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Season 3 | Episode 6 – Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades - Guests: Drs. Jody Guarino and Chepina Rumsey
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.
BIOGRAPHIESChepina 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.
RESOURCESNurturing 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.
TRANSCRIPTMike 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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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.
BIOGRAPHIESChepina 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.
RESOURCESNurturing 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.
TRANSCRIPTMike 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.
© 2024 The Math Learning Center | www.mathlearningcenter.org
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