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# Dedicated to Teachers

What are some words that you would use to describe your math classroom?

Today we’re joined by Jen Munson, Faith Kwon, and Mary Trinkle. If your classroom looks anything like theirs, you might say things like collective, trusting, and vulnerable. Sound surprising?

Their new book, The Collaborative Math Classroom, guides readers through implementing easy and approachable suggestions to launch a truly student-centered environment, where students learn not just mathematical ideas, but also how to do mathematics together. They pose and make sense of problems and experience themselves and one another as mathematical thinkers.

Below is a transcript of this episode.

Jen: This is Jen. A collaborative math classroom is a lot of things. I think that's the first thing is that there isn't just one way that it looks, unlike when I was a child. I think I was in 10 different math classrooms before I went to college, and they all looked the same. The children got bigger, but the rows stayed the same. But a collaborative math classroom is this buzzing busy space where kids are working in lots of different ways simultaneously with one another on mathematics that they're trying to figure out. It's kind of messy and noisy, and you can't always pick the teacher out because that teacher's usually crouched down next to kids or sitting on the floor somewhere rather than standing at the front by a blackboard. And kids are using lots of different materials. There's lots of different activity going on at the same time, from building things to writing things and creating charts and asking questions, and there's lots of movement and figuring out.

I've been in both faith and Mary's classrooms many times and they both look like that no matter what the day is. But on any given day, they still look a little different.

Mary: This is Mary. The thing that I would add is that students are really in this space of being curious and they are moving about the room, like Jen said, moving about the thinking process in ways that make sense for them and they're working together to do that.

Faith: This is Faith, and I would agree with everything that Jen and Mary have described, and also to say that while my collaborative math classroom was often pretty noisy, there was always quite a bit going on, I would add that it's also quite structured, and I think that if you were to ask any student or adult or person in the room, what they were doing, their response would be really purposeful like, I'm getting this tool because we are stuck on this problem and I think this thing will help us, or we're finding a more quiet spot to work, or I'm checking back on this chart because I am trying to remember a strategy that we tried yesterday. And so I would say too, that in addition to all of the wonderful chaos, that there's also a lot of structure and a lot of purpose in a collaborative math classroom.

Mary: Yes, very much facilitated by the teacher.

Steph: Mary, you just mentioned there's a lot of facilitation by the teacher. And Jen, you said something interesting that you can't always pick the teacher out in a collaborative math classroom. So what exactly is the role of a teacher in this type of classroom setting?

Mary: The collaborative math classroom has shared authority with the teachers, the teacher and the students. So the teachers are in the work with the students rather than doing the work for the students. It positions the teacher as a support for the students who are in the thinking. So students are doing the work of mathematics and collaboration together with each other, and the role of the teacher is to support them through that process and to make their thinking visible. But it is a different way of thinking about teaching, where the teacher is not the information provider, but the support of the student thinking in the room.

Jen: I would add that the primary role of the teacher shifts to facilitator, but also includes being a designer. A lot of the work that teachers do happens ahead of time when they design or intentionally choose the activity, the task that kids are going to engage in. And they set the stage by launching that activity in a really purposeful way, by arranging the furniture in ways that allow kids to collaborate, by making materials familiar and available. And so a lot of these things are just setting the stage for collaboration. And then during collaboration, they're doing the work that Mary described really to facilitate and be alongside students on their journey to learn with mathematics, through mathematics and with one another. And so it does change the way we think about the role of the teacher, about what happens before and what happens during instruction.

Steph: You just mentioned how important collaboration is. Collaboration and interaction come up so much in this book. So why is collaboration and interaction so important and what are some types of those interactions that you might see in your classrooms?

Jen: I would say first and foremost, and then I would like to have Faith and Mary narrate some examples of this, but learning is social. We learn by interacting with others, whether it's mathematics or how to fix a car. We learn by interaction, we interact with materials, with tasks and with one another. Talk does a lot of things. It allows learners to test, revise, and justify their thinking. It allows them to differentiate the learning experience for themselves, to get support from one another when they need it, to bounce ideas off of one another, to build those ideas together so that no one has to have a complete idea at the beginning, but that they can generate ideas together and in collaboration. Talk and interaction are just so important.

But we think about interaction in a really broad way. We think about it as interactions between kids, between kids and teachers, but also interactions between kids and their learning environment and between kids and mathematics. And that all of those interactions are getting layered together at the same time to create a really rich experience from which kids can learn. But it can look like a lot of different things, and that depends on what they're learning, the activity they're engaged in and how old they are.

Faith: Yeah, I think the way that I think about interactions is a lot more expansive than I think we typically think about it. Just like Jen said, the way that we frame interactions in this text is not just like how are they interacting with each other and with the teacher? But how are they interacting with the math and how are they interacting, I think really importantly, with the environment? And I think all of it is built on this understanding that we learn and we learn better when we are engaged with and interacting with others.

I'm just thinking about some concrete examples from my first grade classroom would include in terms of interacting with the environment, making decisions on where's the best place to work, really having feeling ownership over the materials and feeling like they can make those kinds of decisions for themselves and with each other about where they want to sit, what they want to work with, how that work looks, and then also with each other, how those kinds of interactions are fostered through the kind of environment that the teacher helps set up, the routines that we support and facilitate, to agree and disagree, to make decisions, to decide on a strategy.

All of those things are happening as the students are interacting with each other and with the math and with the environment that they're in.

Mary: Totally agree with what both of you have said. What I'll add on is that a collaborative math classroom is very complex. There's a lot going on. Having students be the authorities of their work, being responsible for what it means to be a learner, means that we are valuing all of the things that go into that. So just like Faith and Jen were speaking to, we value how students are interacting in the space as much as we value how they're interacting with the task and with each other. Because all of it makes the work meaningful and it supports students in internalizing the work that we're doing and supporting them to be guiding us through this process. And I think thinking about it in this way of interaction also helps teachers to think about all of the things that they should be thinking through in order for this space to be productive.

So Jen spoke about earlier, there's so much planning work and designing work that happens before students even enter the room. And by thinking about it from this perspective of interaction, it really highlights all of these things that feel unseen. You come into this space, you might not know that we spent X amount of days diving into how to get manipulatives out of the bookshelf or how to choose a seat or how to interact with paper or manipulatives. There's so much going on and by thinking about it from this perspective, I just really feel like it supports all of the folks that are in the space, students and teachers, to know how to navigate all of the complex work.

Jen: Really the mathematical side of these interactions sounds a lot like kids negotiating ideas. And some of the richest conversations I've heard in collaborative classrooms are when kids have these sort of partial suggestions, like "Maybe we could use the blocks?" And they build on the ideas.

"I think we could use the base 10 blocks."

"Yeah and then maybe we could count out the T-shirts,"

"Yeah, and then maybe we could ..."

And this sort of, and maybe we could, and maybe we could, and they layer on these ideas to create strategies with one another that they test out, that they've built together and take ownership over. Also, sometimes kids have a clear idea of what they want to do and they say, "yeah, here's what we're going to do. And the partner says, "Wait, wait, wait, what?" But those clarifying questions that they ask one another, or even revoicing back, I've seen this a lot in Faith's classroom when they say, "I think I hear you saying this," pushes kids to explain their thinking in increasingly precise ways, to make it more visible to others and then also to make it negotiable.

So it's really very interesting when you hear kids negotiating ideas, they don't do it exactly in the way that adults do. It is very much the pursuit of children, but they learn how to build ideas, clarify ideas, explain reasoning to one another, which makes it much easier for you to ask them to explain it to the class later or to participate in a fuller discussion, to share a fully formed strategy. But there's a lot of interaction that goes into getting to that fully formed strategy, getting to the answer. And so a collaborative classroom isn't just about the end product, but about every one of those negotiations along the way and all of the learning that it takes to be able to negotiate with a fellow seven-year-old about something as meaningful as place value.

Steph: I think this segues really nicely into another question I had. You identify in the book seven principles that are really at the core of a collaborative classroom. I was hoping that you could take listeners through those seven principles and how they show up in the classroom.

Faith: So the first principle is that teachers trust students and themselves. Students trust themselves, each other and the teacher. And I think there's a lot that feels circular about this principle, but I think each piece is really important, beginning with teachers, really trusting students because I think that it's absolutely critical to this notion of sharing authority and sharing the space and sharing the floor with your students, is to really trust them and trust that they can go there and trust that they can take the seeds that we're planting through the way the environment is set up, the way that we're structuring our math walk, the way that we are introducing and facilitating practice of interactional routines, that inherent in all of that is that we're giving them the floor and we're giving them really the agency and authority to take that and grow it in a way that makes sense for them and their learning and for the classroom community.

And I think also teachers trusting themselves is as important, because I think oftentimes we as teachers are positioned as the conduit of sharing information, but we're not the ones making decisions about how we teach or what we teach or what the pace with which we teach. And I think that there's a lot of forces present in our profession that implicitly message that we shouldn't really trust ourselves and trust our expertise. But what we're saying is that we do trust you. We trust that you know your students, that you trust them as mathematicians, as people, and you have that same trust in yourself that especially in those moments that feel messy.

Jen: The second principle of a collaborative math classroom is that students feel safe to bring their whole selves to doing and learning mathematics. The truth is that children have been doing mathematics since they were born. They've been noticing patterns. They've been trying to quantify things and to reason about their whole environment and make sense of it. They're sense makers. But we often define mathematics so narrowly that it's only what happens in school and often within textbooks and worksheets and within curricular programs. But mathematics is much larger than that. And students have vast repertoires of mathematical skills in their lives and they have their own experiences that can inform the way that they make sense of math and they interact around math. And rather than excluding those as not mathematical, our classrooms become more collaborative and richer when students can bring their whole mathematical selves into the space, bring all of their previous experiences, bring their curiosities, their interests and their questions into the classroom and that it's safe for them to do so.

So the principle that Faith was talking about about trust, begets this idea that you feel safe to bring your whole self and your questions into the classroom. And that mathematics doesn't become something with these tight walls around it that starts at 10:00 AM and ends at 11:00 AM but it's a highly permeable activity that includes your whole life and that it's okay to bring yourself into it, but also it's great when you bring your mathematical experience from your lives into the classroom and you wonder aloud and you use ideas about patterning from playing basketball or from cooking with your grandmother, and then that helps you with fractions. All of these kinds of things, we want them to come into the classroom because they not only help children to do mathematics, they help them to learn from one another's experiences and they make mathematics a joyful activity.

Mary: The third principle is all student voices are worthy. And this principle really positions all student thinking to be an important part of the collaborative math classroom. So in a traditional math classroom, students that are able to memorize, students that are able to solve problems quickly are the voices that we hear over and over again. And we are saying that we want to hear all student thinking, whether complete or incomplete in process, is what we're looking for. And we want to hear what students are questioning about the task or questioning about the environment or questioning about the manipulatives that they're using, that we value anything that's coming to the table and that we follow that inquiry that we are wanting to hear what students are thinking about mathematics in the classroom.

Faith: And I would say that the fourth principle goes even deeper on this notion that all student voices are worthy. So principle four is that teacher center the needs and voices of vulnerable students. And it was really important to all of us authors that this is something that we named explicitly, especially given what we know about historically and traditionally about math classrooms and math learning spaces, is that they're often spaces where hierarchies about smartness and goodness and who's brilliant, really get reinforced. And there's so much that's present in the way that we think about how students learn math or what kind of math that they get to do, that is really hierarchical, that puts a lot of doors in faces of children who really deserve the kind of expansive and exciting intellectual opportunities that we want our students to be having in those math learning spaces.

Jen: And the fifth Principle is that mathematics focuses on reasoning, sense making and mathematical ideas. And so here we really want to redefine mathematics so that it aligns with what we know from research about what mathematics really is. As Mary had said earlier, and Faith was alluding to just now, that oftentimes mathematics focused on the doing mathematics quickly, performing procedures with fidelity to something that was taught and demonstrated, and doing so quickly and accurately and being the first one to get your hand in the air. And that oftentimes that those were the same students again and again, creating patterns about who was successful and then by definition who was less successful. But that is a paltry, really anemic definition of mathematics. We want mathematics to be something that is really about thinking, that mathematics is a way of being in the world so that you can reason about problems and that we all encounter problems that can be mathematized in our daily lives, whether they're at the grocery store or they're playing with a sibling or doing a game of any kind.

Mathematics is really everywhere. And we want children to be experienced in reasoning through the problems that they encounter, both in math classrooms and in daily life. And that's what we're teaching. We're teaching kids how to reason and think and make sense.

Faith: And then the last principle is that the physical environment is designed to serve students in their work with one another and mathematics. And this also felt important to name as a principal because it's so central to what we're talking about when we're talking about building and sustaining a collaborative math classroom, that it's not just about the interactions with each other and how they're interacting with the math in isolation, but it's really how is this all happening and being supported in the physical context of the classroom? And I think it's another way to think about how are you sharing authority? How are you sharing power? How are you positioning students to really take charge and get what they need and find what they need and in order to do the kind of math work that feels exciting and that feels like they can really take the lead on it.

So I think that something like not just having a variety of manipulatives out for them so that they can choose, but thinking about are they actually accessible to them? Are they perhaps, especially for younger students, labeled in a way that they know what they're looking for and exactly how to find it? Have we as a class practiced routines for how to ask ourselves what would be a useful tool, find that tool, get what we need, and safely carry it back to our work area? And in all of those things, of course, the teacher is there as a facilitator, the teacher's the one setting up the space, but the student is the one who's able to really navigate that physical environment and make decisions about how they want to do the math work, how they want to engage with each other, and how they want to work that day.

Steph: And so for an educator who's listening, has heard all this and is like, that sounds incredible, but I don't know where to get started, what would their first steps be?

Jen: Well, I think it depends on whether you're going to start in the middle of the year or at the beginning of the year. But if you already know your students, the first thing I would do is ask yourself what your students already know about collaborating. And I wouldn't just look at math, I would look how they collaborate during reading, how they collaborate on the playground. I would look for how they solve problems together about whose turn it is during indoor rainy day recess, all those kinds of places where kids are constantly negotiating with one another. I'd look to see what they already know how to do, and then I'd begin to name those skills as things that they can use in mathematics too. And the second thing that I would do, I would think really hard about what you want their collaborations to actually look like.

Faith, and Mary and I have talked a lot about the possibility of even story boarding these. Imagine you give kids a task, pick a task that you might give your children and actually play out what the interactions might sound like. The truth is, oftentimes we don't have a clear vision of what we want collaboration to look like. So it becomes really hard to teach kids to collaborate in ways that are a little fuzzy even to us. It might be about getting things really, really concrete. Things like, well, you might need to sit side by side so that you can share a piece of paper and nobody's trying to read upside down. It might mean that you need to be looking at each other in the eyes and facing one another. It might mean that you need to figure out how the materials can go in the middle.

Even these kinds of things that we take for granted about how we might share space and materials and turn taking, this is active work for young children. And so we want to name all of those skills, some of which kids probably already know how to do in other spaces, but they may not realize they need to bring those skills into the math classroom. So I would begin by those two things I think are really critical, of figuring out what your children already know how to do and which of those skills you really want them to be using in the math classroom so that you can be as explicit as possible. If there's a space for telling in the collaborative math math classroom, it is where you tell kids the kinds of things that you want them to be doing and saying to one another when they are interacting.

And so that's the beginning of things. And then I think Faith has talked a little bit about the environment. The other thing is to think a little bit about what messages your environment is sending. If you walk into your classroom after school when there are no kids there, does it look like a space where kids can talk together and work together on mathematics? And I would really think carefully about how you could revise your classroom environment to make it a little bit easier for kids to work together. We want to make it as easy as possible in a classroom for kids to collaborate. We want that to be the natural state. So how can your environment support those things? Those are some immediate places to begin, but of course it's a long journey and there's lots to do.

Mary: I would add on to that, thinking about what you want students to be able to do, right? What does end goal collaboration look like in your classroom? Once you've assessed the environment and you've thought through the space and is it welcoming? Does it allow for collaboration? Really thinking about, okay, what do you want to see if somebody was to walk in, see and hear in your classroom? And then going back to the beginning and thinking, okay, what are the things that we need to explore as a class to build those skills? Knowing that their skills, all of the things that Jen was speaking to, and so many more things, are things that students need to practice and try and revise and in order for them to internalize. So thinking kind of big picture and then backwards mapping to what does that look like for the first six weeks of doing this, whether it's the beginning of the school year in December or in March, really allowing yourself that time to think about the process that it takes for students to try on these skills.

Faith: Yeah, I would echo everything that's been said and say too, that if the idea of getting started the way that you want to get started, feels overwhelming to you as the teacher, then that launch is definitely also going to feel overwhelming for the students. And I think that's something our book really aims to do, is to break it down into, just like Mary was giving examples of, of really possible chunks. Because by the end of the day, this collaborative math classroom, there's so many moving parts, that it makes sense for every person in the room to really understand and feel comfortable with each piece of it. So when Mary talked about the first six weeks, and I think back on how I would launch a school year, we often didn't get to novel math concepts until well after those six weeks. And sometimes the math task in the beginning of the year would just be to explore a certain manipulative.

But I would say something that wound up being very critical in my building and sustaining a collaborative math classroom and also supporting my students and really feeling comfortable and emboldened and excited by the space, was this notion of reflection and really asking ourselves questions about how did that go? How did that feel? What felt important? What do we need to ...? What should we name as a shared value or a norm that we all agreed to? Or what can we try differently tomorrow to help this time go more productively or more towards our shared goals? And I think that that practice of reflection individually as the teacher, in the moment, after that math block, but also with your students, is a really, I think, important component in building and constructing this collaborative math space that is ultimately grounded in trust, in safety and in shared authority.

Jen: I just want to amplify something that Faith just said because I just think it's so important, that engaging in reflection with your students is a kind of vulnerability as a teacher that we just often don't have. That requires a lot of mutual trust, but that's how communities grow and that's what we're trying to foster when we have a collaborative classroom, is it's not just kids collaborating with one another. It's truly you collaborating with your children to create a space where they can learn and you can learn along with them.

Jen Munson is an assistant professor of learning sciences at Northwestern University studying interactions and learning in the math classroom. She is a  former classroom teacher and professional developer who has worked with teachers and school leaders across the U.S. to develop responsive, equitable mathematics instruction.  Jen received her PhD in mathematics education from Stanford University. She is also coauthor of the Mindset Mathematics curriculum series.

Faith Kwon is a doctoral candidate at Stanford University’s Graduate School of Education, studying race, inequality, and language in elementary mathematics education. Formerly, she was a primary grades teacher, instructional coach, and district professional developer.

Mary Trinkle has been in education for more than ten years as a classroom teacher and instructional coach. She is currently the Transitional Kindergarten through fifth grade math coach for her school district, supporting all teachers in the elementary schools.