inadvertently condition kids to see the teacher coming and look and stop and potentially look for the teacher to say something. We actually do want to avoid that. We want to see their thinking. 

The other piece is the difference between, as you said, potentially dropping a question and interjecting, as you said, Myuriel, biting your tongue and letting them persist through—whether it's an idea they're grappling with or a struggle for what to do next—that there's so much information in those moments that we can learn or that might help us think about what's next. It's a challenge, I think, because math culture in the United States is such that we're kind of trained to see something that looks like a mistake. “Let's get in there.” And I hear you giving people permission to say, “Actually, it's OK to step back and watch their thinking and watch them try to make sense of things because there's a big payoff there.”

Sue: Absolutely. Yeah. 

Myuriel: Yes. And, Mike, I think we as teachers—you feel the need of having to address every single “mistake” per either individual student or per partnership. And sometimes you feel like, “I have 30 students, how can I possibly do that?” And I think that's where the power of doing a share out from what you've observed, bringing everyone together, learning from what was in the room, right? Because just like Sue was saying, it's not that you don't ever set up kids with knowledge of what you've observed, but you bring the power. It's what you're bringing, what's in the room, what you've noticed. But you share it out, or you have students share it out, with everyone so that everyone is moving forward.

Mike: I have a follow-up question for you all about goals for partnerships. I'm wondering how you think about the potential for partnerships as a way to help develop language, be it academic or social, for students. Are there particular practices that you imagine educators could take up if language development was one of their goals?

Myuriel: I'm so glad you're asking that question because I don't think we can learn math without language. I don't think we can learn anything without language. And I think that working in partnerships provides such an authentic, meaningful way of developing language because students are in conversations with each other. And we know that conversation is one way that ideas develop conversations or even sharing your thinking. Sometimes we notice that as students are sharing their thinking, and they're listening to themselves, they catch themselves making a mistake, and they are able to revise their thinking based on what they are saying. So again, I think it is the perfect opportunity for students to mathematically learn counting sequence or socially learn how to negotiate and make sense of what they're going to represent, when they're counting, or to explain their thinking. And we know, of course, that one of the mathematical practices is justifying, explaining your thinking. So, it's important to provide those opportunities for students to do that in this kind of structural way.

I also think that working in partnerships provides this opportunity for teachers to listen and notice if there's any language that students are starting to use that can be shared with others. So again, this idea that you hear it from someone in the room and that's going to help everybody else grow. Or that if students are doing something and you can name it, provide those terms to students. So, for example, just like I mentioned, somebody's explaining their thinking and through that they change their mind. They revised their thinking. Actually sharing that with the whole class and naming it: “Oh, they were revising their thinking” or sharing how they were explaining something with academic language so that others can also use that language as they're explaining their own thinking. So, I think that those are powerful ways to provide opportunities for everyone's academic language or social skills through language to be developed.

Sue: Yeah, I think that another big idea that comes out of that language piece is just how kids are learning to make sense of how to be partners, especially our younger students, our younger mathematicians. They're really needing to figure out like, “Oh, what does it mean to take turns to speak about this and how I use my words in this way versus another?” And I think that's another big opportunity for kids to build those skills because we can't just assume that kids come into our classrooms knowing how to talk in these ways, how to address each other, how to engage respectfully, that they can disagree respectfully, even in partnerships. And we want them to have the time and space to be able to develop those skills through language as well.

Mike: You know, I think the mental movie that I have for the point in time after children have engaged in any kind of partnership task, be it counting collections or something else, has really shifted. Because I think beforehand the way the movie ended was potentially sharing a student's representation if they had represented something on a piece of paper that showed what they had physically done with their things. And I still think that's valid and important, particularly if that's one of your goals. 

But you're making me think a lot more about the potential of images of students at work as they're going through the process or video and how closing, or potentially opening the next time, with that really just kind of expands this idea of what's happening. Being able to look at a set of hands that are on a set of materials or in the process of moving materials or listening to language that's emerging from students in the form of a short video. There's a lot of richness that you could capture, and it's also a little bit more of a diverse way of showing what's going on. And it feels like another way to really position what you're doing—not just the output in the form of the paper representation—but what you're actually doing is valuable, and it's a contribution. And I think that just feels like there's a lot of potential in what you all are describing.

Sue: I think you hit the nail on the head. We're trying, and it's hard work. But to be open to these ideas, to these possibilities. And like you said, it's positioning kids so drastically different than how we've been doing it for so many years. And how you're actually inviting kids to be contributors of this work that they are now. They have the knowledge. They are the ones that hold the knowledge in the room. And how we frame kids and what they're doing is I think very critical because kids learn from that, and kids have so many things to offer that we need to really be able to think about how we want to create those opportunities for kids.

Myuriel: And, Mike, something that you said also made me think of just like we want to provide those opportunities for students to be creative and to show what they know. What you were talking about, having this new perspective, makes me think about also teachers being creative with how they use counting collections, right? There isn't just the one way. It doesn't mean that at the end of every counting collection, I have to have a share out right at the end and decide at that moment. I could start the day that way. I could start the next session that way. I could use a video. I could use a picture. I could have students share it. So, you can get creative. And I think that's the beauty also, because I think as a teacher, it's not only the students that are learning; you are learning along with them.

Mike: That's a great place to stop. This has been an absolutely fabulous conversation. Thank you both so much for joining us.

Myuriel: Thank you. Thank you so much for this opportunity.

Sue: Thank you. Thanks for having us.

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

© 2025 The Math Learning Center | www.mathlearningcenter.org

 

Dr. Kasi Allen, Breaking the Cycle of Math Trauma   ROUNDING UP: SEASON 3 | EPISODE 9

If you are an educator, you’ve likely heard people say things like “I’m a math person.” While this may make you cringe, if you dig a bit deeper, many people can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you are a math person as well. Today we’re talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it.

BIOGRAPHY

Kasi Allen serves as the vice president of learning and impact at The Ford Family Foundation. She holds a PhD degree in educational policy and a bachelor’s degree in mathematics and its history, both from Stanford University.

RESOURCES

“Jo Boaler Wants Everyone to Love Math” — Stanford Magazine

R-RIGHTS

Learning to Love Math by Judy Willis

TRANSCRIPT

Mike Wallus: If you're an educator, I'm almost certain you've heard people say things like, “I am not a math person.” While this may make you cringe, if you dig a bit deeper, many of those folks can identify specific experiences that convinced them that this was true. In fact, some of you might secretly wonder if you're actually a math person. Today we're talking with Dr. Kasi Allen about math trauma: what it is and how educators can take steps to address it. 

Well, hello, Kasi. Welcome to the podcast.

Kasi Allen: Hi, Mike. Thanks for having me. Great to be here.

Mike: I wonder if we could start by talking about what drew you to the topic of math trauma in the first place?

Kasi: Really good question. You know, I've been curious about this topic for almost as long as I can remember, especially about how people's different relationships with math seem to affect their lives and how that starts at a very early age. I think it was around fourth grade for me probably, that I became aware of how much I liked math and how much my best friend and my sister had an absolutely opposite relationship with it—even though we were attending the same school, same teachers, and so on. And I really wanted to understand why that was happening. And honestly, I think that's what made me want to become a high school math teacher. I was convinced I could do it in a way that maybe wouldn't hurt people as much. Or it might even make them like it and feel like they could do anything that they wanted to do. 

But it wasn't until many years later, as a professor of education, when I was teaching teachers how to teach math, that this topic really resurfaced for me [in] a whole new way among my family, among my friends. And if you're somebody who's taught math, you're the math emergency person. And so, I had collected over the years stories of people's not-so-awesome experiences with math. But it was when I was asked to teach an algebra for elementary teachers course, that was actually the students’ idea. And the idea of this course was that we'd help preservice elementary teachers get a better window into how the math they were teaching was planting the seeds for how people might access algebra later. 

On the very first day, the first year I taught this class, there were three sections. I passed out the syllabus; in all three sections, the same thing happened. Somebody either started crying in a way that needed consoling by another peer, or they got up and left, or both. And I was just pretty dismayed. I hadn't spoken a word. The syllabi were just sitting on the table. And it really made me want to go after this in a new way. I mean, something—it just made me feel like something different was happening here. This was not the math anxiety that everybody talked about when I was younger. This was definitely different, and it became my passion project: trying to figure how we disrupt that cycle.

Mike: Well, I think that's a good segue because I've heard you say that the term “math anxiety” centers this as a problem that's within the person. And that in fact, this isn't about the person. Instead, it's about the experience, something that's happened to people that's causing this type of reaction. Do I have that right, Kasi?

Kasi: One hundred percent. And I think this is really important. When I grew up and when I became a teacher, I think that was an era when there was a lot of focus on math anxiety, the prevalence of math anxiety. Sheila Tobias wrote the famous book Overcoming Math Anxiety. This was especially a problem among women. There were dozens of books. And there were a number of problems with that work at the time, and that most of the research people were citing was taking place outside of math education. The work was all really before the field of neuroscience was actually a thing. Lots of deficit thinking that something is wrong with the person who is suffering this anxiety. And most of these books were very self-helpy. And so, not only is there something wrong with you, but you need to fix it yourself. So, it really centers all these negative emotions around math on the person that's experiencing the pain, that something's wrong with them. 

Whereas math trauma really shifts the focus to say, “No, no, no. This reaction, this emotional reaction, nobody's born that way.” Right? This came from a place, from an experience. And so, math trauma is saying, “No, there's been some series of events, maybe a set of circumstances, that this individual began to see as harmful or threatening, and that it's having long-lasting adverse effects. And that those long-lasting effects, this kind of triggering that starts to happen, is really beginning to affect that person's functioning, their sense of well-being when they're in the presence, in this case, of mathematics.” And I think the thing about trauma is just that. And I have to say in the early days of my doing this research, I was honestly a little bit hesitant to use that word because I didn't want to devalue some of the horrific experiences that people have experienced in times of war, witnessing the murder of a parent or something.

But it's about the brain. It's how the brain is responding to the situation. And what I think we know now, even more than when I started this work, is that there is simply trauma [in] everyday life. There are things that we experience that cause our brains to be triggered. And math is unfortunately this subject in school that we require nearly every year of a young person's life. And there are things about the way it's been taught over time that can be humiliating, ridiculing; that can cause people to have just some really negative experiences that then they carry with them into the next year. And so that's really the shift. The shift is instead of labeling somebody as math anxious—“Oh, you poor thing, you better fix yourself”—it's like, “No, we have some prevalence of math trauma, and we've got to figure out how people's experiences with math are causing this kind of a reaction in their bodies and brains.”

Mike: I want to take this a little bit further before we start to talk about causes and solutions. This idea that you mentioned of feeling under threat, it made me think that when we're talking about trauma, we are talking about a physiological response. Something is happening within the brain that's being manifested in the body. And I wonder if you could talk just a little bit about what happens to people experiencing trauma? What does that feel like in their body?

Kasi: So, this is really important and our brains have evolved over time. We have this incredible processing capacity, and it's coupled with a very powerful filter called the amygdala. And the amygdala [has been] there from eons ago to protect us. It's the filter that says, “Hey, do not provide access to that powerful processor unless I'm safe, unless my needs are met. Otherwise, I gotta focus on being well over here.” So, we're not going to give access to that higher-order thinking unless we're safe. And this is really important because modern imaging has given us really new insights into how we learn and how our body is reacting when our brain gets fired in this way. 

And so, when somebody is experiencing math trauma, you know it. They sweat. Their face turns red. They cry. Their body and brain are telling them, “Get out. Get away from this thing. It will hurt you.” And I just feel like that is so important for us to remember because the amygdala also becomes increasingly sensitive to repeat negativity. So, it's one thing that you have a bad day in math, or you maybe have a teacher that makes you feel not great about yourself. But day after day, week after week, year after year, that messaging can start to make the amygdala hypersensitive to these sorts of situations. Is that what you were getting at with your question?

Mike: It is. And I think you really hit on something. There's this idea of repeat negativity causing increased sensitivity, I think has real ramifications for classroom culture or the importance of the way that I show up as an educator. It's making me think a lot about culture and norms related to math in schools. I'm starting to wonder about the type of traumatizing traditions that we've had in math education that might contribute to this type of experience. What does that make you think?

Kasi: Oh, for sure. Unfortunately, I think the list is a little long of the things that we may have been doing completely inadvertently. Everybody wants their students to have a great experience, and I actually think our practices have evolved. But culturally, I think there are some things about math that contribute to these “traumatizing traditions,” is what I've called them. 

Before we go there, I do want to say just one other thing about this trauma piece, and that is that we've learned about some things about trauma in childhood. And a lot of the trauma in childhood is about not a single life-altering event. But childhood trauma is often about these things that happened repeatedly where a child was being ridiculed, being treated cruelly. And it's about that repetition that is really seeding that trauma so deeply and that sense that they can't stop it, that they don't have control to stop the thing that is causing them pain or suffering. So, I just wanted to make sure that I tagged that because I think there is something about what we've learned about the different forms of childhood trauma that's especially salient in this situation. 

And so, I'll tie it to your question, which is, think about some of the things we've done in math historically. We don't do them in every place, but the ability grouping that has happened over time, it seems to go in and out of fashion. When a kid is told they're in the lower class, “Oh, this is something you're not good [at]—the slower math.” We often use speed to measure understanding, and so smarter is not faster. And there's some great quotes, Einstein among them. So that's a thing. When you gotta do it right now, it has to be one-hundred-percent right. It has to be superfast. We've often prioritized individual work over collaboration. So, you're all alone in this. In fact, if you're working with others, somehow that's cheating as opposed to collaborating. 

We teach kids tricks rather than teaching them how to think. And I think we deprive kids of the opportunity to have an idea. It's really hard to get excited about something where all you're doing is reproducing—reproducing something that somebody else thought of as quickly as possible and [it] needs to be one-hundred-percent [accurate]. You don't get to bring your own spin to it. And so, we focus on answers rather than people's reasoning behind the answers. That can be something that happens as well. And I think one of the things that's always gotten me is that there's only one way. Not only is there only one right answer, but there's only one way to get there, which also contributes to this idea of having to absorb somebody else's thinking rather than actualizing your own. 

And I absolutely know that most teachers are working to not do as much of these things in their math classrooms. And I want to be sure in having this conversation that—you know, I'm a lover of education and teachers, I taught teachers for many years. This is not about the teachers so much as the sort of culture of math and math education that we were all brought up in. And we've got to figure out how to make math something more so that kids can see themselves in it. And that it's not something that happens in a vacuum and is this performance course rather than a class where you get to solve cool problems that no one knows the exact answer to, or there's the exact right way, or that you get to get your own questions answered. Things you wonder about. That it's a chance to explore. 

So, I mean, ultimately, I think we just know that there's a lot of negativity that happens around math, and we accept it. And that is perhaps the most traumatizing tradition of all because that kind of repeat negativity we know affects the amygdala. It affects people's ability to access math in the long run. So, we gotta have neutral or better.

Mike: So, in the field of psychology, there's this notion of generational trauma, and it's passed from generation to generation. And you're making me wonder if we're facing something similar when it comes to the field of math education. I'm wondering what you think educators might be able to do to reclaim math for themselves, especially if they're a person who potentially does have a traumatic mathematics experience and maybe some of the ways that they might create a different type of experience for their students.

Kasi: Yeah, let's talk about each of those. I'm going to talk about one, the multigenerational piece, and then let's talk about how we can help ourselves and our students. One is, I think it's really very possible that that's what we're looking at in terms of math trauma. Culturally, I think we've known for a while that this is happening, with respect to math, that—you know, I've had parents come to back-to-school night and tell me that they're just not a math family. And even jokingly say, “Oh, we're all bad at math, don't be too hard on us,” and all the other things. And so, kids inherit that. And it's very common for kids to have the same attitude towards math that their parents do and also that their teachers do. 

And that's where I think in my mind, I really want to help every elementary teacher fall in love with math because if we look at the data, I think of any undergraduate major, it's those who major in education who report the highest rates of math anxiety and math trauma. And so, when you think about folks who feel that way about math, then being in charge of teaching it to kids in the early years, that's a lot to carry. And so, we want to give those teachers and anyone who has had this experience with math an opportunity to reclaim, regroup. And in my experience, what I've found is actually simply shifting the location of the problem is a really strong first step. When people understand that they actually aren't broken, that the feelings that they have about math don't reflect some sort of flaw in them as a human, but that it's a result of something they've experienced, a lot is unlocked. And most folks that I have worked with over my time working on this issue, they know. They know exactly the moment. They know the set of experiences that led to the reactions that they feel in their body. They can name it, and with actually fairly startling detail. So, in my teaching—and I think this is something anybody can do—is they would write a “mathography.” What is the story of your life through a math lens? What has been the story of your relationship with math over the course of your life and what windows does that give you into the places where you might need to heal? We've never had more tools to go back and sort of relearn areas of math that we thought we couldn't learn. And so often the trauma points are as math becomes more abstract. So many people have something that happened around fractions or multidigit multiplication and division. When we started—we get letters involved in math. I had somebody say, “Math was great as long as it was numbers. Then we got letters involved, and it was terrible.”

And so, if people can locate, “This is where I had the problem. It's not me. I can go back and relearn some things.” I feel like that's a lot of the healing, and that, in fact, if I'm a teacher or if I'm a parent, I love my kids, whether they're my children or my students, and I'm going to work on me so that they have a better experience than I had. And I've found so many teachers embrace that idea and go to work. So, some of the things that can happen in classrooms that I think fall from this is that, first of all, the recognition that emotional safety, you can't have cognition and problem solving without it. If you have kids in your classroom who have had these negative experiences in math, you're going to need to help them unpack those and level set in order to move on. And “mathography” is also a good tool for that. Some people use breathing. 

Making sure that when you encounter kids that are exhibiting math anxiety, that you help them localize the problem outside of them. No one is born with math anxiety. It's the math of school that creates it. And if we ignore it, it's just going to get worse. So, some people feel like they can kind of smooth it over. I think we need to give kids the tools to unpack it and move beyond it. But it's so widespread, and I've encountered teachers who were afraid to go there. It's like the Pandora's box. My advice to them is that if you'll open the box and heal what's inside, the teaching becomes much easier. Whereas if you don't, you're fighting that uphill battle all the time. 

You know, students will feel more safe in classrooms where mistakes are opportunities to learn; where they're not a bad thing and where they see each other as resources, where they are not alone, and where they can collaborate and really take responsibility for each other's learning. So, some of the most powerful classrooms I've seen where there were a lot of kids who had very negative experiences with math, a teacher had succeeded in creating this learning environment, this community of learners where all the kids seem to recognize that somebody would have a good day, someone else would have a not good day, but it would be their turn for a good day a few days from now. [chuckles] So, we're all just going to take care of each other as we go. 

I think some things that teachers can keep a particular eye on is being sure that kids are given authentic work to do in math. It's really easy to start giving kids what we've called busywork, but work that really isn't engaging their brain. And it turns out that that boredom cycle triggers the negativity cycle, which can actually get your amygdala operating in a way that is not as far from trauma as we might all like to think. And so, while it isn't the same kind of math trauma that we're talking about here, it does affect the amygdala. And so that's something we should be aware of. And so, this is something—I think kids should learn about their brains in school. I don't know if it's the math teacher's job. But if they haven't learned about their brains yet, when you get them, I would recommend teaching kids about their brains, teaching them strategies for when they feel that kind of shutdown, that headache, like “I can't think.” Because most of the time, they actually can't. And they need to have some kind of reset. 

Another tip, just in terms of disrupting that trauma cycle in the classroom, is that by the time kids get to be third, fourth grade and up, they know who is good at math, or they've labeled each other. You know, “Who's good at math? Who's struggled?” Even if they are not tracked and sorted, they've assessed each other. Sometimes they've put those labels on themselves. And so, if a teacher has the skills to assign competence to those students that may be being labeled as low status mathematically in their classroom—and it takes a teacher that knows their students well. But if you happen to see that a student that maybe has low status with computation, but wow, they are really good at developing the visuals for a math problem, or they're really great at illustrating a story or drawing others out in a collaborative group, but finding an area of competence that's authentic. 

Sorry to go on and on. I could sit here and talk to you about this all day, but those are some of the things I would recommend.

Mike: Well, I think there's a few things that jump out, and I wanted to take them in little bits. I'm going to try to summarize, and then I want to come back and pick these up a little bit. So, one of the pieces that you named really struck a chord with me, which is recognizing as an educator that I have a story about mathematics that is playing out maybe just under the level of consciousness that bubbles up here and there. When you mentioned the traumatic experiences, my head went back to third grade with multiplication tables, and I can see myself sitting in the seat. And when you mentioned fractions, again, I could see myself facing the board in third grade looking down at a workbook where we were supposed to be adding fractions with denominators that were not common. And I had this moment of just dread in my stomach because I remember just thinking, “I don't know what is happening at all.”

And I'll say biographically, I think I spent the first seven or eight years of my teaching career carrying those things with me in the way that I approach students. I knew that they weren't good for me, but I didn't really have a compelling sense of what could be different until I actually took some mathematics education courses and really started to understand mathematics and how children's ideas develop. And it did allow me to decenter the problem for myself and say, “Actually, I can make a lot of meaning out of mathematics.” What I experienced was not mathematics. It was memorizing a bunch of stuff and practicing a bunch of procedures. This idea that decentering where the problem is from the educator or in classrooms from the student, really, really feels powerful. I think it's a huge gift that we can give to our students and also to ourselves.

The other piece that I'm really thinking about is this idea of positioning students and finding competency. That really stands out as something that I could attend to as a classroom teacher. I suspect that people who are listening can think about their own class of students. You as an educator probably know who the other kids think of as good at math, and I suspect you also know who they think isn't good at math. Knowing that kids know those stories as well, I could do something about that. I could look at the students who have low status and think about ways that I could raise them up. That feels really tangible. I could take and start thinking about that when I ask students to share their ideas, and I could do that tomorrow. It doesn't take a master's-level course in mathematics to do that. Does that make sense, Kasi?

Kasi: I love all of that so much. One hundred percent. You know, when I was observing teachers—and this tended to happen more with elementary teachers just because of their own histories with math as you were saying here—but the difference between saying, “OK, everybody, we get to do math now. Clear your desks!” and “OK, everybody, I know it's hard, but it's time for math. We're strong. We're going to do it.” But there is this underlying kind of, “I don't really like this either, but we gotta do it.” as opposed to “We're going to discover something new today!” And so really just kind of listening to some of those implicit messages in the words that we choose, that's something we can change in a moment as well.

Mike: Well, I think you and I could probably go on and on and continue this conversation for a long time. If I'm someone who's listening, are there resources you would recommend for someone who wants to continue learning about these ideas?

Kasi: Yeah, absolutely. For me, the OG of this line of thinking is Jo Boaler, who most math teachers will know. She's the first person I ever heard use the word “math-traumatized.” And before I embarked and dove deeper into my math trauma research, I went down to Stanford and met with her, and she was wonderful and encouraging of, like, “Oh, no, no, no. Go, go, go, go. This is great.” 

There's a woman named Ebony McGee, who's the founder of R-RIGHTS. [She was] a professor at Vanderbilt. She's doing some work with math identity that I think touches on this subject in a valuable way. I mean, I think this whole area of developing positive math identity is tightly connected to the math trauma work. And honestly, anyone who is doing work around child trauma and neuroscience and how we are seeing the development of the brain is going to provide some interesting resources. 

I have to say, my all-time favorite is a book that I believe [...] is out of print, so it might be a thrift books purchase. But Dr. Judy Willis wrote a book called Learning to Love Math. Looks like you might be familiar with it. And I really think she did a lovely job in that book in a way that is absolutely targeting teachers to help us see how these very small actions that we take in the classroom could make a really big difference in terms of how our students see and experience the subject that we care about so much.

Mike: I think that's a great place to stop. Thank you so much for joining us, Kasi. It's really been a pleasure talking with you.

Kasi: Oh, my goodness, Mike, thank you so much. It's really been an honor to be here. Thanks for having me.

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

© 2025 The Math Learning Center | www.mathlearningcenter.org

Dr. Rebecca Ambrose, Helping Our Students Build a Meaningful Understanding of Geometry   ROUNDING UP: SEASON 3 | EPISODE 8

As a field, mathematics education has come a long way over the past few years in describing the ways students come to understand number, quantity, place value, and even fractions. But when it comes to geometry, particularly concepts involving shape, it’s often less clear how student thinking develops. Today, we’re talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry.

BIOGRAPHIES

Rebecca Ambrose researches how children solve mathematics problems and works with teachers to apply what she has learned about the informal strategies children employ to differentiate and improve instruction in math. She is currently a professor at the University of California, Davis in the School of Education.

RESOURCES

Geometry Resources Curated by Dr. Ambrose

Seeing What Others Cannot See

Opening the Mind's Eye 

TRANSCRIPT

Mike Wallus: As a field, mathematics education has come a long way over the past few years in describing the ways that students come to understand number, place value, and even fractions. But when it comes to geometry, especially concepts involving shape, it's often less clear how student thinking develops. Today, we're talking with Dr. Rebecca Ambrose about ways we can help our students build a meaningful understanding of geometry. 

Well, welcome to the podcast, Rebecca. Thank you so much for joining us today.

Rebecca Ambrose: It's nice to be here. I appreciate the invitation.

Mike: So, I'd like to start by asking: What led you to focus your work on the ways that students build a meaningful understanding of geometry, particularly shape?

Rebecca: So, I taught middle school math for 10 years. And the first seven years were in coed classrooms. And I was always struck by especially the girls who were actually very successful in math, but they would tell me, “I like you, Ms. Ambrose, but I don't like math. I'm not going to continue to pursue it.” And I found that troubling, and I also found it troubling that they were not as involved in class discussion. And I went for three years and taught at an all-girls school so I could see what difference it made. And we did have more student voice in those classrooms, but I still had some very successful students who told me the same thing. So, I was really concerned that we were doing something wrong and that led me to graduate school with a focus on gender issues in math education. And I had the blessing of studying with Elizabeth Fennema, who was really the pioneer in studying gender issues in math education.

And as I started studying with her, I learned that the one area that females tended to underperform males on aptitude tests—not achievement tests, but aptitude tests—was in the area of spatial reasoning. And you'll remember those are the tests, or items that you may have had where you have one view of a shape and then you have a choice of four other views, and you have to choose the one that is the same shape from a different view. And those particular tasks we see consistent gender differences on. I became convinced it was because we didn't give kids enough opportunity to engage in that kind of activity at school. You either had some strengths there or not, and because of the play activity of boys, that may be why some of them are more successful at that than others. 

And then the other thing that informed that was when I was teaching middle school, and I did do a few spatial activities, kids would emerge with talents that I was unaware of. So, I remember in particular this [student,] Stacy, who was an eighth-grader who was kind of a good worker and was able to learn along with the rest of the class, but she didn't stand out as particularly interested or gifted in mathematics. And yet, when we started doing these spatial tasks, and I pulled out my spatial puzzles, she was all over it. And she was doing things much more quickly than I could. And I said, “Stacy, wow.” She said, “Oh, I love this stuff, and I do it at home.” And she wasn't the kind of kid to ever draw attention to herself, but when I saw, “Oh, this is a side of Stacy that I didn't know about, and it is very pertinent to mathematics. And she needs to know what doorways could be open to her that would employ these skills that she has and also to help her shine in front of her classmates.” So, that made me really curious about what we could do to provide kids with more opportunities like that little piece that I gave her and her classmates back in the day. So, that's what led me to look at geometry thinking. And the more that I have had my opportunities to dabble with teachers and kids, people have a real appetite for it. There are always a couple of people who go, “Ooh.” But many more who are just so eager to do something in addition to number that we can call mathematics.

Mike: You know, I'm thinking about our conversation before we set up and started to record the formal podcast today. And during that conversation you asked me a question that involved kites, and I'm wondering if you might ask that question again for our listeners.

Rebecca: I'm going to invite you to do a mental challenge. And the way you think about it might be quite revealing to how you engage in both geometric and spatial reasoning. So, I invite you to picture in your mind's eye a kite and then to describe to me what you're seeing.

Mike: So, I see two equilateral triangles that are joined at their bases—although as I say the word “bases,” I realize that could also lead to some follow-up questions. And then I see one wooden line that bisects those two triangles from top to bottom and another wooden line that bisects them along what I would call their bases.

Rebecca: OK, I'm trying to imagine with you. So, you have two equilateral triangles that—a different way of saying it might be they share a side?

Mike: They do share a side. Yes.

Rebecca: OK. And then tell me again about these wooden parts.

Mike: So, when I think about the kite, I imagine that there is a point at the top of the kite and a point at the bottom of the kite. And there's a wooden piece that runs from the point at the top down to the point at the bottom. And it cuts right through the middle. So, essentially, if you were thinking about the two triangles forming something that looked like a diamond, there would be a line that cut right from the top to the bottom point.

Rebecca: OK.

Mike: And then, likewise, there would be another wooden piece running from the point on one side to the point on the other side. So essentially, the triangles would be cut in half, but then there would also be a piece of wood that would essentially separate each triangle from the other along the two sides that they shared.

Rebecca: OK. One thing that I noticed was you used a lot of mathematical ideas, and we don't always see that in children. And I hope that the listeners engaged in that activity themselves and maybe even stopped for a moment to sort of picture it before they started trying to process what you said so that they would just kind of play with this challenge of taking what you're seeing in your mind's eye and trying to articulate in words what that looks like. And that's a whole mathematical task in and of itself. And the way that you engaged in it was from a fairly high level of mathematics. 

And so, one of the things that I hope that task sort of illustrates is how a.) geometry involves these images that we have. And that we are often having to develop that concept image, this way of imagining it in our visual domain, in our brain. And almost everybody has it. And some people call it “the mind's eye.” Three percent of the population apparently don't have it—but the fact that 97 percent do suggests for teachers that they can depend on almost every child being able to at least close their eyes and picture that kite. I was strategic in choosing the kite rather than asking you to picture a rectangle or a hexagon or something like that because the kite is a mathematical idea that some mathematicians talk about, but it's also this real-world thing that we have some experiences with. 

And so, one of the things that that particular exercise does is highlight how we have these prototypes, these single images that we associate with particular words. And that's our starting point for instruction with children, for helping them to build up their mathematical ideas about these shapes. Having a mental image and then describing the mental image is where we put language to these math ideas. And the prototypes can be very helpful, but sometimes, especially for young children, when they believe that a triangle is an equilateral triangle that's sitting on, you know, the horizontal—one side is basically its base, the word that you used—they've got that mental picture. But that is not associated with any other triangles. So, if something looks more or less like that prototype, they'll say, “Yeah, that's a triangle.” But when we start showing them some things that are very different from that, but that mathematicians would call triangles, they're not always successful at recognizing those as triangles. And then if we also show them something that has curved sides or a jagged side but has that nice 60-degree angle on the top, they'll say, “Oh yeah, that's close enough to my prototype that we'll call that a triangle.” 

So, part of what we are doing when we are engaging kids in these conversations is helping them to attend to the precision that mathematicians always use. And that's one of our standards. And as I've done more work with talking to kids about these geometric shapes, I realize it's about helping them to be very clear about when they are referring to something, what it is they're referring to. So, I listen very carefully to, “Are they saying ‘this’ and ‘that’ and pointing to something?” That communicates their idea, but it would be more precise as like, I have to ask you to repeat what you were telling me so that I knew exactly what you were talking about. And in this domain, where we don't have access to a picture to point to, we have to be more precise. And that's part of this geometric learning that we're trying to advance.

Mike: So, this is bringing a lot of questions for me. The first one that I want to unpack is, you talked about the idea that when we're accessing the mind's eye, there's potentially a prototype of a shape that we see in our mind's eye. Tell me more about what you mean when you say “a prototype.”

Rebecca: The way that that word is used more generally, as often when people are designing something, they build a prototype. So, it's sort of the iconic image that goes with a particular idea.

Mike: You're making me think about when I was teaching kindergarten and first grade, we had colored pattern blocks that we use quite often. And often when we talked about triangles, what the students would describe or what I believed was the prototype in their mind's eye really matched up with that. So, they saw the green equilateral triangle. And when we said trapezoid, it looked like the red trapezoid, right? And so, what you're making me think about is the extent to which having a prototype is useful, but if you only have one prototype, it might also be limiting.

Rebecca: Exactly. And when we're talking to a 3- or a 4-year-old, and we're pointing to something and saying, “That's a triangle,” they don't know what aspect of it makes it a triangle. So, does it have to be green? Does it have to be that particular size? So, we’ll both understand each other when we're talking about that pattern block. But when we're looking at something that's much different, they may not know what aspect of it is making me call it a triangle” And they may experience a lot of dissonance if I'm telling them that—I'm trying to think of a non-equilateral triangle that we might all, “Oh, well, let’s”—and I'm thinking of 3-D shapes, like an ice cream cone. Well, that's got a triangular-ish shape, but it's not a triangle. But if we can imagine that sort of is isosceles triangle with two long sides and a shorter side, if I start calling that a triangle or if I show a child that kind of isosceles triangle and I say, “Oh, what's that?” And they say, “I don't know.”

So, we have to help them come to terms with that dissonance that's going to come from me calling something a triangle that they're not familiar with calling a triangle. And sadly, that moment of dissonance from which Piaget tells us learning occurs, doesn't happen enough in the elementary school classroom. Kids are often given equilateral triangles or maybe a right triangle. But they're not often seeing that unusual triangle that I described. So, they're not bumping into that dissonance that'll help them to work through, “Well, what makes something a triangle? What counts and what doesn't count?” And that's where the geometry part comes in that goes beyond just spatial visualization and using your mind's eye, but actually applying these properties and figuring out when do they apply and when do they not apply.

Mike: I think this is probably a good place to shift and ask you: What do we know as a field about how students' ideas about shape initially emerge and how they mature over time?

Rebecca: Well, that's an interesting question because we have our theory about how they would develop under the excellent teaching conditions, and we haven't had very many opportunities to confirm that theory because geometry is so overlooked in the elementary school classroom. So, I'm going to theorize about how they develop based on my own experience and my reading of the literature on very specific examples of trying to teach kids about squares and rectangles. Or, in my case, trying to see how they describe three-dimensional shapes that they may have built from polydrons. So, their thinking tends to start at a very visual level. And like in the kite example, they might say, “It looks like a diamond”—and you actually said that at one point—but not go farther from there. 

So, you decomposed your kite, and you decomposed it a lot. You said it has two equilateral triangles and then it has those—mathematicians would call [them] diagonals. So, you were skipping several levels in doing that. So, I'll give you the intermediate levels using that kite example. So, one thing a child might say is that “I'm seeing two short sides and two long sides.” So, in that case, they're starting to decompose the kite into component parts. And as we help them to learn about those component parts, they might say, “Oh, it's got a couple of different angles.” And again, that's a different thing to pay attention to. That's a component part that would be the beginning of them doing what Battista called spatial structuring. Michael Battista built on the van Hiele levels to try to capture this theory about how kids’ thinking might develop. So, attention to component parts is the first place that we see them making some advances. 

And then the next is if they're able to talk about relationships between those component parts. So, in the case of the kite, they might say, “Oh, the two short sides are equal to each other”—so, there's a relationship there—“and they're connected to each other at the top.” And I think you said something about that. “And then the long sides are also connected to each other.” And that's looking at how the sides are related to the other sides is where the component parts start getting to become a new part. So, it's like decomposing and recomposing, which is part of all of mathematics. 

And then the last stage is when they're able to put the shapes themselves into the hierarchy that we have. So, for example, in the kite case, they might say, “It's got four sides, so it's a quadrilateral. But it's not a parallelogram because none of the four sides are parallel to each other.” So now I'm not just looking at component parts and their relations, but I'm using those relations to think about the definition of that shape. So, I would never expect a kid to be able to tell me, “Oh yeah, a kite is a quadrilateral that is not a parallelogram,” and then tell me about the angles and tell me about the sides without a lot of experience describing shapes.

Mike: There are a few things that are popping out for me when I'm listening to you talk about this. One of them is the real importance of language and attempting to use language to build a meaningful description or to make sense of shape. The other piece that it really makes me think about is the prototypes, as you described them, are a useful starting place. They’re something to build on. 

But there's real importance in showing a wide variety of shapes or even “almost-shapes.” I can imagine a triangle that is a triangle in every respect except for the fact that it's not a closed shape. Maybe there's an opening or a triangle that has wavy sides that are connected at three points. Or an obtuse triangle. Being able to see multiple examples and nonexamples feels like a really important part of helping kids actually find the language but also get to the essence of, “What is a triangle?” Tell me if I'm on point or off base when I'm thinking about that, Rebecca.

Rebecca: You are right on target. And in fact, Clements and Sarama wrote a piece in the NCTM Teaching Children Mathematics in about 2000 where they describe their study that found exactly what you said. And they make a recommendation that kids do have opportunities to see all kinds of examples. And one way that that can happen is if they're using dynamic geometry software. So, for example, Polypad, I was just playing with it, and you can create a three-sided figure and then drag around one of the points and see all these different triangles. And the class could have a discussion about, “Are all of these triangles? Well, that looks like a weird triangle. I've never seen that before.” And today I was just playing around with the idea of having kids create a favorite triangle in Polypad and then make copies of it and compose new shapes out of their favorite triangle.

What I like about that task, and I think can be a design principle for a teacher who wants to play around with these ideas and get creative with them, is to give kids opportunities to use their creativity in making new kinds of shapes and having a sense of ownership over those creations. And then using those creations as a topic of conversation for other kids. So, they have to treat their classmates as contributors to their mathematics learning, and they're all getting an opportunity to have kind of an aesthetic experience. I think that's the beauty of geometry. It's using a different part of our brain. Thomas West talks about Seeing What Others Cannot See, and he describes people like Einstein and others who really solved problems visually. They didn't use numbers. They used pictures. And Ian Robertson talks about Opening the Mind's Eye. So, his work is more focused on how we all could benefit from being able to visualize things. And actually, our fallback might be to engage our mind's eye instead of always wanting to talk [chuckles] about things. 

That brings us back to this language idea. And I think language is very important. But maybe we need to stretch it to communication. I want to engage kids in sharing with me what they notice and what they see, but it may be embodied as much as it is verbal. So, we might use our arms and our elbow to discuss angle. And well, we'll put words to it. We're also then experiencing it in our body and showing it to each other in a different way than [...] just the words and the pictures on the paper. So, people are just beginning to explore this idea of gesture. But I have seen, I worked with a teacher who was working with first graders and they were—you say, “Show us a right angle,” and they would show it to us on their body.

Mike: Wow. I mean, this is so far from the way that I initially understood my job when I was teaching geometry, which was: I was going to teach the definition, and kids were going to remember that definition and look at the prototypical shape and say, “That's a triangle” or “That's a square.” Even this last bit that you were talking about really flips that whole idea on its head, right? It makes me think that teaching the definitions before kids engage with shapes is actually having it backwards. How would you think about the way that kids come to make meaning about what defines any given shape? If you were to imagine a process for a teacher helping to build a sense of triangle-ness, talk about that if you wouldn't mind.

Rebecca: Well, so I'm going to draw on a 3-D example for this, and it's actually something that I worked with a teacher in a third grade classroom, and we had a lot of English language learners in this classroom. And we had been building polyhedra, which are just three-dimensional shapes using a tool called the polydrons. And our first activities, the kids had just made their own polyhedra and described them. So, we didn't tell them what a prism was. We didn't tell them what a pyramid was or a cube. Another shape they tend to build with those tools is something called an anti-prism, but we didn't introduce any of those terms to them. They were familiar with the terms triangle and square, and those are within the collection of tools they have to work with. But it was interesting to me that their experience with those words was so limited that they often confused those two. And I attributed it to all they'd had was maybe a few lessons every year where they were asked to identify, “Which of these are triangles?” They had never even spoken that word themselves. So, that's to have this classroom where you are hearing from the kids and getting them to communicate with each other and the teacher as much as possible. I think that's part of our mantra for everything.

But we took what they built. So, they had all built something, and it was a polyhedra. That was the thing we described. We said it has to be closed. So, we did provide them with that definition. You have to build a closed figure with these shapes, and it needs to be three-dimensional. It can't be flat. So, then we had this collection of shapes, and in this case, I was the arbiter. And I started with, “Oh wow, this is really cool. It's a pyramid.” And I just picked an example of a pyramid, and it was the triangular pyramid, made out of four equilateral triangles. And then I pulled another shape that they had built that was obviously not any—I think it was a cube. And I said, “Well, what do you think? Is this a pyramid?” And they'd said, “No, that's not a pyramid.” “OK, why isn't it?” And by the way, they did know something about pyramids. They'd heard the word before. And every time I do this with a class where I say, “OK, tell me, ‘What's a pyramid?’” They'll tell me that it's from Egypt. It's really big. So, they're drawing on the Egyptian pyramids that they're familiar with. Some of them might say a little something mathematical, but usually it's more about the pyramids they've seen maybe in movies or in school. 

So, they're drawing on that concept image, right? But they don't have any kind of mathematical definition. They don't know the component parts of a pyramid. So, after we say that the cube is not a pyramid, and I say, “Well, why isn't it?,” they'll say, “because it doesn't have a pointy top.” So, we can see there that they're still drawing on the concept image that they have, which is valid and helpful in this case, but it's not real defined. So, we have attention to a component part. That's the first step we hope that they'll make. And we're still going to talk about which of these shapes are pyramids. So, we continued to bring in shapes, and they ended up with, it needed to have triangular sides. Because we had some things that had pointy tops, but it wasn't where triangles met. It would be an edge where there were two sloped sides that were meeting there. Let's see. If you can imagine, while I engage your mind's eye again, a prism, basically a triangular prism with two equilateral triangles on each end, and then rectangles that attach those two triangles.

Mike: I can see that.

Rebecca: OK. So, usually you see that sitting on a triangle, and we call the triangles the base. But if you tilt it so it's sitting on a rectangle, now you've got something that looks like a tent. And the kids will say that. “That looks like a tent.” “OK, yeah, that looks like a tent.” And so, that's giving us that Level 1 thinking: “What does it look like?” “What's the word that comes to mind?” And—but we've got those sloped sides, and so when they see that, some of them will call that the pointy top because we haven't defined pointy top.

Mike: Yes.

Rebecca: But when I give them the feedback, “Oh, you know what, that's not a pyramid.” Then the class started talking about, “Hmm, OK. What's different about that top versus this other top?” And so, then they came to, “Well, it has to be where triangles meet.” I could have introduced the word vertex at that time. I could have said, “Well, we call any place where sides meet a vertex.” That might be [a] helpful word for us today. But that's where the word comes from what they're doing, rather than me just arbitrarily saying, “Today I'm going to teach you about vertices. You need to know about vertices.” But we need a word for this place where the sides meet. So, I can introduce that word, and we can be more precise now in what we're talking about. So, the tent thing didn't have a vertex on top. It had an edge on top. So now we could be precise about that.

Mike: I want to go back, and I'm going to restate the thing that you said for people who are listening, because to me, it was huge. This whole idea of “the word comes from the things that they are doing or that they are saying.” Did I get that right?

Rebecca: Yeah, that the precise terminology grows out of the conversation you're having and helps people to be clear about what they're referring to. Because even if they're just pointing at it, that's helpful. And especially for students whose first language might not be English, then they at least have a reference. That's why it's so hard for me to be doing geometry with you just verbally. I don't even have a picture or a thing to refer to. But then when I say “vertex” and we're pointing to this thing, I have to try as much as I can to help them distinguish between, “This one is a vertex. This one is not a vertex.”

Mike: You brought up earlier supporting multilingual learners, particularly given the way that you just modeled what was a really rich back-and-forth conversation where children were making comparisons. They were using language that was very informal, and then the things that they were saying and doing led to introducing some of those more precise pieces of language. How does that look when you have a group of students who might have a diverse set of languages that they're speaking in the same classroom?

Rebecca: Well, when we do this in that environment, which is most of the time when I'm doing this, we do a lot of pair-share. And I like to let kids talk to the people that they communicate best with so that if you have two Spanish speakers, for example, they could speak in Spanish to each other. And ideally the classroom norms have been established so that that's OK. But that opportunity to hear it again from a peer helps them to process. And it slows things down. Like, often we're just going so fast that people get lost. And it may be a language thing; it may be a concept thing. So, whatever we can do to slow things down and let kids hear it repeatedly—because we know that that repeated input is very helpful—and from various different people.

So, what I'll often do, if I want everybody to have an opportunity to hear about the vertex, I'm going to invite the kids to retell what they understood from what I said. And then that gives me an opportunity to assess those individuals who are doing the retell and also gives the other students a chance to hear it again. It's OK for them to see or hear the kind of textbook explanation for vertex in their preferred language.

But again, only when the class has been kind of grappling with the idea, it's not the starting point. It emerges as needed in that heat of instruction. And you don't expect them to necessarily get it the first time around. That's why these building tasks or construction tasks can be done at different levels. So, we were talking about the different levels the learner might be at. Everybody can imagine a kite, and everybody could draw a kite. So, I'm sort of differentiating my instruction by giving this very open-ended task, and then I'm trying to tune into what am I seeing and hearing from the different individuals that can give me some insight into their geometrical reasoning at this point in time. But we're going to keep drawing things, and we're going to keep building things, and everybody's going to have their opportunity to advance. But it's not in unison.

Mike: A few things jumped out. One, as you were describing the experiences that you can give to students, particularly students who might have a diversity of languages in the same classroom, it strikes me that this is where nonverbal communication like gesturing or using a visual or using a physical model really comes in handy. 

I think the other piece that I was reminded of as I was listening to you is, we have made some progress in suggesting that it's really important to listen to kids' mathematical thinking. And I often think that that's taken root, particularly as kids are doing things like adding or subtracting. And I think what you're reminding [me] is, that holds true when it comes to thinking about geometry or shape; that it's in listening to what kids are saying, that they're helping us understand, “What's next?” “Where do we introduce language?” “How can we have kids speaking to one another in a way that builds a set of ideas?” 

I think the big takeaway for me is that sometimes geometry has kind of been treated like this separate entity in the world of elementary mathematics. And yet some of the principles that we find really important in things like number or operation, they still hold true.

Rebecca: Definitely, definitely. And again, as I said, when you are interested in getting to know your children, seeing who's got some gifts in this domain will allow you to uplift kids who might otherwise not have those opportunities to shine.

Mike: I think that's a great place to stop. Rebecca, thank you so much for joining us. It's been a pleasure talking to you.

Rebecca: This has really been fun. And I do want to mention one thing: that I have developed a list of various articles and resources. Most of them come from NCTM, and I can make that available to you so that people who are interested in learning more can get some more resources.

Mike: That's fantastic. We'll link those to our show notes. Thank you again very much for helping us make sense of this really important set of concepts.

Rebecca: You're welcome.

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

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

Dr. James Brickwedde, How You Say It Matters: Teacher Language Choices That Support Number Sense ROUNDING UP: SEASON 3 | EPISODE 7

Carry the 1. Add a 0. Cross multiply.

All of these are phrases that educators heard when they were growing up. This language is so ingrained that many educators use it without even thinking. But what’s the long-term impact of language like this on the development of our students’ number sense? Today, we’re talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense.

BIOGRAPHIES

James Brickwedde is the director of the Project for Elementary Mathematics. He served on the faculty of Hamline University’s School of Education & Leadership from 2011–2021, supporting teacher candidates in their content and pedagogy coursework in elementary mathematics.

RESOURCES

The Project for Elementary Mathematics

TRANSCRIPT

Mike Wallus: Carry the 1, add a 0, cross multiply. All of these are phrases that educators heard when they were growing up. This language is so ingrained, we often use it without even thinking. But what's the long-term impact of language like this on our students’ number sense? Today we're talking with Dr. James Brickwedde about the impact of language and the ways educators can use it to cultivate their students’ number sense. 

Welcome to the podcast, James. I'm excited to be talking with you today.

James Brickwedde: Glad to be here.

Mike: Well, I want to start with something that you said as we were preparing for this podcast. You described how an educator’s language can play a critical role in helping students think in value rather than digits. And I'm wondering if you can start by explaining what you mean when you say that.

James: Well, thinking first of primary students—so, kindergarten, second grade, that age bracket—kindergartners, in particular, come to school thinking that numbers are just piles of ones. They're trying to figure out the standard order. They're trying to figure out cardinality. There are a lot of those initial counting principles that lead to strong number sense that they are trying to integrate neurologically. And so, one of the goals of kindergarten, first grade, and above is to build the solid quantity sense—number sense—of how one number is relative to the next number in terms of its size, magnitude, et cetera. And then as you get beyond 10 and you start dealing with the place value components that are inherent behind our multidigit numbers, it's important for teachers to really think carefully of the language that they're using so that, neurologically, students are connecting the value that goes with the quantities that they're after. So, helping the brain to understand that 23 can be thought of not only as that pile of ones, but I can decompose it into a pile of 20 ones and three ones, and eventually that 20 can be organized into two groups of 10. And so, using manipulatives, tracking your language so that when somebody asks, “How do I write 23?” it's not a 2 and a 3 that you put together, which is what a lot of young children think is happening. But rather, they realize that there's the 20 and the 3.

Mike: So, you're making me think about the words in the number sequence that we use to describe quantities. And I wonder about the types of tasks or the language that can help children build a meaningful understanding of whole numbers, like say, 11 or 23.

James: The English language is not as kind to our learners [laughs] as other languages around the world are when it comes to multidigit numbers. We have in English 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And when we get beyond 10, we have this unique word called “eleven” and another unique word called “twelve.” And so, they really are words capturing collections of ones really then capturing any sort of tens and ones relationship.

There's been a lot of wonderful documentation around the Chinese-based languages. So, that would be Chinese, Japanese, Korean, Vietnamese, Hmong follows the similar language patterns, where when they get after 10, it literally translates as “10, 1,” “10, 2.” When they get to 20, it's “2, 10”—”2, 10, 1,” “2, 10, 2.” And so, the place value language is inherent in the words that they are saying to describe the quantities. The teen numbers, when you get to 13, a lot of young children try to write 13 as “3, 1” because they're trying to follow the language patterns of other numbers where you start left to right. And so, they're bringing meaning to something, which of course is not the social convention. So, the teens are all screwed up in terms of English. 

Spanish does begin to do some regularizing when they get to 16 because of the name “diez y seis,” so “ten, six.” But prior to that you have, again, sort of more unique names that either don't follow the order of how you write the number or they're unique like 11 and 12 is. 

Somali is another interesting language in that—and I apologize to anybody who is fluent in that language because I'm hoping I'm going to articulate it correctly—I believe that there, when they get into the teens, it's “1 and 10,” “2 and 10,” is the literal translation. So, while it may not be the “10, 1” sort of order, it still is giving … the fact that there's ten-ness there as you go. 

So, for the classrooms that I have been in and out of—both [in] my own classroom years ago as well as the ones I still go in and out of now—I try to encourage teachers to tap the language assets that are among their students so that they can use them to think about the English numbers, the English language, that can help them wire that brain so that the various representations—the manipulatives, expanded notation cards or dice, the numbers that I write, how I break the numbers apart, say that 23 is equal to 20 plus 3—all of those models that you're using, and the language that you use to back it up with, is consistent so that, neurologically, those pathways are deeply organized. 

Piaget, in his learning theory, talks about young children—this is sort of the 10 years and younger—can only really think about one attribute at a time. So that if you start operating on multidigit numbers, and I'm using digitized language, I'm asking that kindergartner, first [grader], second grader to think of two things at the same time. I'm, say, moving a 1 while I also mean 10. What you find, therefore, is when I start scratching the surface of kids who were really procedural-bound, that they really are not reflecting on the values of how they've decomposed the numbers or are reconfiguring the numbers. They're just doing digit manipulation. They may be getting a correct answer, they may be very fast with it, but they've lost track of what values they're tracking. There's been a lot of research on kids’ development of multidigit operations, and it's inherent in that research about students following—the students who are more fluid with it talk in values rather than in digits. And that's the piece that has always caught my attention as a teacher and helped transform how I talked with kids with it. And now as a professional development supporter of teachers, I'm trying to encourage them to incorporate in their practice.

Mike: So, I want to hang on to this theme that we're starting to talk about. I'm thinking a lot about the very digit-based language that as a child I learned for adding and subtracting multidigit numbers. So, phrases like, “Carry the 1” or “Borrow something from the 6.” Those were really commonplace. And in many ways, they were tied to this standard algorithm, where a number was stacked on top of another number. And they really obscured the meaning of addition and subtraction. 

I wonder if we can walk through what it might sound like or what other models might draw out some of the value-based language that we want to model for kids and also that we want kids to eventually adopt when they're operating on numbers.

James: A task that I give adults, whether they are parents that I’m out doing a family math night with or my teacher candidates that I have worked with, I have them just build 54 and 38, say, with base ten blocks. And then I say, “How would you quickly add them?” And invariably everybody grabs the tens before they move to the ones. Now your upbringing, my upbringing is the same and still in many classrooms: Students are directed only to start with the ones place. And if you get a new 10, you have to borrow and you have to do all of this exchange kinds of things. 

But the research shows when school gets out of the way [chuckles] and students and adults are operating on more of their natural number sense, people start with the larger and then move to the smaller. And this has been found around the world. This is not just unique to US classrooms that have been working this way. If, in the standard algorithms—which really grew out of accounting procedures that needed to save space in ledger books out of the 18th, 19th centuries—they are efficient, space-saving means to be able to accurately compute. But in today's world, technology takes over a lot of that bookkeeping type of thing. An analogy I like to make is, in today's world, Bob Cratchit out of [A] Christmas Carol, Charles Dickens’s character, doesn't have a job because technology has taken over everything that he was in charge of. So, in order for Bob Cratchit to have a job, [laughs] he does need to know how to compute. But he really needs to think in values. 

So, what I try to encourage educators to loosen up their practice is to say, “If I'm adding 54 plus 38, so if you keep those two numbers in your mind, [chuckles] if I start with the ones and I add 4 and 8, I can get 12.” There's no reason, if I'm working in a vertical format, to not put 12 fully under the line down below, particularly when kids are first learning how to add. But then language-wise, when they go to the tens place, they're adding 50 and 30 to get 80, and the 80 goes under the 12. Now, many teachers will know that's partial sums. That's not the standard algorithm. That is the standard algorithm. The difference between the shortcut of carrying digits is only a space-saving version of partial sums. Once you go to partial sums in a formatting piece, and you're having kids watch their language—and that's a phrase I use constantly in my classrooms—is, it's not a 5 and 3 that you are working with, it's a 50 and a 30. So when you move to the language of value, you allow kids to initially, at least, get well-grounded in the partial sums formatting of their work, the algebra of the connectivity property pops out, the number sense of how I am building the quantities, how I'm adding another 10 to the 80, and then the 2, all of that begins to more fully fall into place.

There are some of the longitudinal studies that have come out that students who were using more of the partial sums approach for addition, their place value knowledge fell into place sooner than the students who only did the standard algorithm and used the digitized language. So, I don't mind if a student starts in the ones place, but I want them to watch their language. So, if they're going to put down a 2, they're not carrying a 1—because I'll challenge them on that—is “What did you do to the 12 to just isolate the 2? What's left?” “Oh, you have a 10 up there and the 10 plus the 50 plus the 30 gives me 90.” So, the internal script that they are verbalizing is different than the internal digitized script that you and I and many students still learn today in classrooms around the country. So, that's where the language and the values and the number sense all begin to gel together. And when you get to subtraction, there's a whole other set of language things. So, when I taught first grade and a student would say, “Well, you can't take 8 from 4,” if I still use that 54 and 38 numbers as a reference here, my challenge to them is, “Who said?”

Now, my students are in Minnesota. So, Minnesota is at a cultural advantage of knowing what happens in wintertime when temperatures drop below 0. [laughs] And so, I usually have as a representation model in my room, a number line that’s swept around the edges of the room, that started from negative 35 and went to 185. And so, there are kids who've been puzzling about those other numbers on the other side of 0. And so, somebody pops up and says, “Well, you'll get a negative number.” “What do you mean?” And then they whip around and start pointing at that number line and being able to say, “Well, if you're at 4 and you count back 8, you'll be at negative 4.” So, I am not expecting first graders to be able to master the idea of negative integers, but I want them to know the door is open. And there are some students in late first grade and certainly in second grade who start using partial differences where they begin to consciously use … the idea of negative integers. 

However, there [are] other students, given that same scenario, who think going into the negative numbers is too much of The Twilight Zone. [laughs] They'll say, “Well, I have 4 and I need 8. I don't have enough to take 8 from 4.” And another phrase I ask them is, “Well, what are you short?” And that actually brings us back to the accounting reference point of sort of debit-credit language of, “I'm short 4.” “Well, if you're short 4, we’ll just write ‘minus 4.’” But if they already have subtracted 30 from 50 and have 20, then the question becomes, “Where are you going to get that 4 from?” “Well, you have 20 cookies sitting on that plate there. I'm going to get that 4 out of the 20.” So again, the language around some of these strategies in subtractions shifts kids to think with alternative strategies and algorithms compared to the American standard algorithm that predominates US education.

Mike: I think what's interesting about what you just said too is you're making me think about an article. I believe it was “Rules That Expire.” And what strikes me is that this whole notion that you can't take 8 away from 4 is actually a rule that expires once kids do begin to work in integers. And what you're suggesting about subtraction is, “Let's not do that. Let's use language to help them make meaning of, “Well, what if?” As a former Minnesotan, I can definitely validate that when it's 4 degrees outside and the temperature drops 8 degrees, kids can look at a thermometer and that context helps them understand. I suppose if you're a person listening to this in Southern California or Arizona, that might feel a little bit odd. But I would say that I have seen first graders do the same thing.

James: And if you are more international travelers, as soon as, say, people in Southern California or southern Arizona step across into Mexico, everything is in Celsius. If those of us in the northern plains go into Canada, everything is in Celsius. And so, you see negative numbers sooner [laughs] than we do in Fahrenheit, but that's another story.

Mike: This is a place where I want to talk a little bit about multiplication, particularly this idea of multiplying by 10. Because I personally learned a fairly procedural understanding of what it is to multiply by 10 or 100 or 1,000. And the language of “add a 0” was the language that was my internal script. And for a long time when I was teaching, that was the language that I passed along. You're making me wonder how we could actually help kids build a more meaningful understanding of multiplying by 10 or multiplying by powers of 10.

James: I have spent a lot of time with my own research as well as working with teachers about what is practical in the classroom, in terms of their approach to this. First of all, and I've alluded to this earlier, when you start talking in values, et cetera, and allow multiple strategies to emerge with students, the underlying algebraic properties, the properties of operations begin to come to the surface. So, one of the properties is the zero property, [laughs], right? What happens when you add a number to 0 or a 0 to a number? I'm now going to shift more towards a third-grade scenario here. When a student needs to multiply four groups of 30. “I want 30 four times,” if you're using the times language. And they'd say, “Well, I know 3 times 4 is 12 and then I just add a 0.” And that's where I as a teacher reply, “Well, I thought 12 plus 0 is still 12. How could you make it 120?” And they’d say, “Well, because I put it there.” So, I begin to try to create some cognitive dissonance [laughs] over what they're trying to describe, and I do stop and say this to kids: “I see that you recognize a pattern that's happening there. But I want us to explore, and I want you to describe why does that pattern work mathematically?” 

So, with addition and subtraction, kids learn that they need to decompose the numbers to work on them more readily and efficiently. Same thing when it comes to multiplication. I have to decompose the numbers somehow. So if, for the moment, you come back to, if you can visualize the numbers four groups of 36. Kids would say, “Well, yeah, I have to decompose the 36 into 30 plus 6.” But by them now exploring how to multiply four groups of 30 without being additive and just adding above, which is an early stage to it. But as they become more abstract and thinking more in multiples, I want them to explore the fact that they are decomposing the 30 into factors.

Now, factors isn't necessarily a third-grade standard, right? But I want students to understand that that's how they are breaking that number apart. So, I'm left with 4 times 3 times 10. And if they've explored, in this case, the associate of property of multiplication, “Oh, I did that. So, I want to do 4 times 3 because that's easy. I know that. But now I have 12 times 10.” And how can you justify what 12 times 10 is? And that's where students who are starting to move in this place quickly say, “Well, I know 10 tens are 100 and 2 tens are 20, so it's 120.” They can explain it. The explanation sometimes comes longer than the fact that they are able to calculate it in their heads, but the pathway to understanding why it should be in the hundreds is because I have a 10 times a ten there.

So that when the numbers now begin to increase to a double digit times a double digit—so now let's make it 42 groups of 36. And I now am faced with, first of all, estimating how large might my number be? If I've gotten students grounded in being able to pull out the factors of 10, I know that I have a double digit times a double digit, I have a factor of 10, a factor of 10. My answer's going to be in the hundreds. How high in the hundreds? In this case with the 42 and 36, 1,200. Because if I grab the largest partial product, then I know my answer is at least above 1,200 or one thousand, two hundred. Again, this is a language issue. It's breaking things into factors of 10 so that the powers of 10 are operated on. So that when I get deeper into fourth grade, and it's a two digit times a three digit, I know that I'm going to have a ten times a hundred. So, my answer's at least going to be up in the thousands. I can grab that information and use it both from an estimation point of view, but also, strategically, to multiply the first partial product or however you are decomposing the number. Because you don't have to always break everything down into their place value components. That's another story and requires a visual [laughs] work to explain that. 

But going back to your question, the “add the 0,” or as I have heard, some teachers say, “Just append the 0,” they think that that's going to solve the mathematical issue. No, that doesn't. That's still masking why the pattern works. So, bringing students back to the factors of 10 anchors them into why a number should be in the hundreds or in the thousands.

Mike: What occurs to me is what started as a conversation where we were talking about the importance of speaking in value really revealed the extent to which speaking in value creates an opportunity for kids to really engage with some of the properties and the big ideas that are going to be critical for them when they get to middle school and high school. And they're really thinking algebraically as opposed to just about arithmetic.

James: Yes. And one of the ways I try to empower elementary teachers is to begin to look at elementary arithmetic through the lens of algebra rather than the strict accounting procedures that sort of emerge. Yes, the accounting procedures are useful. They can be efficient. I can come to use them. But if I've got the algebraic foundation underneath it, when I get to middle school, it is—my foundation allows for generative growth rather than a house of cards that collapses, and I become frustrated. And where we see the national data in middle school, there tends to be a real separation between who [is] able to go on and who gets stuck. Because as you mentioned before, the article … “Rules That Expire,” too many of them expire when you have to start thinking in rates, ratios, proportionality, et cetera.

Mike: So, for those of you who are listening who want to follow along, we do have a visual aid that's attached to the show notes that has the mathematics that James is talking about. I think that's a great place to stop. 

Thank you so much for joining us, James, it has really been a pleasure talking with you.

James: Well, thanks a lot, Mike. It was great talking to you as well.

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

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

Summer Pettigrew and Megan Williams, Building Asset-Focused Professional Learning Communities ROUNDING UP: SEASON 3 | EPISODE 5

Professional learning communities have been around for a long time, in many different iterations. But what does it look like to schedule and structure professional learning communities that help educators understand and respond to their students' thinking in meaningful ways? Today we’re talking with Summer Pettigrew and Megan Williams from the Charleston County Public Schools about building asset-focused professional learning communities.

BIOGRAPHIES

Summer Pettigrew serves as an instructional coach at Springfield Elementary School in Charleston, South Carolina. 

Megan Williams serves as principal at Springfield Elementary School in Charleston, South Carolina.

RESOURCES

OGAP website

TRANSCRIPT

Mike Wallus: Professional learning communities have been around for a long time and in many different iterations. But what does it look like to schedule and structure professional learning communities that actually help educators understand and respond to their students’ thinking in meaningful ways? Today we're talking with Summer Pettigrew and Megan Williams from the Charleston Public Schools about building asset-focused professional learning communities. 

Hello, Summer and Megan. Welcome to the podcast. I am excited to be talking with you all today about PLCs.

Megan Williams: Hi!

Summer Pettigrew: Thanks for having us. We're excited to be here.

Mike: I'd like to start this conversation in a very practical place: scheduling. So, Megan, I wonder if you could talk just a bit about when and how you schedule PLCs at your building.

Megan: Sure. I think it's a great place to start too, because I think without the structure of PLCs in place, you can't really have fabulous PLC meetings. And so, we used to do our PLC meetings once a week during teacher planning periods, and the teachers were having to give up their planning period during the day to come to the PLC meeting. And so, we created a master schedule that gives an hour for PLC each morning. So, we meet with one grade level a day, and then the teachers still have their regular planning period throughout the day. 

So, we were able to do that by building a time for clubs in the schedule. So, first thing in the morning, depending on your day, so if it's Monday and that's third grade, then the related arts teachers—and that for us is art, music, P.E., guidance, our special areas—they go to the third-grade teachers’ classrooms. The teachers are released to go to PLC, and then the students choose a club. And so, those range from basketball to gardening to fashion to [STEM]. We've had Spanish Club before. So, they participate with the related arts teacher in their chosen club, and then the teachers go to their PLC meeting. And then once that hour is up, then the teachers come back to class. The related arts teachers are released to go get ready for their day. So, everybody still has their planning period, per se, throughout the day.

Mike: I think that feels really important, and I just want to linger a little bit longer on it. One of the things that stands out is that you're preserving the planning time on a regular basis. They have that, and they have PLC time in addition to it.

Megan: Mm-hmm, correct. And that I think is key because planning time in the middle of the day is critical for making copies, calling parents, calling your doctor to schedule an appointment, using the restroom—those kind of things that people have to do throughout the day. And so, when you have PLC during their planning time, one or the other is not occurring. Either a teacher is not taking care of those things that need to be taken care of on the planning period or they're not engaged in the PLC because they're worried about something else that they've got to do. So, building that time in, it's just like a game changer.

Mike: Summer, as a person who’s playing the role of an instructional coach, what impact do you think this way of scheduling has had on educators who are participating in the PLCs that you're facilitating?

Summer: Well, it's huge. I have experienced going to a PLC on our planning [period] and just not being one-hundred-percent engaged. And so, I think having the opportunity to provide the time and the space for that during the school day allows the teachers to be more present. And I think that the rate at which we're growing as a staff is expedited because we're able to drill into what we need to drill into without worrying about all the other things that need to happen. So, I think that the scheduling piece has been one of the biggest reasons we've been so successful with our PLCs.

Mike: Yeah, I can totally relate to that experience of feeling like I want to be here, present in this moment, and I have 15 things that I need to do to get ready for the next chunk of my day. So, taking away that “if-then,” and instead having an “and” when it comes to PLCs, really just feels like a game changer.

Megan: And we were worried at first about the instructional time that was going to be lost from the classroom doing the PLC like this. We really were because we needed to make sure instructional time was maximized and we weren't losing any time. And so, this really was about an hour a week, right, where the teachers aren't directly instructing the kids. 

But it has not been anything negative at all. Our scores have gone up, our teachers have grown. They love—the kids love going to their clubs. I mean, even the attendance on the grade-level club day is so much better because they love coming in. They start the day really getting that SEL instruction. I mean, that's really a lot of what they're getting in clubs. They're hanging out with each other. They're doing something they love.

Mike: Maybe this is a good place to shift and talk a little bit about the structure of the PLCs that are happening. So, I've heard you say that PLCs, as they're designed and functioning right now, they're not for planning; they're instead for teacher collaboration. So, what does that mean?

Megan: Well, there's a significant amount of planning that does happen in PLC, but it's not a teacher writing his or her lesson plans for the upcoming week. So, there's planning, but not necessarily specific lesson planning, like, “On Monday I'm doing this; on Tuesday I'm doing this.” It's more looking at the standards, looking at the important skills that are being taught, discussing with each other ways that you do this. “How can I help kids that are struggling? How can I push kids that are higher?” So, teachers are collaborating and planning, but they're not really producing written lesson plans.

Mike: Yeah. One of the pieces that you all talked about when we were getting ready for this interview, was this idea that you always start your PLCs with a recognition of the celebrations that are happening in classrooms. I'm wondering if you can talk about what that looks like and the impact it has on the PLCs and the educators who are a part of them.

Summer: Yeah. I think our teachers are doing some great things in their classrooms, and I think having the time to share those great things with their colleagues is really important. Just starting the meeting on that positive note tends to lead us in a more productive direction.

Mike: You two have also talked to me about the impact of having an opportunity for educators to engage in the math that their students will be doing or looking at common examples of student work and how it shows up in the classroom. I wonder if you could talk about what you see in classrooms and how you think that loops back into the experiences that are happening in PLCs.

Summer: Yeah. One of the things that we start off with in our PLCs is looking at student work. And so, teachers are bringing common work examples to the table, and we're looking to see, “What are our students coming with? What's a good starting point for us to build skills, to develop these skills a little bit further to help them be more successful?” And I think a huge part of that is actually doing the work that our students are doing. And so, prior to giving a task to a student, we all saw that together in a couple of different ways. And that's going to give us that opportunity to think about what misconceptions might show up, what questions we might want to ask if we want to push students further, reign them back in a little bit. Just that pre-planning piece with the student math, I think has been very important for us. 

And so, when we go into classrooms, I'll smile because they kind of look like little miniature PLCs going on. The teacher’s facilitating, the students are looking at strategies of their classmates and having conversations about what's similar, what's different. I think the teachers are modeling with their students that productive practice of looking at the evidence and the student work and talking about how we go about thinking through these problems.

Mike: I think the more that I hear you talk about that, I flash back to, Megan, what you said earlier about [how] there is planning that's happening, and there's collaboration. They're planning the questions that they might ask. They're anticipating the things that might come from students. So, while it's not, “I'm writing my lesson for Tuesday,” there is a lot of planning that's coming. It’s just perhaps not as specific as, “This is what we'll do on this particular day.” Am I getting that right?

Megan: Yes. You're getting that one-hundred-percent right. Summer has teachers sometimes [take] the assessment at the beginning of a unit. We'll go ahead and take the end-of-unit assessment and the information that you gain from that, just with having the teachers take it and knowing how the kids are going to be assessed, then just in turn makes them better planners for the unit. And there's a lot of good conversation that comes from that.

Mike: I mean, in some ways, your PLC design, the word that pops into my head is almost like a “rehearsal” of sorts. Does that analogy seem right?

Meghan: It seems right.

Summer: And just to add on to that, I think too again, providing that time within the school day for them to look at the math, to do the math, to think about what they want to ask, is like a mini rehearsal. Because typically, when teachers are planning outside of school hours, it's by themselves in a silo. But this just gives that opportunity to talk about all the possibilities together, run through the math together, ask questions if they have them. So, I think that's a decent analogy, yeah.

Mike: Yeah. Well, you know what it makes me think about is competitive sports like basketball. As a person who played quite a lot, there are points in time when you start to learn the game that everything feels so fast. And then there are points in time when you've had some experience when you know how to anticipate, where things seem to slow down a little bit. And the analogy is that if you can kind of anticipate what might happen or the meaning of the math that kids are showing you, it gives you a little bit more space in the moment to really think about what you want to do versus just feeling like you have to react.

Summer: And I think, too, it keeps you focused on the math at hand. You're constantly thinking about your next teacher move. And so, if you've got that math in your mind and you do get thrown off, you've had an opportunity, like you said, to have a little informal rehearsal with it, and maybe you're not thrown off as badly. [laughs]

Mike: Well, one of the things that you’ve both mentioned when we've talked about PLCs is the impact of a program called OGAP. I'm wondering if you can talk about what OGAP is, what it brought to your educators, and how it impacted what’s been happening in PLCs.

Megan: I'll start. In terms—OGAP stands for “Ongoing Assessment Project”. Summer can talk about the specifics, but we rolled it out as a whole school. And I think there was power in that: everybody in your school taking the same professional development at the same time, speaking the same language, hearing the same things. And for us, it was just a game changer.

Summer: Yeah, I taught elementary math for 12 years before I knew anything about OGAP, and I had no idea what I was doing until OGAP came into my life. All of the light bulbs that went off with this very complex elementary math that I had no idea was a thing, it was just incredible. 

And so, I think the way that OGAP plays a role in PLCs is that we're constantly using the evidence in our student work to make decisions about what we do next. We're not just plowing through a curriculum, we're looking at the visual models and strategies that Bridges expects of us in that unit. We're coupling it with the content knowledge that we get from OGAP and how students should and could move along this progression. And we're planning really carefully around that, thinking about, “If we give this task and some of our students are still at a less sophisticated strategy and some of our students are at a more sophisticated strategy, how can we use those two examples to bridge that gap for more kids?” And we're really learning from each other's work. It's not the teacher up there saying, “This is how you'd solve this problem.” But it's a really deep dive into the content. And I think the level of confidence that OGAP has brought our teachers as they've learned to teach Bridges has been like a powerhouse for us.

Mike: Talk a little bit about the confidence that you see from your teachers who have had an OGAP experience and who are now using a curriculum and implementing it. Can you say more about that?

Summer: Yeah. I mean, I think about our PLCs, the collaborative part of it, we're having truly professional conversations. It's centered around the math, truly, and how students think about the math. And so again, not to diminish the need to strategically lesson plan and come up with activities and things, but we're talking really complex stuff in PLCs. And so, when we look at student work and we sort that work on the OGAP progression, depending on what skill we're teaching that week. We're able to really look at, “Gosh, the kid is, he's doing this, but I'm not sure why.” And then we can talk a little bit about, “Well, maybe he's thinking about this strategy, and he got confused with that part of it.” 

So, it really, again, is just centered around the student thinking. The evidence is in front of us, and we use that to plan accordingly. And I think it just one-ups a typical PLC because our teachers know what they're talking about. There's no question in, “Why am I teaching how to add on an open number line?” We know the reasoning behind it. We know what comes before that. We know what comes after that, and we know the importance of why we're doing it right now.

Mike: Megan, I wanted to ask you one more question. You are the instructional leader for the building, the position you hold is principal. I know that Summer is a person who does facilitation of the PLCs. What role do you play or what role do you try to play in PLCs as well?

Megan: I try to be present at every single PLC meeting and an active participant. I do all the assessments. I get excited when Summer says, “We're taking a test.” I mean, I do everything that the teachers do. I offer suggestions if I think that I have something valuable to bring to the table. I look at student work. I just do everything with everybody because I like being part of that team.

Mike: What impact do you think that that has on the educators who are in the PLC?

Megan: I mean, I think it makes teachers feel that their time is valuable. We're valuing their time. It's helpful for me too. When I go into classrooms, I know what I'm looking for. I know which kids I want to work with. Sometimes I'm like, “Ooh, I want to come in and see you do that. That's exciting.” It helps me plan my day, and it helps me know what's going on in the school. And I think it also is just a nonjudgmental, nonconfrontational time for people to ask me questions. I mean, it's part of me trying to be accessible as well.

Mike: Summer, as the person who’s the facilitator, how do you think about preparing for the kind of PLCs that you've described? What are some of the things that are important to know as a facilitator or to do in preparation?

Summer: So, I typically sort of rehearse myself, if you will, before the PLC kicks off. I will take assessments, I will take screeners. I'll look at screener implementation guides and think about the pieces of that that would be useful for our teachers if they needed to pull some small groups and reengage those kids prior to a unit. 

What I really think is important though, is that vertical alignment. So, looking at the standards that are coming up in a module, thinking about what came before it: “What does that standard look like in second grade?” if I'm doing a third grade PLC. “What does that standard look like in fourth grade?” Because teachers don't have time to do that on their own, and I think it's really important for that collective efficacy, like, “We're all doing this together. What you did last year matters. What you're doing next year matters, and this is how they tie together.”

I kind of started that actually this year, wanting to know more myself about how these standards align to each other and how we can think about Bridges as a ladder among grade levels. Because we were going into classrooms, and teachers were seeing older grade levels doing something that they developed, and that was super exciting for them. And so, having an understanding of how our state standards align in that way just helps them to understand the importance of what they're doing and bring about that efficacy that we all really just need our teachers to own. It's so huge. And just making sure that our students are going to the next grade prepared.

Mike: One of the things that I was thinking about as I was listening to you two describe the different facets of this system that you've put together is how to get started. Everything from scheduling to structure to professional learning. There's a lot that goes into making what you all have built successful. I think my question to you all would be, “If someone were listening to this, and they were thinking to themselves, ‘Wow, that's fascinating!’ What are some of the things that you might encourage them to do if they wanted to start to take up some of the ideas that you shared?”

Megan: It's very easy to crash and burn by trying to take on too much. And so, I think if you have a long-range plan and an end goal, you need to try to break it into chunks. Just making small changes and doing those small changes consistently. And once they become routine practices, then taking on something new.

Mike: Summer, how about you?

Summer: Yeah, I think as an instructional coach, one of the things that I learned through OGAP is that our student work is personal. And if we're looking at student work without the mindset of, “We're learning together,” sometimes we can feel a little bit attacked. And so, one of the first things that we did when we were rolling this out and learning how to analyze student work is we looked at student work that wasn't necessarily from our class. We asked teachers to save student work samples. I have folders in my office of different student work samples that we can practice sorting and have conversations about. And that's sort of where we started with it. Looking at work that wasn't necessarily our students’ gave us an opportunity to be a little bit more open about what we wanted to say about it, how we wanted to talk about it. And it really does take some practice to dig into student thinking and figure out, “Where do I need to go from here?” And I think that allowed us to play with it in a way that wasn't threatening necessarily.

Mike: I think that's a great place to stop, Megan and Summer. I want to thank you so much for joining us. It's really been a pleasure talking to both of you.

Megan: Well, thank you for having us. 

Summer: Yeah, thanks a lot for having us.

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

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

Drs. Jody Guarino and Chepina Rumsey, Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades ROUNDING UP: SEASON 3 | EPISODE 6

Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergartners, first-, or second graders. What would it look like to encourage these practices with our youngest learners? Today, we’ll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2.

BIOGRAPHIES

Chepina Rumsey, PhD, is an associate professor of mathematics education at the University of Northern Iowa (UNI). 

Jody Guarino is currently a mathematics coordinator at the Orange County Department of Education and a lecturer at the University of California, Irvine. 

RESOURCES

Nurturing Math Curiosity with Learners in Grades K–2

Nurturing Math Curiosity on X/Twitter

Tools to support K–2 students in mathematical argumentation. Teaching Children Mathematics, 25(4), 208–217.

TRANSCRIPT

Mike Wallus: Argumentation, justification, conjecture. All of these are practices we hope to cultivate, but they may not be practices we associate with kindergarten, first-, and second graders. What would it look like to encourage these practices with our youngest learners? Today, we'll talk about this question with Jody Guarino and Chepina Rumsey, authors of the book Nurturing Math Curiosity with Learners in Grades K–2. 

Welcome to the podcast, Chepina and Jody. Thank you so much for joining us today. 

Jody Guarino: Thank you for having us. 

Chepina Rumsey: Yeah, thank you. 

Mike: So, I'm wondering if we can start by talking about the genesis of your work, particularly for students in grades K–2. 

Jody: Sure. Chepina had written a paper about argumentation, and her paper was situated in a fourth grade class. At the time, I read the article and was so inspired, and I wanted to use it in an upcoming professional learning that I was going to be doing. And I got some pushback with people saying, “Well, how is this relevant to K–2 teachers?” And it really hit me that there was this belief that K–2 students couldn't engage in argumentation. Like, “OK, this paper's great for older kids, but we're not really sure about the young students.” And at the time, there wasn't a lot written on argumentation in primary grades. So, we thought, “Well, let's try some things and really think about, ‘What does it look like in primary grades?’ And let's find some people to learn with.” 

So, I approached some of my recent graduates from my teacher ed program who were working in primary classrooms and a principal that employed quite a few of them with this idea of, “Could we learn some things together? Could we come and work with your teachers and work with you and just kind of get a sense of what could students do in kindergarten to second grade?” So, we worked with three amazing teachers—Bethany, Rachael, and Christina—in their first years of teaching, and we worked with them monthly for two years. We wanted to learn, “What does it look like in K–2 classrooms?” And each time we met with them, we would learn more and get more and more excited. Little kids are brilliant, but also their teachers were brilliant, taking risks and trying things. I met with one of the teachers last week, and the original students that were part of the book that we've written now are actually in high school. So, it was just such a great learning opportunity for us. 

Mike: Well, I'll say this, there are many things that I appreciated about the book, about Nurturing Math Curiosity with Learners in Grades K–2, and I think one of the first things was the word “with” that was found in the title. So why “with” learners? What were y'all trying to communicate? 

Chepina: I'm so glad you asked that, Mike, because that was something really important to us when we were coming up with the title and the theme of the book, the message. So, we think it's really important to nurture curiosity with our students, meaning we can't expect to grow it in them if we're not also growing it in ourselves. So, we see that children are naturally curious and bring these ideas to the classroom. So, the word “with” was important because we want everyone in the classroom to grow more curious together. So, teachers nurturing their own math curiosity along with their students is important to us. 

One unique opportunity we tried to include in the book is for teachers who are reading it to have opportunities to think about the math and have spaces in the book where they can write their own responses and think deeply along with the vignettes to show them that this is something they can carry to their classroom. 

Mike: I love that. 

I wonder if we could talk a little bit about the meaning and the importance of argumentation? In the book, you describe four layers: noticing and wondering, conjecture, justification, and extending ideas. Could you share a brief explanation of those layers? 

Jody: Absolutely. So, as we started working with teachers, we'd noticed these themes or trends across, or within, all of the classrooms. So, we think about noticing and wondering as a space for students to make observations and ask curious questions. So, as teachers would do whatever activity or do games, they would always ask kids, “What are you noticing?” So, it really gave kids opportunities to just pause and observe things, which then led to questions as well. 

And when we think about students conjecturing, we think about when they make general statements about observations. So, an example of this could be a child who notices that 3 plus 7 is 10 and 7 plus 3 is 10. So, the child might think, “Oh wait, the order of the addends doesn't matter when adding. And maybe that would even work with other numbers.” So, forming a conjecture like, “This is what I believe to be true.”

The next phase is justification, where a student can explain either verbally or with writing or with tools to prove the conjecture. So, in the case of the example that I brought up, 3 plus 7 and 7 plus 3, maybe a student even uses their fingers, where they're saying, “Oh, I have these 3 fingers and these 7 fingers, and whichever fingers I look at first, or whichever number I start with, it doesn't matter. The sum is going to be the same.” So, they would justify in ways like that. I've seen students use counters, just explaining it. Oftentimes, they use language and hand motions and all kinds of things to try to prove what they're saying works. Or sometimes they'll find, just, really look for, “Can I find an example where that doesn't work?” So, just testing their conjecture would be justifying. 

And then the final stage, extending ideas, could be extending that idea to all numbers. So, in the idea of addition in the commutative property, and they come to discover that they might realize, “Wait a minute, it also works for 1 plus 9 and 9 plus 1.” They could also think, “Does it work for other operations? So, not just with addition, but maybe I can subtract like that too. Does that make a difference if I'm subtracting 5 take away 2 versus 2 take away 5.” So, just this idea of, “Now [that] I've made sense of something, what else does it work with or how can I extend that thinking?” 

Mike: So, the question that I was wondering about as you were talking is, “How do you think about the relationship between a conjecture and students’ justification?” 

Jody: I've seen a lot of kids—so, sometimes they make conjectures that they don't even realize are conjectures, and they're like, “Oh, wait a minute, this pattern's happening, and I think I see something.” And so often they're like, “OK, I think that every time you add two numbers together, the sum is greater than the two numbers.” And so, then this whole idea of justifying, we often ask them, “How could you convince someone that that's true?” Or, “Is that always true?” And now they actually have to take and study it and think about, “Is it true? Does it always work?” Which, Mike, in your question, often leads back to another conjecture or refining their conjecture. It's kind of this cyclical process. 

Mike: That totally makes sense. I was going to use the words “virtuous cycle,” but that absolutely helps me understand that. 

I wonder if we can go back to the language of conjecture, because that feels really important to get clear on and to both understand and start to build a picture of. So, I wonder if you could offer a definition of conjecture for someone who’s unfamiliar with the term or talk about how students understand conjecture. 

Chepina: Yeah. So, a conjecture is based on our exploration with the patterns and observations. So, through that exploration, we might have an idea that we believe to be true. We are starting to notice things and some language that students start to use—things like, “Oh, that's always going to work” or “Sometimes we can do that.” So, there starts to be this shift toward an idea that they believe is going to be true. It's often a work in progress, so it needs to be explored more in order to have evidence to justify why that's going to be true. And through that process, we can modify our conjecture, or we might have an idea, like this working idea of a conjecture, that then when we go to justify it, we realize, “Oh, it's not always true the way we thought, so we have to make a change.” So, the conjecture is something that we believe to be true, and then we try to convince other people. 

So, once we introduce that with young mathematicians, they tend to latch on to that idea that it's this really neat thing to come up with a conjecture. And so, then they often start to come up with them even when we're not asking and get excited about, “Wait, I have a conjecture about the numbers and story problems,” where that wasn't actually where the lesson was going, but then they get excited about it. And that idea that we can take our patterns and observations, create a conjecture, and have this cyclical thing that happens. We had a second grade student make what she called a “conjecture cycle.” So, she drew a circle with arrows and showed, “We can have an idea, we can test it, we can revise it, and we can keep going to create new information.” So, those are some examples of where we've seen conjectures and kids using them and getting excited and what they mean. And yeah, it's been really exciting. 

Mike: What is hitting me is that this idea of introducing conjectures and making them, it really has the potential to change the way that children understand mathematics. It has the potential to change from, “I'm seeking a particular answer” or “I'm memorizing a procedure” or “I'm doing a thing at a discrete point in time to get a discrete answer.” It feels culturally very different. It changes what we're talking about or what we're thinking about. Does that make sense to the two of you? 

Chepina: Yeah, it does. And I think it changes how they view themselves. They're mathematicians who are creating knowledge and seeking knowledge rather than memorizing facts. Part of it is we do want them to know their facts—but understand them in this deep way with the structure behind it. And so, they're creating knowledge, not just taking it in from someone else. 

Mike: I love that. 

Jody: Yeah, I think that they feel really empowered. 

Mike: That's a great pivot point. I wonder if the two of you would be willing to share a story from a K–2 classroom that could bring some of the ideas we've been talking about to life for people who are listening. 

Jody: Sure, I would love to. I got to spend a lot of time in these teachers' classrooms, and one of the days I spent in a first grade [classroom], the teacher was Rachael Gildea, and she had led a choral count with her first graders. And they were counting by [10s] but starting with 8. So, like, “8, 18, 28, 38, 48… .” And as the kids were counting, Rachael was charting. And she was charting it vertically. So, below 8 was written 18, and then 28. And she wrote it as they counted. And one of her students paused and said, “Oh, they're all going to end with 8.” And Rachael took that student's conjecture. So, a lot of other conjectures or a lot of other ideas were shared. Students were sharing things they noticed. “Oh, looking at the tens place, it's counting 1, 2, 3,” and all sorts of things. But this one particular student, who said, “They're all going to end in 8,” Racheal took that student’s—the actual wording—the language that the student had used, and she turned it into the task that the whole class then engaged in. Like, “Oh, this student thought or thinks it's always going to end in 8. That's her conjecture; how can we prove it?”

And I happened to be in her classroom the day that they tested it. And it was just a wild scene. So, students were everywhere: at tables, laying down on the carpet, standing in front of the chart, they were examining it or something kind of standing with clipboards. And there was all kinds of buzz in the classroom. And Rachael was down on the carpet with the students listening to them. And there was this group of girls, I think three of them, that sort of screamed out, “We got it!” And Rachael walked over to the girls, and I followed her, and they were using base ten blocks. And they showed her, they had 8 ones, little units, and then they had the 10 sticks. And so, one girl would say, they'd say, “8, 18, 28,” and one of the girls was adding the 10 sticks and almost had this excitement, like she discovered, I don't know, a new universe. It was so exciting. And she was like, “Well, look, you don't ever change them. You don't change the ones, you just keep adding tens.” And it was so magical because Rachael went over there and then right after that she paused the class and she's like, “Come here, everyone. Let's listen to these girls share what they discovered.” And all of the kids were sort of huddled around, and it was just magical. And they had used manipulatives, the base ten blocks, to make sense of the conjecture that came from the choral count. And I thought it was beautiful. And so, I did choral counts in my classroom and never really thought about, “OK, what's that next step beyond, like, ‘Oh, this is exciting. Great things happen with numbers.’”

Mike: What's hitting me is that there's probably a lot of value in being able to use students' conjectures as reference points for potential future lessons. I wonder if you have some ideas or if you've seen educators create something like a public space for conjectures in their classroom. 

Chepina: We've seen amazing work around conjectures with young mathematicians. In that story that Jody was telling us about Rachael, she used that lesson—she used that conjecture in the next lesson to bring it together. It fit so perfectly with the storyline for that unit, and the lesson, and where it was going to go next. 

But sometimes ideas can be really great, but they don't quite fit where the storyline is going. So, we've encouraged teachers and [have] seen this happen in the classrooms we've worked in, where they have a conjecture wall in their classroom, where ideas can be added with Post-it Notes, have a station where [there are] Post-it Notes and pencil right there. And students can go and write their idea, put their name on it, stick it to the wall. And so, conjectures that are used in the lesson can be put up there, but ones that aren't used yet could be put up there. 

And so, if there was a lesson where a great idea emerges in the middle, and it doesn't quite fit in, the teacher could say, “That's a great idea. I want to make sure we come back to it. Could you add it to the conjecture wall?” And it gives that validation that their idea is important, and we're going to come back to it instead of just shutting it down and not acknowledging it at all. So, we have them put their names on to share. It's their expertise. They have value in our classroom. They add something to our community. Everyone has something important to share. So, that public space, I think, is really important to nurture that community where everyone has something to share. And we're all learning together. We're all exploring, conjecturing.

Jody: And I've been too in those classrooms, that Chepina is referring to with conjecture walls, and kids actually will come in, they'll be doing math, and they'll go to recess or lunch and come back in and ask for a Post-it to add a conjecture like this—I don't know, one of my colleagues uses the [words] “mathematical residue.” They continue thinking about this, and their thoughts are acknowledged. And there's a space for them. 

Mike: So, as a former kindergarten/first grade teacher, I'm seeing a picture in my head. And I'm wondering if you could talk about setting the stage for this type of experience, particularly the types of questions that can draw out conjectures and encourage justification? 

Jody: Yeah. So, as we worked with teachers, we found so many rich opportunities. And now looking back, those opportunities are probably in all classrooms all the time. But I hadn't realized in my experience that I'm one step away from this. (chuckles)

So, as teachers engaged in instructional routines, like the example of choral counting I shared from Rachael's classroom, they often ask questions like: “What do you notice?,” “Why do you think that's happening?,” “Will that always happen?,” “How do you know?,” ”How can you prove it will always work?,” “How can you convince a friend?” And those questions nudge children naturally to go to that next step when we're pushing, asking an advancing question in response to something that a student said. 

Mike: You know, one of the things that occurs to me is that those questions are a little bit different even than the kinds of questions we would ask if we were trying to elicit a student's strategy or their conceptual understanding, right? In that case, it seems like we want to understand the ideas that were kind of animating a student's strategy or the ideas that they were using or even how they saw a mental model unfolding in their head. But the questions that you just described, they really do go back to this idea of generalizing, right? Is there a pattern that we can recognize that is consistently the same or that doesn't change? And it's pressing them to think about that in a way that's different even than conceptual-based questions. Does that make sense? 

Jody: It does, and it makes me think about—I believe it's Vicki Jacobs and Joan Case who do a lot of work with questioning. They ask this question too: “As a teacher, what did that child say that gave you permission to ask that question?” Where often, I want to take my question somewhere else, but really all of these questions are nudging kids in their own thinking. So, when they're sharing something, it's like, “Well, do you think that will always work?” It's still grounded in what their ideas were but sort of taking them to that next place. 

Mike: So, one of the things that I'm also wondering about is a scaffold called “language frames.” How do students or a teacher use language frames to support argumentation? 

Chepina: Yeah, I think that communication is such a big part of argumentation. And we found language frames can help support students to share their ideas by having this common language that might be different than the way they talk about other things with their friends or in other subjects. So, using the language frames as a scaffold that supports students in communicating by offering them a model for that discussion. When I've been teaching lessons, I will have the language frame written out in a space where everyone can see, and I'll use it to model my discussion. And then students will use it as they're sharing their ideas. And that's been really helpful to get language at all grade levels. 

Mike: Can you share one or two examples of a language frame that's something you would use in say, a K, 1, or 2 classroom, Chepina? 

Chepina: Yeah. We've had something like, we'll put, “I notice” and then a blank line. And so, we'll have them say, “I notice…” and then they'll fill it in or “I wonder…” or “I have a different idea.” So, helping to model how [...] you talk in a community of learners when you're sharing ideas, or if you have a different idea and you're disagreeing. So, we'll have that actually written out, and we can use it ourselves or help students to restate what they've said using that model so that then they can pick up that language. 

Mike: One of the things that stands out for me is that these experiences with argumentation and conjecture, they obviously have benefits for individual students’ conceptual understanding and for their communication. But I suspect that they also have a real benefit for the class as a collective. Can you talk about the impact that you've seen in K–2 classrooms that are thinking about argumentation and putting some of these practices into place? 

Jody: Sure. I've been really fortunate to get to spend so much time in classrooms really learning from the teachers that we worked with. And one of the things I noticed about the classrooms is the ongoing curiosity and wonder. Students were always making sense of things and investigating ideas. And the other thing that I really picked up on was how they listened to each other, which, coming from a primary background, is challenging for kids to listen to each other. But they were really attentive and attuned, and they saw themselves as problem solvers, and they thought their role was to figure things out. That's just what they do at school. 

But they thought about other kids in those ways too. “Well, let me see what other people think” or “Let me hear Chepina’s idea because maybe there's something that's useful for me.” So, they really engaged in learning, not as an isolated, sort of, “Myself as a learner” but as part of a community. The classrooms were also buzzing all the time. There was noise and movement. And the kids, the word I would say is “intellectually engaged.” So, not just engaged, like busy doing things, but really deeply thinking. 

Chepina: The other thing we've seen that has been also really exciting is the impact on the teachers as they become more curious along with the students. So, in our first group, we had the teachers, the K–2 teachers, and we saw that they started to say things when we would meet because we would meet monthly. And they would start to say things like, “I noticed this” and “I wonder if this is what my student was thinking?” So, when they were talking about their own students and their own lessons and the mathematics behind the problems, we saw teachers start to use that language and become more curious too. So, it's been really exciting to see that aspect as we work with teachers. 

Mike: So, I suspect that we have many listeners who are making sense of the ideas that you're sharing and are going to want to continue learning about argumentation and conjecture. Are there particular resources that you would recommend that might help an educator continue down this path? 

Chepina: Yeah. We are both so excited that our first book [Nurturing Math Curiosity With Learners in Grades K–2] just came out in May. And we took all the things that we had learned in this project, exploring alongside teachers, and we have more examples. There are strategies; [there are] examples of the routines that we think it's often, we kind of stop too soon. Like, “Here are some ideas of how to keep going with these instructional routines,” and we have templates to support teachers as they take those common routines further. So, we also have some links [to] our recent articles, and we have some social media pages. We can share those. 

Mike: That's fabulous. We will post all of those links and also a link to the book that you all have written. I think this is probably a great place to stop. Chepina and Jody, I want to thank you both so much for joining us. It's really been a pleasure talking with you. 

Jody: Thank you for the opportunity. It's been great to share some of the work that we've learned from classrooms, from students and teachers. 

Chepina: Yeah. Thank you, Mike. It's been so fun to talk to you. 

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

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