Show notes
ROUNDING UP: Nurturing Mathematical Curiosity: Supporting Mathematical Argumentation in the Early Grades with Drs. Jody Guarino and Chepina Rumsey 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…
Full show notes at the publisher 
Our TOPPODCAST Picks 
Stay Connected
