Grouping Practices That Promote Efficacy & Knowledge Transfer with Dr. Peter Liljedahl

We know from research that student collaboration can have a powerful impact on learning. That said, how we group students for collaboration matters — a lot. Today we're talking with Dr. Peter Liljedahl, author of Building Thinking Classrooms in Mathematics , about how educators can form productive, collaborative groups in their classrooms.

Peter Liljedahl is a professor of mathematics education in the Faculty of Education at Simon Fraser University and author of the best-selling book Building Thinking Classrooms in Mathematics, Grades K–12: 14 Teaching Practices for Enhancing Learning .

Building Thinking Classrooms

Mike Wallus: We know from research that student collaboration can have a powerful impact on learning. That said, how we group students for collaboration matters — a lot. Today we're talking with Dr. Peter Liljedahl, author of Building Thinking Classrooms in Mathematics, about how educators can form productive, collaborative groups in their classrooms.

Hello, Peter. Welcome to the podcast.

Peter Liljedahl: Thanks for having me.

Mike: So, to offer our listeners some background, you've written a book, called Building Thinking Classrooms in Mathematics , and I think it's fair to say that it's had a pretty profound impact on many educators. In the book, you address 14 different practices. And I'm wondering if you could weigh in on how you weigh the importance of the different practices that you addressed?

Peter: Well, OK, so, first of all, 14 is a big number that publishers don't necessarily like. When we first started talking with Corwin about this, they were very open. But I know if you think about books, if there's going to be a number in the title, the number is usually 3, 5, or 7. It's sometimes 8 — but 14 is a ridiculous number. They can't all be that valuable.

What's important about the fact that it's 14, is that 14 is the number of core practices that every teacher does. That's not to say that there aren't more or less for some teachers, but these are core routines that we all do. We all use tasks. We all create groups for collaboration. We all have the students work somewhere. We all answer questions. We do homework, we assign notes, we do formative, summative assessment. We do all of these things. We consolidate lessons. We launch lessons. These are sort of the building blocks of what makes our teaching. And through a lot of time in classrooms, I deduced this list of 14. Robert Kaplinsky, in one of his blog posts, actually said that he thinks that that list of 14 probably accounts for 95 percent of what happens in classrooms. And my research was specifically about, how do we enact each of those 14 so that we can maximize student thinking? So, what kind of tasks get students to think? How can we create groups so that more thinking happens? How can we consolidate a lesson so we get more thinking? How can we do formative and summative assessments so the students are thinking more? So, the book is about responding to those 14 core routines and the research around how to enact each of those to maximize thinking.

Your question around which one is, "How do we put weight on each of these?" They're all important. But, of course, they're not all equally impactful. Building thinking classrooms is most often recognized visually as the thing where students are standing at whiteboards working. And, of course, that had a huge impact on student engagement and thinking in the classroom: getting them from sitting and working at desks to getting them working at whiteboards. But in my opinion, it's not the most impactful. It is hugely impactful, but the one that actually makes all of thinking classroom function is how we form collaborative groups, which is Chapter 2. And it seems like that is such an inconsequential thing. We've been doing groups for forever, and we got this figured out. We know how to do this. But do we really? Do we really have it figured out? Because my research really showed that if we want to get students thinking, then the ways we've been doing it aren't working.

Mike: I think that's a great segue. And I want to take a step back, Peter. Before we talk about grouping, I want to ask what might be an obvious question. But I wonder if we can talk about the "why" behind collaboration. How would you describe the value or the potential impact of collaboration on students' learning experiences?

Peter: That's a great question. We've been doing collaborative work for decades. And by and large, we see that it is effective. We have data that shows that it's effective. And when I say "we," I don't mean me or the people I work with. I mean "we, in education," know that collaboration is important. But why? What is it about collaboration that makes it effective? There are a lot of different things. It could be as simple as it breaks the monotony of having to sit and listen. But let's get into some really powerful things that collaboration does.

Number one, about 25 years ago, we all were talking about metacognition. We know that metacognition is so powerful and so effective, and if we get students thinking about their thinking, then their thinking actually improves. And metacognition has been shown time and time again to be impactful in learning. Some of the listeners might be old enough to remember the days where we were actually trying to teach students to be metacognitive, and the frustration that that created because it is virtually impossible. Being reflective about your thinking while you're thinking is incredibly hard to do because it requires you to be both present and reflective at the same time. We're pretty good at being present, and we're pretty good about reflecting on our experiences. But to do both simultaneously is incredibly hard to do. And to teach someone to do it is difficult.

But I think we've also all had that experience where a student puts up their hand, and you start walking over to them, and just as you get there, they go, "Never mind." Or they pick up their book, and they walk over to you, and just as they get to you, they just turn around and walk back. I used to tell my students that they're smarter when they're closer to me. But what's really going on there is, as they've got their hand up, or as they're walking across the room toward you as a teacher, they're starting to formulate their thoughts to ask a question. They're preparing to externalize their thinking. And that is an incredibly metacognitive process.

One of the easiest forms of metacognition, and one of the easiest ways to access metacognition, is just to have students collaborate. Collaborating requires students to talk. It requires them to organize their thoughts. It requires them to prepare their thinking and to think about their thinking for the purposes of externalization. It is an incredibly accessible way of creating metacognition in your classroom, which we already know is effective. So, that's one reason I think collaboration is really, really vital.

Another one comes from the work on register. So, "register" is the level of sophistication with which we speak about something. So, if I'm in a classroom, and I'm talking to kindergarten students, I set a register that is accessible to them. When I talk to my undergraduates, I use a different register. My master's students, my PhD students, my colleagues, I'm using different registers. I can be talking about the same thing, but the level of sophistication with which I'm going to talk about those things varies depending on the audience. And as much as possible, we try to vary our register to suit the audience we have. But I think we've also all had that instructor who's completely incapable of varying their register, the one who just talks at you as if you're a third-year undergraduate when you're really a grade 8 student. And the ability to vary our register to a huge degree is going to define what makes us successful as a teacher. Can we meet our learners where they're at? Can we talk to them from the perspective that they're at? Now we can work at it, and very adept teachers are good at it. But even the best teachers are not as good at getting their register to be the same as students. So, this is another reason collaboration is so effective. It allows students to talk and be talked to at their register, which is the most accessible form of communication for them.

And I think the third reason that collaboration is so important is the difference between, what I talk in my book about, the difference between absolute and tentative knowledge. So, I'm going to make two statements. You tell me which one is more inviting to add a comment to. So, statement number one is, "This is how to do it, or this is what I did." That's statement number one. Statement number two is, "I think that one of the ways that we may want to try, I'm wondering if this might work." Which one is more inviting for you to contribute to?

Mike: Yes, statement number two, for many, many reasons, as I'm sitting here thinking about the impact of those two different language structures.

Peter: So, as teachers, we tend to talk in absolutes. The absolute communication doesn't give us anything to hold onto. It's not engaging. It's not inviting. It doesn't bring us into the conversation. It's got no rough patches — it's just smooth.

But when that other statement is full of hedging, it's tentative. It's got so many rough patches, so many things to contribute to, things I want to add to, maybe push back at or push further onto. And that's how students talk to each other. When you put them in collaborative groups, they talk in tentative discourse, whereas teachers, we tend to talk in absolutes. So, students are always talking to each other like that. When we put them in collaborative groups, they're like, "Well, maybe we should try this." "I'm wondering if this'll work." "Hey, have we thought about this?" "I wonder if…" Right? And it's so inviting to contribute to.

Mike: That's fascinating. I'm going to move a little bit and start to focus on grouping. So, in the book, you looked really closely at the way that we group students for collaborative problem solving and how that impacts the way students engage in a collaborative effort. And I'm wondering if you could talk a little bit about the type of things that you were examining.

Peter: OK. So, you don't have to spend a lot of time in classrooms before you see the two dominant paradigms for grouping. So, the first one we tend to see a lot at elementary school. So, that one is called "strategic grouping." Strategic grouping is where the teacher has a goal and then they're going to group their students to satisfy that goal. So, maybe my goal is to differentiate, so I'm going to make ability groups. Or maybe my goal is to increase productivity, so I'm going to make mixed-ability groups. Or maybe my goal is to just have peace and quiet, so I'm going to keep those certain students apart. Whatever my goal is, I'm going to create the groups to try to achieve that goal, recognizing that how students behave in the classroom has a lot to do with who they're partnered with. So that's strategic grouping. It is the dominant grouping paradigm we see in elementary school.

By the time we get to high school, we tend to see more of teachers going, "Work with who you want." This is called "self-selected groupings." And this is when students are given the option to group themselves any way they want. And alert: They don't group themselves for academic reasons, they group themselves for social reasons. And I think every listener can relate to both of those forms of grouping. It turns out that both of those are highly ineffective at getting students to think. And ironically, for the exact same reason. We surveyed hundreds of students who were in these types of grouping settings: strategic grouping or self-selected groupings. We asked one question: "If you knew you were going to work in groups today, what is the likelihood you would offer an idea?" That was it. And 80 percent of students said that they were unlikely or highly unlikely to offer an idea, and that was the exact same, whether they were in strategic groupings or self-selected groupings. The data cut the same.

Mike: That's amazing, Peter.

Peter: Yeah, and it's for the same reason, it turns out; that whether students were being grouped strategically or self-selected, they already knew what their role was that day. They knew what was expected of them. And for 80 percent of the students, their role is not to think. It's not to lead. Their role is to follow, right? And that's true whether they're grouping themselves socially, where they already know the social hierarchy of this group, or they're being grouped strategically. We interviewed hundreds of students. And after grade 3, every single student could tell us why they were in the group this teacher placed them in. They know . They know what you think of them. You're communicating very clearly what you think their abilities are through the way you group them, and then they live down to that expectation.

So, that's what we were seeing in classrooms was that strategic grouping may be great at keeping the peace. And self-selected grouping may be fabulous for getting students to stop whining about collaboration. But neither of them was effective for getting students to think. In fact, they were quite the opposite. They were highly ineffective for getting students to think.

Mike: So, I want to keep going with this. And I think one of the things that stood out for me as I was reading is this notion that regardless of the rationale that a teacher might have for grouping, there's almost always a mismatch between what the teacher's goals are and what the students' goals are. I wonder if you could just unpack this and maybe explain this a bit more.

Peter: So, when you do strategic grouping, do you really think the students are with the students that they want to be with? One of the things that we saw happening in elementary school was that strategic grouping is difficult. It takes a lot of effort to try to get the balance right. So, what we saw was teachers largely doing strategic grouping once a month. They would put students into a strategic group, and they would keep them in that group for the entire month. And the kids care a lot about who they're with, when you're going to be in a group for a month. And do you think they were happy with everybody that was in that group? If I'm going to be with a group of students for a month, I'd rather pick those students myself. So, they're not happy. You've created strategic groupings. And, by definition, a huge part of strategic grouping is keeping kids who want to be together away from each other. They're not happy with that.

Self-selected groupings, the students are not grouping themselves for academic reasons. They're just grouping themselves for social reasons so that they can socialize, so they talk, so they can be off topic, and all of these things. And yes, they're not complaining about group work, but they're also not being productive. So, the students are happy. But do you think the teacher's happy? Do you think the teacher looks out across that room and goes, "Yeah, there were some good choices made there." No, nobody's happy, right? If I'm grouping them strategically, that's not matching their goals. That's not matching their social goals. When they're grouping themselves in self-selected ways, that's matching their social goals but not matching my academic goals for them.

So, there's always going to be this mismatch. The teacher, more often than not, has academic goals. The students, more often than not, have social goals. There are some overlaps, right? There are students who are like, "I'm not happy with this group. I know I'm not going to do well in this group. I'm not going to be productive." And there are some teachers who are going, "I really need this student to come out of their shell, so I need to get them to socialize more." But other than that, by and large, our goals as teachers are academic in nature. The goals as students are social in nature.

Mike: I think one of the biggest takeaways from your work on grouping, for me at least, was the importance of using random groups. And I have to admit, when I read that there was a part of me, thinking back to my days as a first grade teacher, that felt a little hesitant. As I read, I came to think about that differently. But I'm wondering if you can talk about why random groups matter, the kind of impact that they have on the collaborative experience and the learning experience for kids.

Peter: All right, so going back to the previous question. So, we have this mismatch. And we have also that 80 percent of students are not thinking; 80 percent of students are entering into that group not prepared to offer an idea. So those are the two problems that we're trying to address here. So, random groups. Random wasn't good enough. It had to be visibly random. The students had to see the randomness because when we first tried it, we said, "Here's your random groups." They didn't believe we were being random. They just thought we were being strategic. So, it has to be visibly random, and it turns out it has to be frequent as well. About once every 45 to 75 minutes. See, when students are put into random groups, they don't know what their role is. So, we're solving this problem. They don't know what their role is. When we started doing visibly random groups frequently, within three weeks we were running that same survey: "If you know you're going to work in groups today, what is the likelihood you would offer an idea?" Remember the baseline data was that 80 percent of students said that they were unlikely or highly unlikely, and, all of a sudden, we have 100 percent of students saying that they're likely or highly likely. That was one thing that it solved. It shifted this idea that students were now entering groups willing to offer an idea, and that's despite 50 percent of them saying, "It probably won't lead to a solution, but I'm going to offer an idea." Now why is that? Because they don't know what their role is. So, right on the surface, what random groups does, is it shatters this idea of preconceived roles and then preconceived behaviors. So, now they enter the groups willing to offer an idea, willing to be a contributor, not thinking that their role is just to follow. But there's a time limit to this because within 45 to 75 minutes, they're going to start to fall into roles.

In that first 45 minutes, the roles are constantly negotiated. They're dynamic. So, one student is being the leader, and the others are being the follower. And now, someone else is a leader, the others are following. Now everyone is following. They need some help from some external source. Now everyone is leading. We've got to resolve that. But there is all of this [dynamism] and negotiation going on around the roles. But after 45 to 75 minutes, this sort of stabilizes, and now you have sort of a leader and followers, and that's when we need to randomize again so that the roles are dynamic and that the students aren't falling into sort of predefined patterns of nonthinking behavior.

Mike: I think this is fascinating because we've been doing some work internally at MLC around this idea of status or the way that, the stories that kids tell about one another or the labels that kids carry either from school systems or from the community that they come from, and how those things are subtle. They're unspoken, but they often play a role in classroom dynamics in who gets called on. What value kids place on a peer's idea if it is shared. What you're making me think is there's a direct line between this thing that we've been thinking about and what happens in small groups as well.

Peter: Yeah, for sure. So, you mentioned status. I want to add to that identity and self-efficacy and so on and so forth. One of the interesting pieces of data that came out of the research into random groups was, we were interviewing students several weeks into this. And we were asking them questions around this, and the students were saying things like, "Oh, the teacher thinks we're all the same, otherwise they wouldn't do random groups. The teacher thinks we're all capable, otherwise they wouldn't do random groups." So, what we're actually talking about here is that we're starting — just simply through random groups — to have a positive impact on student self-efficacy. One of the things that came out of this work, that I wrote about in a separate paper, was that we've known for a long time that student self-efficacy has a huge impact on student performance. But how do we increase, how do we improve student self-efficacy?

There are a whole bunch of different ways. The work of [Albert] Bandura on this is absolutely instrumental. But it comes down to a couple of things. From a classroom teacher perspective, the first thing, in order for a student to start on this journey from low self-efficacy to high self-efficacy, they have to encounter a teacher who believes in them. Except students don't listen to what we say. They listen to what we do. So, simply telling our students that we have confidence in them doesn't actually have much impact. It's how we show them that we have confidence in them. And it turns out that random groups actually have a huge impact on that. By doing the random groups, we're actually showing the kids that we believe in them and then they start to internalize this. So that's one thing.

The work of Bandura about how we can start to shift student self-efficacy through mastery experiences, where they start to, for example, be successful at something, and that starts to have an impact — that is amplified when students start to be successful in front of others, when they are the ones who are contributing in a small group. And that group is now successful. And that success is linked in some small or great part to your contributions. That self-efficacy is amplified because not only am I being successful, I'm being successful in a safe environment, but in front of others.

Now, self-efficacy contributes to identity, and identity has an interesting relationship with status. And you mentioned status. So, self-efficacy is what I think of myself. Status is what others think of me. I can't control my status. I can't shift my status. Status is something that is bestowed on me by others. And, of course, it's affected by their interactions with me in collaborative spaces. So, how they get to see me operate is going to create a status for me, on me, by others. But the status gets to be really nicely evenly distributed in thinking classrooms when we're doing these random groups because everybody gets to be seen as capable. They all get to be someone who can be mathematical and someone who can contribute mathematically.

Mike: I want to shift back for a moment to this idea of visibly random groups. This idea that for kids, they need to believe that it's not just a strategic grouping that I've called random for the sake of the moment. What are some of the ways that you've seen teachers visibly randomize their groups so that kids really could see the proof was right out there in front of them?

Peter: So, we first started with just cards. So, we got 27 kids. We're going to use playing cards, we're going to have three aces, three 2s, three 3s, three 4s, and so on. We would just shuffle the deck, and the kids would come and take a card. And if you're a 4, you would go to the board that has a 4 on it. Or maybe that fourth 4 is there, so to speak.

We learned a whole bunch of things. It has to be visible. And however way we do it, the randomization doesn't just tell them what group they're in, it tells them where to go. That's an efficiency thing. You don't want kids walking around the classroom looking for their partners and then spending 5 minutes deciding where they want to work. Take a card, you got a 7, you go to the 7 board. You got an ace, you go to the ace board. And that worked incredibly well. Some teachers already had Popsicle sticks in their classroom, so they started using those: Popsicle sticks with students' names. So, they would pull three Popsicle sticks and they would say, "OK, these students are together. These students are together."

At first, we didn't see any problems with that. That seemed to be pretty isomorphic to using a playing card. Some teachers got frustrated with the cards because with a card, sometimes what happens is that they get ripped or torn or they don't come back. Or they come back, and they're sweaty or they're hot. And it's like, "OK, where were you keeping this card? I don't want to know." It's hot, it's dirty. They got ink on it. The cards don't come back. The kids are swapping cards. And teachers were frustrated by this. So, they started using digital randomizers, things like Flippity and ClassDojo and Picker Wheel and Team Shake and Team Maker. There were tons of these digital randomizers, and they all work pretty much the same. But there was a bit of a concern that the students may not perceive the randomness as much in these methods. And you can amplify that by, for example, bringing in a fuzzy [die], a big one, and somebody gets to roll it. And if a 5 comes up, they get to come up and hit the randomized button five times. And now there's a greater perception of randomness that's happening. With Flippity, that turns out actually it'd be true. Turns out that the first randomization is not purely random, and the kids spot that pattern. And we thought, "OK, perfect. That's fine. As long as the students perceive it's random, that it is truly random, that the teacher isn't somehow hacking this so that they are able to impose their own bias into this space." So, it's seemingly random, but not purely random.

And everything was running fine until about six to eight months ago. I was spending a lot of time in classrooms. I think in the last 14 months I've been in 144 different classrooms, coteaching or teaching. So, I was spending a lot of time in classrooms, and for efficiency's sake, a lot of these teachers were using digital randomizers. And then I noticed something. It had always been there, but I hadn't noticed it. This is the nature of research. It's also the nature of just being a fly on the wall, or someone who's observing a classroom or a teacher. There's so much to notice we can't notice it all. So, we notice the things that are obvious. The more time we spend in spaces, the more nuanced things we're able to notice. And about six to eight months ago, I noticed something that, like I said, has always been there, but I had never really noticed it.

Teacher hits a randomized button, and all the students are standing there watching, waiting for the randomized groups to appear on the screen. And then somebody goes, "Ugh." It's so small. Or somebody laughs. Or somebody's like, "Nooo." And it's gone. It's in a moment, it's gone. Sometimes others snicker about it, but it's gone. It's a flash. And it's always been there, and you think it's not a big deal. Turns out it's a huge deal because this is a form of microbullying. This is what I call it, "microbullying." Because when somebody goes, "Ugh," everybody in the room knows who said it. And looking at the screen, they know who they said it about. And this student, [themself], knows who said it, and they know that they're saying it about them. And what makes this so much worse than other overt forms of bullying is that they also are keenly aware that everybody in the room just witnessed and saw this happen, including the teacher. And it cuts deeply. And the only thing that makes bullying worse is when bullying happens in front of someone who's supposed to protect you, and they don't; not because we're evil, but because it's so short, it's so small, it's over in a flash. We don't really see the magnitude of this. But this has deep psychological effects and emotional effects on these students. Not just that they know that this person doesn't like them. But they know that everybody knows that they don't like them. And then what happens on the second day? The second day, whoever's got that student, that victimized student in their group, when the randomization happens, they also go, "Ugh," because this has become acceptable now. This is normative. Within a week, this student might be completely ostracized. And it's just absolutely normal to sort of hate on this one student.

It's just not worth it. It cuts too deeply. Now you can try to stop it. You can try to control it, but good luck, right? I've seen teachers try to say, "OK, that's it. You're not allowed to say anything when the randomization happens. You're not allowed to cheer, you're not allowed to grunt, you're not allowed to groan, you're not allowed to laugh. All you can do is go to your boards." Then they hit the random, and immediately you hear someone go, "Ugh." And they'll look at them, and the student will go, "What? That's how I breathe." Or "I stubbed my toe" or "I thought of something funny." It's virtually impossible to shut it down because it's such a minor thing. But seemingly minor. In about 50 percent of elementary classrooms that I'm in, where a teacher uses that digital randomizer, you don't hear it. But 50 percent you do. Almost 100 percent of high school classrooms I'm in you hear some sort of grunt or groan or complaint. It's not worth it. Just buy more cards. Go to the casino, get free cards. Go to the dollar store, get them cheap. It's just not worth it.

Now, let's get back to the Popsicle stick one. It actually has the same effect. "I'm going to pull three names. I'm going to read out which three names there are, and I'm going to drop them there." And somebody goes, "Ugh." But why does this not happen with cards? It doesn't happen with cards because when you take that card, you don't know what group you're in. You don't know who else is in your group. All you know is where to go. You take that card, you don't know who else is in your group. There's no grunting, groaning, laughing, snickering. And then when you do get to the group, there might be someone there that you don't like working with. So, the student might go, "Ugh." But now there's no audience to amplify this effect. And because there's no audience, more often than not, they don't bother going, "Ugh." Go back to the cards, people. The digital randomizers are fast and efficient, but they're emotionally really traumatizing.

Mike: I think that's a really subtle but important piece for people who are thinking about doing this for the first time. And I appreciate the way that you described the psychological impact on students and the way that using the cards engineers less of the audience than the randomizers [do].

Peter: Yeah, for sure.

Mike: Well, let's shift a little bit and just talk about your recommendations for group size, particularly students in kindergarten through second grade as opposed to students in third grade through fifth grade. Can you talk about your recommendations and what are the things that led you to them?

Peter: First of all, what led to it? It was just so clear, so obvious. The result was that groups of 3 were optimal. And that turned out to be true [in] every setting, every grade. There are some caveats to that, and I'll talk about that in a minute. But groups of 3 were obvious. We saw this in the data almost immediately. Every time we had groups of 3, we heard three voices. Every time we heard groups of 4, we heard three voices. When we had groups of 5, we heard two voices on task, two voices off task, and one voice was silent. Groups of 3 were just that sort of perfect, perfect group size. It took a long time to understand why. And the reason why comes from something called "complexity theory." Complexity theory tells us that in order for a group to be productive, it has to have a balance between diversity and redundancy.

So, redundancy is the things that are the same. We need redundancy. We need things like common language, common notation, common vocabulary, common knowledge. We need to have things in common in order for the collaboration to even start. But if all we have is redundancy, then the group is no better than the individual. We also have to have diversity. Diversity is what every individual brings to the group that's different. And the thing that happens is, when the group sizes get larger, the diversity goes up, but redundancy goes down. And that's bad. And when the group sizes get smaller, the redundancy goes up, but the diversity goes down. And that's bad. Groups of 3 seem to have this perfect balance of redundancy and diversity. It was just the perfect group size. And if you reflect on groups that you've done in your settings, whatever that setting was, you'll probably start to recognize that groups of 3 were always more effective than groups of 4.

But we learned some other things. We learned that in K–2, for example, groups of 3 were still optimal, but we had to start with groups of 2. Why? Because very young children don't know how to collaborate yet. They come to school in kindergarten, they're still working in what we call "parallel," which means that they'll happily stand side by side at a whiteboard with their own marker and work on their own things side by side. They're working in parallel. Eventually, we move them to a state that we call "polite turn-taking." Polite turn-taking is, we can have two students working at a whiteboard sharing one marker, but they're still working independently. So, it's now your turn and you're working on your thing, and now it's my turn, I'm working on my thing. Eventually, we get them to a state of collaboration. And collaboration is defined as "when what one student says or does affects what the other student says or does." And now we have collaboration happening.

Very young kids don't come to school naturally able to collaborate. I've been in kindergarten classrooms in October where half the groups are polite turn-taking, and half the groups are collaborating. It is possible to accelerate them toward that state. But I've also been in grade 2 classrooms in March where the students are still working in parallel or turn-taking. We need to work actively at improving the collaboration that's actually happening. Once collaboration starts to happen in those settings, we nurture it for a while and then we move to groups of 3. So, I can have [kindergartners] by the end of the year working in groups of 3, but I can't assume that grade [2 students] can do it at the beginning of the year. It has a lot to do with the explicit efforts that have been made to foster collaboration in the classroom. And having students sit side by side and pair desks does not foster collaboration. It fosters parallel play.

So, we always say that "[In] K–2, start with groups of 2, see where their level of collaboration is, nurture that, work on it, move toward groups of 3." The other setting that we had to start in groups of 2 were alternate ed settings. Not because the kids can't collaborate, but because they don't trust yet. They don't trust in the process in the educational setting. We have to nurture that. Once they start to trust in working in groups of 2, we can move to groups of 3. But the data was clear on this.

So, if you have a classroom, and let's say you're teaching grade 6, and you don't have a perfect multiple of 3, what do you do? You make some groups of 2. So, rather than groups of 4, make some groups of 2. Keep those groups of 2 close to each other so that they may start to collaborate together. And that was one of the ironies of the research: If I make a group of 4, it's a dumpster fire. If I make two groups of 2 and put them close to each other, and they start to talk to each other, it works great. You start with groups of 2. So, having some extra groups of 2 is handy if you're teaching in high school or any grade, to be honest. But let's say you have 27 students on your roster, but only 24 are there. There's going to be this temptation to make eight groups of 3. Don't do it. Make nine groups, have a couple of groups of 2. Because the minute you get up and running, someone's going to walk in late. And then when they walk in late, it's so much easier to plug them into a group of 2 than to have them waiting for another person to come along so that they can pair them or to make a group of 4.

Mike: Yeah, that makes sense.

Before we close, Peter, I want to talk about two big ideas that I really wish I would've understood more clearly when I was still in the classroom. What I'm thinking about are the notion of crossing social boundaries and then also the concept of knowledge mobility. And I'm wondering if you could talk about each of them in turn and talk about how they relate to one another.

Peter: Certainly. So, when we make our groups, when we make groups, groups are very discreet. I think this comes from that sort of strategic grouping, or even self-selected groupings where the groups are really separate from each other. There are very well-defined boundaries around this group, and everything that happens, happens inside that group, and nothing happens between groups. In fact, as teachers, we often encourage that, when we're like, "No, do your own work in your group. Don't be talking to the other groups," because the whole purpose of doing strategic groups is to keep certain kids away from each other. And that creates a very nonpermeable boundary between the groups.

But what if we can make these boundaries more porous, and so that knowledge actually starts to flow between the groups? This is what's called "knowledge mobility," the idea that we don't actually want the knowledge to be fixed only inside of a group. The smartest person in the room is the room . We gotta get that knowledge moving around the room. It's not groups, it's groups among groups. So, how can we get what one group is achieving and learning to move to another group that's maybe struggling? And this is called "knowledge mobility." The easiest way to increase this is we have the students working at vertical whiteboards. Working at vertical whiteboards creates a space where passive knowledge mobility is really easy to do. It's really easy to look over your shoulder and see what another group is doing and go, "Oh, let's try that. They made a table of values. Let's make a table of values." Or "they've done a graph" or "they drew a picture" or whatever. "We'll steal an idea." And that idea helps us move forward. And that passive can also lead to more active, where it's like, "I wonder what they're doing over there?" And then you go and talk to them, and the teacher can encourage this. And both of these things really help with mobilizing knowledge, and that's what we want. We don't want the only source of knowledge to be the teacher. Knowledge is everywhere. Let's get that moving around the room — within groups, between groups, between students. And that's not to say that the students are copying. We're not encouraging copying. And if you set the environment up right, they don't copy. They're not going to copy. They'll steal an idea, "Oh, let's organize our stuff into a table of values," and then it's back to their own board and working on that.

And the other way that we help make these boundaries more porous is by breaking down the social barriers that exist within a classroom. All classrooms have social barriers. They could be gender, race. They could be status-based. There are so many things that make up the boundaries that exist within classrooms. There are these social structures that exist in schools. And one of the things that random groups does is it breaks down these social barriers because we're putting students together that wouldn't normally be together. And our data really reveals just how much that happens; that after three weeks, the students are coming in, they're socializing with different students, students that hadn't been part of their social structure before. They're sitting together outside of class. I see this at the university where students are coming in, they almost don't know each other at all. Or they're coming in small groups that are in the same class; they know each other from other courses. And within three, four weeks, I'm walking through the hallways at the university and I'm seeing them sitting together, working together, even having lunch together in structures that didn't exist on day one. There are so many social structures, social barriers in classrooms. And if we can just erode those barriers, those group structures are going to become more and more porous, and we're creating more community, and we're reducing the risk that exists within those classrooms.

Mike: I think the other piece that jumps out for me is when I go back to this notion of one random grouping, a random grouping that shifts every 45 to 75 minutes. This idea of breaking those social boundaries — but also, really this idea that knowledge mobility is accelerated jumps out of those two practices. I can really see that in the structure and how that would encourage that kind of change.

Peter: Yeah. And it encourages both passively and actively. Passive in the sense that students can look over the shoulder, active that they can talk to another group. But also passively from the teacher perspective, that random groups does a lot of that heavy lifting. But I can also encourage it actively when a group asks a question. Rather than answering their question, looking around the room going, "You should go talk to the 7s over there." Or "We're done. What do we do next?" "Go talk to the 4s. They know what's next." That sort of, "I as a teacher can be passive and let the random groups do a lot of the heavy lifting. But I can also be active and push knowledge around the room by the way I respond to students' questions."

Mike: Well, and I think what also strikes me is you're really distributing the authority mathematically to the kids as well.

Peter: Yeah. So we're displacing status, we're increasing identity. We're doing all sorts of different things that are de-powering the classroom, decentralizing the classroom.

Mike: Well, before we go, Peter, I'm wondering if there are any steps that you'd recommend to an educator who's listening. They want to start to dabble, or they want to take up some of the ideas that we've talked about. Where would you invite people to make a start?

Peter: So, first of all, one of the things we found in our research was [that] small change is no change. When you make small changes, the classroom as a system will resist that. So, go big. In building thinking classrooms, random groups is not a practice that gets enacted on its own. It's enacted with two other practices: thinking tasks, which is Chapter 1 of my book, random groups, which is Chapter 2. And then getting the students working at vertical whiteboards. These are transformational changes to the classroom. What we're doing in doing that is we're changing the environment in which we're asking students to behave differently. Asking students to behave differently in exactly the same environment that they behaved a certain way for five years already is almost impossible to do. If you want them to behave differently, if you want them to start to think, you're going to have to create an environment that is more conducive to thinking. So, that's part of it.

The other thing is, don't do things by half measures. Don't start doing, "Well, we're going to do random groups on Mondays, but we're going to do strategic groups the rest of the days," or something like this. Because what that communicates to students is that the randomness is something that you don't really value. Go big. "We're doing random groups; we're always doing random groups." Have the courage. Yes, there's going to be some combinations that you're going to go, "Uh-oh." And some of those are going to be really uh-oh combinations. But you're also going to have way more situations where you go "uh-oh" and then it turns out to be amazing. So, have that courage, go with the random groups, and do it persistently and consistently. Because there is going to be resistance. The students are going to resist this thing because at least when you're being strategic, you're being thoughtful about it. But this feels like too much chance. And they start to attribute, they start to map their emotions around being placed in strategic groups, which were often for a month, into this setting. And what we need to do is, we need to show that this is not that by being consistent, doing it randomly, doing it frequently, so they start to realize that this is different . This is not the kind of grouping structures that have happened in the past. And do it. Do it consistently, persistently. Do it for at least 10 days before you start to really see and really reap those benefits.

Mike: I think that's a really great place to stop. Thank you so much for joining us on the podcast, Peter. It really has been a pleasure chatting with you.

Peter: Thanks so much. It's been a great conversation.

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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