Peter Liljedahl: Well, the kinds of tasks that we'd have to give them in that setting need to be very different than if I have them face-to-face in a room and they can work on a whiteboard, or vice versa, if I'd have them in an online setting where they are working on a shared whiteboard or jamboard or something like that. So the nature of the task has to in many ways fit the environment. Then again, I have to think very carefully about what tasks I want to give for them to be able to work collaboratively, as opposed to just sharing their crosstalk.

Kyle Pearce: You're listening to the godfather or the thinking classroom, Peter Liljedahl. We spoke with Peter way back on episode 21, of The Making Math Moments That Matter podcast about how he built the components of the thinking classroom. And we needed to bring him back on because we only got through a small fraction of what we were hoping to dive into.

Jon Orr: If you haven't listened yet to episode 21, we strongly encourage you to push pause on this one and head over to give it a listen. But once you're done listening there, you'll be itching to get back to listen to this one. So I guess it's almost like a Back to the Future moment right now. So to any of you who just did that and are back, welcome back.

Kyle Pearce: Yes, that's right, Jon, stick around because as always, Peter drops some classroom move knowledge bombs. In particular, he shares why groups of three are better than groups of two in your math class, at least when you're face-to-face, how to choose a task to fit the environment instead of modifying your environment to fit the task, four practices to help move group synergy work to an individual knowing and understanding. And finally, why descriptions of effective teaching strategies shouldn't necessarily lead to prescriptions.

Jon Orr: Queue up that music, and let's do it. (Music).

Kyle Pearce: Welcome to The Making Math Moments That Matter podcast, I'm Kyle Pearce from tapintoteenminds.com.

Jon Orr: And I'm Jon Orr from mrorr-isageek.com, we are two math teachers who together-

Kyle Pearce: With you, the community of Math Moment Makers worldwide who want to build and deliver math lessons that spark curiosity, fuel sense making, and ignite your teacher moves. Jon, we are two, count it, two episodes away from our 100th episode of the show, and what better way to spend it than with a guest who joined us way back on episode number 21.

Jon Orr: Yeah. That's right. That's right. We are super proud to bring Dr. Peter Liljedahl back on the show to continue our conversation about building thinking classrooms, and also discuss some ways we can implement those practices during the COVID-19 era.

Kyle Pearce: Yes, Jon, I'm ready to dive into this episode. How about you?

Jon Orr: Yeah, absolutely. But before we do, we want to announce the dates for our 2020 Make Math Moments virtual summit. It is coming up on Saturday, November 7th and Sunday, November 8th, which is open now for registration.

Kyle Pearce: You're absolutely right, Jon, this is one of our favorite times of the year because we get the honor of bringing some amazing minds from the math education space straight to you just like Peter, and actually Peter is going to be joining us, you'll hear in just a moment and we continue to find ways to manage to do it for free. That means free registrations for all you Math Moment Makers from around the world.

Jon Orr: You got it. If you want some amazing math professional learning, just like we did last year from the comfort of your couch, we encourage you to pause this episode right now and head to makemathmoments.com/summit to register for the 2020 Make Math Moments virtual summit.

Kyle Pearce: Yes, that's right. We are running our second annual free online math professional development summit for K through 12 math educators. The date again Saturday, November 7th, and Sunday November 8th. Peter, our guest today and the wonderful Judy Larson will be one of our over 20 sessions being presented.

Jon Orr: Coolest part yet, some of the sessions will be happening live over Zoom while others are pre-recorded for you to enjoy your convenience over the weekend.

Kyle Pearce: Go ahead register for this year's summit at makemathmoments.com/summit.

Jon Orr: Listening to this episode after this year's summit you can still head to makemathmoments.com/summit and make the wait list for the next Make Math Moments virtual summit.

Kyle Pearce: All right, Jon, let's dive into this great conversation, conversation number two, or should I say the continuation of-

Jon Orr: Right, part two.

Kyle Pearce: ... conversation number one with Peter.

Jon Orr: Hey there, Peter, welcome back to The Making Math Moments That Matter podcast. It's been a while since the last time we spoke with you here on the podcast. We chatted with you way back, in our early episode, it was episode, I think 21. This episode I think is going to be 98. So it's been a while. So over a year, I think. You shared your background on teaching and how you built the idea of the thinking classroom. We want to chat some more about that with you here this time. But how are you doing these days? What's going on?

Peter Liljedahl: First of all, thanks for having me back. What am I doing? Well, I've been like everybody else, March came with a big surprise. And I had to shift my teaching online. It wasn't so bad for me, because I only had three weeks left in my term teaching at the university. So shifting into online where you already had a really strong rapport with your group wasn't horrible. I spent a lot of time in those three weeks, so trying many different things every week just to get a feel for what was possible and what the students, and when I say students, these were practicing teachers and pre-service teachers had tolerance and patience for.
And then I taught a course in the summer. But again, I was very lucky because it was with a cohort of masters students in Grand Prairie who I already had a very strong personal rapport with establishing face-to-face settings. So it was just extending that way. And since then, I haven't been teaching anything else. But I've spent a lot of time finishing off the book, Building Thinking Classrooms, that's coming out on October 20th. And then, of course, all the edits and the proofs back and forth, and back and forth, and so on, and so forth. So I've been busy.

Kyle Pearce: Oh, busy, busy, busy, we've got lots to chat about in regards to both the book but also backing up a little bit with that surprise you referenced back in March, some people have been joking, and I try not to joke outside of the education community like this. But we just went back to school here in Ontario and I've been joking around at work saying that March break is finally over, because it's been such a long haul here. But again, every one was working very, very hard to try to keep students engaged, especially students that may have maybe not had access to technology or some issues on that front. So I'm wondering, can you dive a little deeper for us on what did it look like for you. So it sounds like you already had that rapport with your students. So transitioning online is probably much less cumbersome. What was that like for you if you could paint us a picture?

Peter Liljedahl: Okay. So what I did was, because when I teach university classes, they're already four or five hours long. First of all, that wasn't going to work. The Zoom fatigue was just going to wear everybody out.

Kyle Pearce: The Zoom NATO, they say.

Peter Liljedahl: Yeah. So what I did to begin with, was I assigned some random groups. So I started playing with this synchronous, asynchronous idea. And I wanted to come up with a hybrid of that. So what I did was, I made random groups, but I had them work asynchronous from the rest of the class but synchronously with each other. So I assigned some tasks for them to do in their groups, but they did them not during our synchronous class time, so they could decide when they wanted to meet, and how they wanted to meet whatever platform they wanted to use. And they were working collaboratively on these tasks. And then I created an asynchronous discussion board where they had to go in and enter some details of their work, not necessarily their answers, but thoughts on what were the hiccups, what were the challenges, how did they get over those challenges, things like that.
So it was this synchronous asynchronous space where they obviously were working synchronously in their groups, but asynchronously from the rest of the groups. And then when we were together as a group synchronously, we were able to have whole class discussions about those tasks and so on.

Jon Orr: Yeah. So it's almost like you tried to make the best of both worlds there. Whereas I think some teachers were like, let me just go online and do synchronous lessons. And were with their kids for an hour here or an hour there. And then it sounds like you were like, let's also make use of that time that you're not going to be online. So I think that's pretty great. Whereas I think also a lot of teachers were like, let me go on for office hours. And let me just answer questions. And it sounds like you really made the most out of your opportunity, in that not so great situation. But you mentioned a couple of things I want to riff on a little bit here, especially you said random groups, and I know that we talked a lot about some of your tips for creating thinking classrooms in our last episode on episode 21. But I feel like we didn't dive into that idea of random grouping.
Before we get into that, I'd wanted to bring up a couple of the things that we did chat about, we did talk about choosing tasks and what that looks like. We did talk about what room organization would look like. I mean, you had some really great quotes, like, the one that we wrote down for the episode was, there was these 14 ways to build a thinking classroom, and you said it, but there's 100 ways to wreck it. And that definitely resonates with us, because I know there's lots of things you can do right in the classroom, but so many things you can do wrong. But I think we just riffed on a couple of these 14, but Peter, let's chat about random grouping. Because I know that there's so many people ask us about random groups, like how do you make groups? Do you make groups of two, do you make groups of three? Last time you talked four wasn't great, but three was better. But maybe, why? Why is that the case? And then how do you go about making those groups?

Peter Liljedahl: This is a great question and really relevant right now, actually. So one of the things that happened in my research with Thinking Classrooms was I always had answers before I had theory or explanations. So relatively quickly, it emerged that groups of three were optimal for intermediate and not. The primary, the very younger students, groups of two was best, because they were still playing in parallel and just learning how to negotiate that social space. But groups of three were optimal, groups of four usually devolved into a group of three plus one outsider. And that was really clear, we could see that in the data over and over and over again, no idea why.
Since then, I've spent some time thinking about a theory that's called complexity theory. Now complexity theory says that in order for a group to be generative, it needs to have both redundancy and diversity. So redundancy are the things that we have in common that allow us to talk to each other. So a common vocabulary, maybe some common notation. If we don't have that, we can't even get off the ground. Diversity are the things that we carry with us that are different from the other members of the group. And if you don't have diversity, and all you have is redundancy, you might as well be working by yourself. But having that diversity really allows the group to be generative.
The problem is, when there was too many students, there was a social layer that laid over top that created some noise in the system. So the theory is that groups of three brings the right amount of diversity, and the right amount of redundancy and the groups can be generative. But one of the things that has been really interesting to see is what happened in the spring, when all of a sudden everybody had to go to this online teaching. One of the things we discovered pretty quickly was that groups of three didn't seem to be working. And there was several reasons for this. One of them, I believe, was that silence in these online settings was... So if you have a group of three, and one student is silent, then you don't have enough diversity in the group.
So what we started to notice was that we needed to artificially increase the diversity when we went into this online mode by having groups of four or five and not just that, but doing some things. And you mentioned, there's 100 ways to ruin a thinking classroom. When we were doing the face-to-face research, one of the things we found was really problematic was when we would do this when students got to think about a task individually before they went into their group, often, when that happened, the group would never really gel. And it was because I think alone time created too much diversity. Students were entering that group knowing the answer, and some members hadn't gotten off the ground yet. And now the diversity was too great. But in the online environment, we also started playing with this, we'll spend some time thinking on your own, again, to increase the diversity because it's almost like the online setting was a diversity depleting space.

Kyle Pearce: It's interesting, because when Jon and I are running any of our online workshops and courses, and in our academy, we find that when we group, or sometimes people will ask to be grouped with almost a little pod that they can learn alongside. And we'll randomly select these groups. And in our own experience, we found in the online space that getting that five to seven range, it sounds like you were saying four or five seems like a good number. We found if the number was too small, it's almost like that silence, as you mentioned, and as soon as you said it, it popped into my mind. When it's silent too long, it's almost like the energy is gone. Or maybe there was no energy to begin with. And I'm really interested in this redundancy and diversity piece as well. That's going to be fresh in my mind the next time through as we're about to enter into another course here with a new cohort of educators.
And that's really interesting to me, and I find that in-person and face-to-face, that group of three, as soon as you were saying it, that number as it gets larger, there's some social layer that's affecting things, I can completely see that. It's almost like the safety in the group starts to decrease. And now you're feeling a little bit more it's a little more risky to maybe speak out. Whereas in the online space, I find like I just did a webinar today with kindergarten educators, and we had 50 people in the room, and I'm telling you, it was not energetic like it would be in a face-to-face environment. So that's really interesting. And I'm really excited for us to continue through some of these ideas in your list of 14, but then also to have a bit of a compare and contrast to the COVID era.

Peter Liljedahl: Right. And that's one of the things that I've been thinking a lot about this summer is what is transferable and what doesn't need to change, and what needs to change when you start shifting into these, either an online synchronous or asynchronous or a restricted face-to-face where there are some protocols in place or a hybrid? How does that work?

Jon Orr: I'm going to go back to something you said to that really struck something with me about this idea of when you're online, when you gave think time, it killed the group think and when they came together, and I felt like this is the opposite experience of what we've learned as teachers in face-to-face. Think pair share has always worked great. It's like think first, now join up and start on your whiteboards. I've always felt that when I asked my students to think first and then share to the big group, I've gotten way more engagement, and quick to work than say, just jump into group work, or even just think on your own. So I find that constantly, that comparison to being online is just all of a sudden, that's just out the window. And I agree with you, what else is transferable and what's not?
So I'm wondering, what would you see as being optimal for trying to navigate group work? And I want to spin it a couple of ways too, in this pandemic world because some teachers are fully online, depending on their school board, or a blend. Kyle school board's in a blend model, but my school board's fully face-to-face, but we're trying to social distance in our rooms. So it's like how do I do group work when we're social distancing? Can we do it? Is it okay? I know these are all big questions that not all of us have answers to, because it's so new. But what do you see as being optimal?

Peter Liljedahl: Oh. Optimal is a word that I use for something that emerges out of the data. So this is purely speculation.

Jon Orr: Yeah, for sure.

Peter Liljedahl: But I want to come back to-

Kyle Pearce: You're just gambling at this point.

Peter Liljedahl: Yeah, that's right. I want to come back to something that we had talked about in the previous episode, which was this notion of thinking tasks. So one of the things I've been seeing has been bypassed in many ways is thinking about how a task fits a collaborative environment. So in the spring, on Twitter, I spent a lot of time watching how teachers were trying to shape the environment to match the activity they wanted to do rather than to shape the activity to match the environment that they were forced to have. So for example, if I'm teaching in an online setting, and the students are operating in, I can put them in a collaborative group, let's say, but they don't have any way to collaborate through notation.
So there is no shared workspace, all they can do is talk to each other. Well, the kinds of tasks I would have to give them in that setting need to be very different than if I have them face-to-face in a room and they can work on a whiteboard. Or vice versa, if I'd have them in an online setting where they are working on a shared whiteboard or jamboard or something like that. So the nature of the task has to in many ways fit the environment. So if all the students can do collaboratively is talk to each other, then I need to give a task that doesn't require a lot of notation, but requires a lot of discussion. And if even a worse scenario, all they could do is collaborate through text, then again, I have to think very carefully about what tasks I want to give for them to be able to work collaboratively as opposed to just sharing their individual answers that they have come up with. So the difference between thinking together and thinking alone and sharing their work.

Kyle Pearce: Interesting. And when you say that, that makes perfect sense. Now I'm wondering if I'm a math educator, and I'm listening to the podcast and I'm going, "Okay, that's great. Can that work for me?" And I'm seeing ways that it can be I can ask questions that are about almost like in the consolidation phase of a particular type of task or unit or concept. But what did that look like for you? So when you were working, let's say, with your students, did you find that you were unable to give certain tasks that you would have liked to have given? And if so, what did you do instead, in place of that?

Peter Liljedahl: So what I did in those spaces was, when I was doing this asynchronous synchronous, I left it to them to find a way to collaborate so that they could collaborate using notation. So some groups would literally just have a personal whiteboard and hold it up. And one person would scribe and they would talk to each other, and they would collaborate that way. So there was a visual along with the bouclé. The questions I would ask, in the space, the asynchronous text space, were more about the task, rather than solving the task. And you could think about that as, so let's say we have students who can only collaborate verbally, I don't necessarily want to ask them to find the zeros of a quadratic or a cubic, but I might ask them about the zeros.

Kyle Pearce: Right. What did you notice about them? Or what was the shape?

Peter Liljedahl: Right. And what do you think would happen to the zeros if I change this coefficient from positive to negative? Or what do you think would happen if I did this? So it becomes about questions, as opposed to solving questions. If they're working only verbally.

Kyle Pearce: I'm also as well, Peter, I'm picturing this collaborative group. If let's say the number is a group of four, or five. And we've actually randomly selected these groups and now with all the software, it really just goes to show how tech companies and how fast they move to see how Zoom and Microsoft Teams and Google Meet, and all these companies, were able to on a dime, add in all kinds of features like Breakout rooms, and teachers are sharing hacks in order to make Breakout rooms when they weren't available in Google, for example.
And I'm picturing as well, there's actually a hidden benefit, even if students are working through a task together. And much like the role of the scribe in a group, I could be holding up that whiteboard or my webcam can be pointed towards my little whiteboard on the wall, or whatever it might be, and the rest of the group could be describing to me in much the same way as they might in a face-to-face setting. Did you have any success with that over this time? Or did you find that there were hindrances there, or nuances there that emerged that may not be obvious to one looking from the outside?

Peter Liljedahl: The groups that did it reported that it worked very well. But I didn't it do enough to be able to say anything definitively about it. But I can imagine that, of course, it's going to be more taxing. If you're trying to speak to a scribe and the scribe is trying to record what it is you're saying, it's very difficult if I can't at some point step in and say, "Well, can I scribe for a bit?" And so on. Which of course is solvable if everybody has a whiteboard in which they could ping back and forth on. There's no school like old school. This idea of being able to just have a whiteboard that you'll hold up to a webcam. But that's not to say that these shared workspaces like jamboard and whiteboard and Zoom don't offer that as well.

Jon Orr: Yeah. I found some success with jamboard in the sharing of collaborative writing, it was just sometimes hard to navigate where you were going to write and where was I going to write? And how are we going to manage all that. Peter, I wouldn't mind chatting a little bit more about some of the other aspects of the thinking classroom. But I know that some listeners are listening going like, "Well, what are they? Which ones did I miss? Which one should I go back and look at from episode 21?" What I'm going to do is I'm just going to read the ones that you've written down, or shared with us, the 14 and then I'll just let everyone know which ones we talked about in episode 21, but I'm going to let you choose one to chat about and then maybe we can also chat about how we've modified or maybe even transferred, or maybe inaudible into today's teaching world.
But the first one you had was problems and tasks. And we talked about that in episode 21 and how to choose those. We just talked a little bit about that here now too. The second one was how we give the problem. Third one was how we answer questions. Fourth one is room organization. We chat a little bit about that in episode 21. The fifth was groups and how they're formed. And we chatted here just now about that. Six was student workspace, we chatted slightly about that in the first episode with you. Seven was autonomy. Eight was how we give notes. Nine was homework, what's that look like. 10 is hints and extensions. 11 is how we consolidate. 12 is formative assessment, 13 summative assessment and 14 is reporting. Lots of there for sure. Where would you like to go to chat about?

Peter Liljedahl: So first of all, those 14 practices are broken into four toolkits. And those toolkits are what I call a pseudo sequence of implementation. So if a teacher is picking this up, for the first time, it comes with a sequence of implementation. And it's called a pseudo sequence because these are clustered and sometimes it matters what order you do them in. And sometimes it doesn't matter what order you do them in.
But I'd like to talk about something, which is what I call the responsibility toolkit. And this was one of the things that emerged in the research, where we started to see, when we were first doing having kids or random groups working on whiteboards on these great tasks, and even on curricular tasks, we were seeing all this amazing synergy in the groups and you've probably seen this too, these groups are just working through these tasks and they're doing so amazing and then a week later, you test them. And it's like, where are you? You're like none of what we saw a week earlier, was transferring into this individual tests.
And I spent a lot of time thinking about and researching what was going on there. And at first, we thought that maybe we were miss seeing what we thought we were seeing in these synergistic moments. And we spent a lot of time looking closer. And I'll give you an example, we're watching a group of three students working through a sequence of tasks on I think it was factoring quadratics. But it could have been another algebra topic. And every member of that group was completely locked into the task and they were solving, there wasn't anybody riding the coattails of the others, and they were piling on each other, this bursting, it's called in the psychology literature, where they're just jumping on each other and taking turns holding the marker. And every single student was able to do those tasks on that day. We were certain that what we saw was comprehension and understanding. But again, that individual task a week later didn't produce the results that we had seen the week before.
So I spent a lot of time trying to understand what is missing? How is it that we're not moving from that group synergistic knowing and doing to an individual knowing and doing? And this was early on in the research and all the tools hadn't developed yet. But as those tools, those practices started to develop and emerge, what started to become clear was that there was four practices that actually helped move that group synergy and the ability to do and know in a group to an individual knowing and doing. And these four practices are consolidation from the bottom, meaningful notes and notes to my future dumber self, check your understanding questions, which is what we formerly called homework, and formative assessment of helping students understand where they are and where they're going. Okay.
So I'll go through each of these in turn. So a group is working in a synergistic way. And I think you've seen that in your classrooms and everyone's engaged, but they're operating at an informal level, they have their own vocabulary and their own informal notation. And they're working through these things. What consolidation does, in particular consolidation from the bottom, is it helps move that synergistic work to a more formal language and a more formal system of notation, and so on and so forth. So the teacher who leads the consolidation is able to take that synergy and formalize it somewhat. So that's the first step.
The second step is now the teacher says to the students, "Okay, now I want you to sit down. And for the next 15 minutes, I want you to write some notes to your future forgetful self. So what do you have to write down now, so that in three weeks, you'll be able to remember what you did today?" And it turns out the note taking one was really, really vital, because this is the first opportunity the student actually has to move that collective knowing and doing to some individual knowing and doing. We can think of it as a personal consolidation. The teacher has done that consolidation for everyone. But now this student gets to sit down and actually think about, "What did I get from today? What can I write down to myself?" And this became really, really important because our research showed that in a traditional note taking sense of what I call I write you write, so I the teacher writes and the student... It's like the world's slowest photocopier is what was missing then, what was happening when we were interviewing the students were that they weren't taking the notes for themselves. And in turn, they weren't using the notes.
So these notes became really mindless, whereas now when we're doing notes to my future forgetful self, they're very mindful. They're thinking deeply about what do I write down? What examples do I include? How do I articulate this to myself? How do I annotate an example so that I could follow it so that I could recreate this knowing and doing three weeks from now? The next one was check your understanding questions. Check your understanding questions. Again, we did a whole bunch of research on homework and we interviewed kids and we surveyed kids. And the number one thing that emerged was who are they doing the homework for? They're doing it for the teacher. Why are they doing it? For marks. Which was completely out of sync with why the teachers were assigning homework. The teachers were assigning homework, because they wanted the students to see what they were doing wrong, and learn from their mistakes and a self assessment. They wanted the homework to be for checking their understanding.
So we shifted that. We shifted from homework, we stopped calling it homework, we stopped calling it practice. And we shifted it to just check your understanding questions to more centrally situated in who it's for, and what it's for. And we gave a whole bunch of autonomy or responsibility around that. We didn't mark it, we gave more questions than necessary, the students got to choose which ones they wanted to do, so on and so forth. But it focused them more on actually checking if that knowing and doing they could do in the group, could they now do it as an individual.

Kyle Pearce: I'm loving this because you're outlining so many important pieces here. And this idea of consolidating a lesson, we talk about that a lot on the podcast and how important it is. And I love how you take that synergy and formalize it and make sure in a lot of ways too, I'm picturing these students and you were painting a picture of students who were right into it and solving problems and doing all kinds of great work. And that's amazing. But at the end of the day, sometimes we forget as educators that just because they're getting answers, and they're doing a great job, sometimes they miss the point. The big idea, that's laying right in front of them. And some groups do pick up on that, and maybe some people in groups pick up on it, but others don't.
And that is really important. And I love this idea that it's sounding like and I'm hoping that you can confirm or deny this, but this consolidation after doing the work, it's again, students are given this opportunity to investigate, to solve, truly solve problems instead of mimicking. I know we'll talk more about your book as well. The term mimicking is something I've been using a lot ever since having the opportunity to review that book. And this idea of them creating a meaningful note at the end, whereas in that slowest photocopier ever example you gave, typically, in a traditional classroom, you would actually do the opposite. The teacher would fire off some examples, there would be this big long note, students are copying it because they have to not because they want to. And then in the end, we're asking them to solve some problems.
So again, it's this flipped classroom idea, Jon and I always say, the real flipped classroom is putting the problem solving first and consolidating after. What does that look like? I'm wondering if you can compare and contrast that, if we were to look at maybe two types of classes. Let's say a higher level course like a pre-calculus or calculus? What might that look like, sound like? Or does it look any different in say a class, I think you would referenced from your career in the first episode, I re-listened to it today on the way home from work and you referenced, I think it was called the essentials class. What might it look like in there? Is it any different? Is it the same? And I guess at what point does the teacher step into help?
And I'm picturing more of a facilitator role than say, the guy the sage on the stage, it would be more of a there to help guide the conversation and help students formalize their thinking. But I'm wondering if you can dive a little deeper on that, because I know there's going to be people at home listening saying like, wow, this sounds awesome. But how am I going to take that and do that with my group, especially if that's maybe outside their comfort zone?

Peter Liljedahl: Right. So first of all, I think a really good consolidation is probably the most skillful thing a teacher can do in the room. So in a thinking classroom, you got to picture this, the students are all working on the whiteboards in their groups, and they're moving through curriculum tasks. So they're moving through tasks, whatever they are, and what you're seeing on the boards there over time, is a lot of really good work, that would be nice to draw attention to as part of a consolidation.
So there's three types of consolidation in a thinking classroom and I can go over them in a minute, but ideally, what you want to do is draw on as much of the student work to honor the thinking that's been happening in the room. So that the thinking doesn't feel like, I'm going to let you guys think for 20, 30, 40 minutes and then we're just going to ignore what you've done and I'll do it my way. So we wanted a consolidation to be an extension of that work that they were doing. So a whole bunch of things have to happen in order for that to work effectively.
Number one, you have to make sure the students don't erase the things that you want to draw attention to for the consolidation, which means that while they're working, you have to walk around with a red marker and lock things in. Draw a box around it and tell the group don't erase this part. And so there's that. You have to lock things in. Number two, you have to see the space. So let's say they're working on a task and nobody's making a graph. And it would be so great if there was a graph for the consolidation. You might go up to a group and say, "Hey, I really like what you're doing, you have a really nice table of values, and you're trying to make a hypothesis here, could I-

Kyle Pearce: Would a graph, maybe-

Peter Liljedahl: ... that you make a graph?" And then when they've made the graph, you go and draw a box around, and say, "Don't erase that." So-

Jon Orr: The orchestrator. Sorry to cut you off there-

Peter Liljedahl: Yeah, yeah.

Jon Orr: ... I'll let you finish, but it's like, you know you've got to have that foresight. You've often referenced the 5 Practices for Orchestrating Productive Mathematics Discussions by Mary Kay Stein and Peg Smith. And that whole anticipation. You got to map out what you need it to look like, so that you can do exactly your trick move there, of bringing out what you need, so that you can make a really great connection stage.

Peter Liljedahl: Right. So this is why I said you have to be so skillful, because as you're trying to manage the thinking classroom, you're also trying to lock things in, you're trying to see the space and anticipate what you need. And the third thing you got to do before you consolidate is now you have to sequence what it is you want to do. And again, this is in the 5 Practices, but you have to think about, "Okay, so, how am I going to take the students on this journey in this consolidation?" And in a typical for like 100 years, consolidation has been what Alan Schoenfeld refers to as leveling to the top. So I was hoping everybody would be able to solve this problem, only three groups solved that problem. So my consolidation is I'm going to go over how to solve that problem.
And it doesn't work. And it doesn't work because of what's called zone of proximal development. You're trying to consolidate a level well above what many other groups got to. So what we did in thinking classrooms was we started experimenting with something called consolidating from the bottom, which is, okay, let's start at the very bottom. Let's start with some of the very fundamental work that the groups did that every group got to. And now let's work our way up and work our way up and work our way up. And what we do then is we're moving actually, through everybody's zone of proximal development. We're able to elevate everybody's thinking above where they got to as a group, but at the same time is honoring the work that they did as a group.

Jon Orr: I totally agree. That skillful consolidation or connect stage is so important. And we get a lot of questions about that too. How do you do that? When do you do that? What's the right time to do that? And people who've asked us and who are listening right now, we just say it depends. It depends on so many things. It depends on what you've seen, what you've anticipated. It also will depend on the time-

Kyle Pearce: What's the big idea you're after.

Jon Orr: ... that you have.

Kyle Pearce: Yep.

Jon Orr: Yeah, it does depend on the time that you have in the room. Even though we'd love to have an unlimited time, it depends on so many factors, which we always say takes practice.

Peter Liljedahl: Oh, yeah.

Jon Orr: You can't be good at it right away, you're going to have to try it and see how it goes. And you have a plan in mind, but you definitely have to orchestrate it. And it's like managing an orchestra, just like your stages that you've outlined there.

Peter Liljedahl: At the beginning, we thought, okay, we're going to create these what we call thresholds. So as soon as every group has gotten across this threshold, we're going to consolidate, and then the first time we tried it, every group shot through that threshold in eight minutes. And we were like," Okay, so now we're moving the threshold." Other times you think, "Okay, here's how far we wanted to get to," but we realize on the ground that that's not going to be achievable. So you have to be nimble. And one of the things we found, the biggest depends really is the energy of the room, there comes a point where the energy is waning, you just can't get them to achieve more. And now it's time for consolidation.

Kyle Pearce: Right.

Jon Orr: We've often said, I know that you've referenced this meaningful note, and sometimes that formalization for you to step in and go, this is where we're going. This is how it connects to what you guys have done and you tie all those people's work in and then you steer them in where the learning goal was for that day. I think a lot of people think that this is purely student driven at that point where it can totally be teacher driven to be, this is where we were going, this is what we want to pull from this. And making it clear. Kyle talked about that too, it's like, we got to make it clear to students because a lot of them might have just been missed the boat on what the main goal was there that day. And I think that's really an important part.

Kyle Pearce: Right. We just impart problem solving. That's their big takeaway.

Jon Orr: Yeah, part of the consolidation.

Peter Liljedahl: And that is, there was this movement in education. It started about 20 years ago in the UK where we have to name the intention at the beginning of the lesson. And what we found is it makes much more sense to name the intention as part of the consolidation. And you just said that this is where we were trying to get to, and this is what it's called, whereas naming it at the beginning, like telling a group of students that today we're going to learn how to factor quadratics, where the leading coefficient is greater than one doesn't mean a lot to them. But if they have a whole bunch of experiences, and then we say, that's what that's called.

Kyle Pearce: Right. Or you give them a problem, and I'm trying to think of the rat cat-

Peter Liljedahl: The cats and the rats problem.

Kyle Pearce: Yeah. That problem. You tell kids upfront that today, we're going to use scaling in tandem and proportional relationships in order to solve problems. Well, first of all, the curiosity, some of that is out the window. But then second of all, the problem solving is out the window, because you're been funneled right to a play. So we've talked about that so much, and how, again, and I love how you've mentioned it a few times just how skillful teachers have to be in not just in the consolidation, although that I would argue, yes, that's like the big event in the lesson to tie it all together. But as you're crafting your lesson from a high level and anticipating and organizing it and planning it, and then finally delivering it, that all has to be in mind.
And I think that's why that shift to naming it at the beginning. They talked about students knowing where the target was, and so forth. But I feel like that was based on a very procedural approach to mathematics. So we're definitely all in line with that as well.

Peter Liljedahl: And it emerged, and you see a lot of this happen in education. Is it's this transference, this tension between what I call description and prescription. So somebody looked at some classrooms and noticed that in some high functioning classrooms, students understood what the intention was. That was a description. And then they said, "Oh, well, that must be a good prescription then. So let's now make it a mandate that you have to name this intention." And it doesn't work that way. It's the same thing with roles and group work. We experimented with this too. And it emerged as a description, somebody noticed that high functioning groups, there was a leader, there was a recorder, there was a timekeeper there was an encourager, and so on, and so forth. Those are descriptions of highly effective groups. That doesn't mean that the prescription is going to work. And we see this happen a lot in education, where we turn descriptions of really effective things into prescriptions for how to do things.

Jon Orr: Yeah. Yeah, that's the-

Kyle Pearce: Big takeaway right there for sure.

Jon Orr: Yeah. Yeah, for sure. And I think there's lots of teachers listening because we've got that question a lot about, we got to write the learning goal of the day on the board, how do I get around that if I have to reveal it later? And our recommendation has always been that write those learning goals as big idea goals. Give them your example about factoring. Maybe the big idea... What's the big idea? Why do we factor? And one reason, there's lots of reasons, but I mean one reason might be we're going to understand the power of equivalent representations or equivalent expressions. And that's it. You don't have to say anything else, as a big general statement. But you're at least saying that's the main goal we're looking at. I'm not going to tell you how, I'm not going to tell you where. We're going to discover that later. I know you did have four practices that help move group synergy into individual knowing we talked about three of them. The last one was that formative assessment.

Peter Liljedahl: Right. So formative assessment, it's interesting when you look across Canada and other curriculum, and so on and so forth. Formative assessment came into Vogue about 20, 17, 20 years ago. And it was described as summative assessment is a gathering of information for the purpose of reporting out. Formative assessment is a gathering of information for the purposes of informing teaching. And I know when formative assessment came out in BC, that's exactly how it was defined. Now who are we missing in that equation? The learner. And so one of the metaphors that I drew on around this has to do with navigation. In order to be able to navigate, you need to know two pieces of information, you need to know where you are, and where you're going. Because if you don't know where you are, you're already lost. And if you don't know where you're going, you won't know when you get there.
So if we want students to navigate their learning, then they need to understand those two pieces of information as well. They need to know where they are and where they're going. So where they are is what they can do and where they're going is what they're required to do that they don't yet know how to do. So how do we provide feedback formative assessment that actually informs those two pieces? Where you are and where you're going. And there's some very technical aspects to that. But that turned out to be probably the most powerful practice in terms of increasing student performance. When we implemented that we saw a 20% increase in student performance in 60% of the kids.
But that turned out to be that final practice that helped move from group knowing and doing to individual knowing and doing. And together, these four, I call them the responsibility toolkit, because aside from consolidation, if you look at all of the other three, the responsibility is on the learner. Notes to my future forgetful self, check your understanding questions that I'm doing for myself to check my understanding, they're not being marked, they're not being collected, I have full autonomy over it. And then that formative self assessment that allows me to gauge where I am, and where I'm going. Those three things place a lot of responsibility on the learner.
And one of the things that was interesting that happened in March was that I was in communication with several teachers who had been working on those aspects of thinking classrooms prior to the pandemic lockdown. And they reported that once they went into that online, synchronous, asynchronous, whatever, their model was, their students were able to cope much better because they had that responsibility to be able to navigate their own learning and through these three practices of meaningful notes, check your understanding questions and knowing where you're on where you're going.

Kyle Pearce: And as I'm looking at this, and it hasn't come up explicitly in this episode, but back in episode 21, anyone who's listening who hasn't listened to 21, yet, and you're this far, in this episode, well, you did it backwards. You should have started there. But it's too late now, if you're already with us this far. But back in episode 21, you gave an example about trig and the unit circle. And we came to this conclusion that you had said that you were, for many years teaching very conceptually. And that's something as well, that I think can be easily missed. Listing the consolidation piece, the meaningful note checking, understanding and formative assessment, I'm sure those would be very helpful as well, even if I'm teaching from a procedural standpoint, or what some call a procedures first approach. But to have true understanding, you really have to conceptually understand what's going on. And that is what I think keeps those wheels turning, especially on the meaningful note to your forgetful self, and that checking, understanding questions.
And then obviously leading into this formative assessment, which also the way you've described it is really not just teacher formative assessment, it's students formative assessment and self reflection, it's really hard for some students to be able to actually do any of these things if they're just mimicking. And I know in your book, you talk a lot about traditional classrooms, and how many classrooms out there are still operating in that way. I know Jon and I, we can relate so well to that, because we were that teacher for so long. And we know what that's like. And that's how we remember learning. And it might not have been our teachers' intent. But at the end of the day, we were memorizing, and we were mimicking, and it made it really hard for us to even attempt doing any of these things, not to mention that this was never on our radar to make our own note, or to do our own checking, understanding questions, but even had we tried in that particular or from that particular approach, it would have been really, really challenging, because we were constantly checking an answer key at the back of the book.
And if you ever have, and all teachers listening, they've all had the student come up and say, "Is this right?" And that's to me a clear sign. Now my response is always to students is, "I don't know, is it?" And if they don't know whether it's right or wrong, then what are we doing this for? And so at this point, I want to shift to the book because I think, based on these two episodes, episode 21 and now this episode number 98, which I'm still shocked that we're at episode 98, I had the chance to review your book after I'd seen you live a number of times, we've had chances to catch up at dinners before and talk in-person, and then also getting to read this book and there were still so many nuances here and you'd think after two episodes that we'd be able to dig through all 14 of your tips for building a thinking classroom but obviously we've come up much short.
I want to give you an opportunity to share, frame this book out for us, does it go through? And I'm asking these questions like I've ever read it before, does it go through all 14 of these tips? And what are people going to learn when they grab the book over, let's say listening to these episodes, or even seeing you live doing a keynote somewhere a full day workshop?

Peter Liljedahl: Okay, well, so the book is called Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning. It's published by Corwin, it comes out October 20th. I wanted the book to capture all of what the thinking classroom was. All 14 practices, the pseudo sequence that emerged from the research as to the best order to implement it in. And then all of the little micro moves, I call them. All the little things that we as teachers can do to make each of the practices even better. And so the book has all of that in it.
The way every chapter is laid out is that, aside from the introduction, there's 14 chapters, one chapter for each of the 14 practices. The chapter starts with, so here's what the issue is, if it's about homework, here's what the issue is. And here's now the problem. So I spent a lot of time in my research, looking at the status quo, what is the most normal or the normative practice? And is that effective? And if not, what is the problem with it? So there's a section of every chapter that talks about the problems with the normative practice of doing a particular move, like homework, or consolidation, and so on, and so forth.
And then it has a section on okay, so that was a normative practice and the problems with it, here's what we learned from thinking classroom and the optimal practice that emerged from the research and how to do it. And then embedded inside of that are a whole bunch of really practical examples, anecdotes, photographs of teachers enacting it, descriptions, and these little micro moves, and so on and so forth. And then there's a summary at the end of the chapter, which highlights a macro move and a micro movement and it gives some questions to think about, and gives you some tasks that you can use to implement this.

Kyle Pearce: Awesome.

Jon Orr: I'm really pumped to get my hands on it. And we'll put the info in the show notes page for when it goes live to purchase on October 20th. Kyle, I think this episode now we're recording this in advance, but I'm pretty sure this is going to go before that. Right, Kyle?

Kyle Pearce: Yeah. I think this episode is actually slated to go live on October 12th. So it will be, yeah, if you're listening to this the day it goes live, you are just days away from being able to put this book in your hands. I'm guessing there may be some pre orders online. I'm not clear on that right now. I don't know Peter, if you have any information on that to share, but feel free to share away.

Peter Liljedahl: If you go to Corwin's website, corwin.com you can search for Building Thinking Classrooms, or you can search for Peter Liljedahl. And it's spelled exactly the way it sounds. Then it'll pop up a page, and you can pre order now.

Jon Orr: Awesome. We'll get that link and we'll throw that in the show notes for sure. So people can go directly there. So check out the show notes page for that. If you're listening. Peter is there anywhere else you want to direct people to go to, to learn more about what you're doing and what you've got coming up?

Peter Liljedahl: So I'm starting to build an accompanying website for this book. I'll put that out on Twitter when it's ready. It's not quite ready yet. And I'm going to start on that website. I think I'm going to start blogging a little bit about building thinking classrooms in the COVID-19 era. So what is it? Some of the things we talked about today. So this is what we learned from building thinking classrooms, 15 years of research and face-to-face settings, what does it now look like in a synchronous online or asynchronous or modified face-to-face or hybrid setting? And start to explore that space conceptually, in small amounts of experimentations and hope to get teachers who are implementing, trying those things to pile on and give feedback on some of those things?

Kyle Pearce: That is awesome, Peter. We are super thrilled to not only have you back on the show again. You know what? We didn't make it through all 14. So you know what that means Peter.

Peter Liljedahl: Yeah.

Kyle Pearce: Maybe sometime in the next nine months or so we'll be knocking on your door out west again, and we'll get you back on here. So we want to thank you so much for taking the time today to share your insights about the building the thinking classroom with all of the Math Moment Makers who are listening to the podcast from around the world. And we're wishing you all the best with your upcoming book launch. I'm sure it's going to be a huge hit and very helpful to so many educators around the world.

Peter Liljedahl: And thanks, guys, thanks for the work you do. And thanks for all the teachers out there who are engaging in these podcasts. The mere fact that you're dialed into this means that you are change makers.

Jon Orr: So true. So true. Thanks, Peter. And we'll be in touch. Wow. Thanks so much, Peter, for joining us on the Making Math Moments that Matter podcast. You as always have dropped a bunch of math PD knowledge bombs on us.

Kyle Pearce: I'm sure we'll have to have him back for yet another episode to continue digging even deeper into his thinking classroom framework. And I'm sure many of you out there are going to be rushing out to go grab that book. It was an honor-

Jon Orr: I am.

Kyle Pearce: ... to be able to review that book as it was in the draft stages. So I encourage you to definitely pick it up.

Jon Orr: Before we head out of this episode, there's a quick reminder that the 2020 Make Math Moments virtual summit is coming soon on Saturday, November 7th, and eighth. So get yourself registered at makemathmoments.com/summit.

Kyle Pearce: That's right Jon, our free virtual summit's going to feature live and pre recorded sessions for you to enjoy and you'll be able to catch the replays for up to a week afterwards. So if the weekend gets busy, catch what you can and then grab some throughout the following week.

Jon Orr: Yeah. Peter from today's episode, will be presenting with a bunch of other amazing speakers. To catch who they are head to makemathmoments.com/summit now, to ensure you get access to this awesome PD from your couch event.

Kyle Pearce: Awesome stuff, Jon. So how about you at home? What was your big takeaway from this particular episode with Peter? Go ahead, share it with a friend, a colleague or send us a message on social media @MakeMathMoments on Twitter, Instagram and Facebook. And don't forget about our private Facebook group, Math Moment Makers K through 12.

Jon Orr: In order to ensure you don't miss out on new episodes as they come out every Monday morning be sure to subscribe on Apple podcasts or your favorite podcast platform.

Kyle Pearce: All right friends show notes and links to resources plus full transcripts from this episode can be found at makemathmoments.com/episode98. Again that's makemathmoments.com/episode98. Well, Math Moment Makers until next time, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And a high five you. (music).