Dr. Juanita Silva, Building a Broader Definition of Participation

Mike Wallus, Vice President for Educator Support


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

But ask yourself, what counts as participation?

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

More Episodes



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


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

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

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

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

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

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

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

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

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

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

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

Juanita: Correct. 

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

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

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

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

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

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

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

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

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

Juanita: On that alone? (laughs)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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