Episode 225: How Building a Thinking Classroom Can Make Math Moments
How can we build thinking classrooms to create memorable math moments? In this special episode we share our opening session of the 2022 Make Math Moments Virtual Summit.
Join Kyle, Jon, and Peter Liljedahl as they unpack how elements of a Thinking Classroom are entwined in the Make Math Moments 3-Part Framework.
- How to structure your lessons so they are engaging and encourage student thinking;
- What classroom structures can you put together to make every moment in your classroom matter;
- What are key questions you can ask students to gain the most insight into their thinking.
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Make Math Moments Framework [Blog Article]
Peter Liljedahl: To kickstart the process, the task needs to really hold all the attention, right? It's got to be where all the engagement is and it's got to hold that attention in the direction we want to hold the students. But as we get better at this, we still want to use good tasks, but the students are now, they're walking alongside us, right? They have some inaudible engagement as they come into the activity. But getting back to your first question, which is, okay, so I'm going into a classroom and what am I going to pick as a task? Well, I often start with a conversation with a teacher about, there's always two questions. One, where are you in your building thinking classroom journey?
Jon Orr: How can we build thinking classrooms to create memorable math moments? In this special episode, we share an opening session of the 2022 Make Math Moments Virtual Summit with you.
Kyle Pearce: Awesome stuff. Join Jon, myself, and our wonderful special guest, Peter Liljedahl Hall, as we unpack how elements of a thinking classroom are entwined in the Make Math moments three part framework.
Welcome to the Making Math Moments that Matter podcast. I'm Kyle Pierce.
Jon Orr: And I'm Jon Orr. We are from makemathmoments.com.
Kyle Pearce: This is the only podcast that coaches you through a six step plan to grow your mathematics program, whether at the classroom level or at the district level.
Jon Orr: And we do that by helping you cultivate and foster your mathematics program like a strong, healthy, and balanced tree.
Kyle Pearce: If you master the six parts of an effective mathematics program, the impact of your math program will grow and reach far and wide.
Jon Orr: Each week here you'll get the insight you need to stop feeling overwhelmed, gain back your confidence, and get back to enjoying the planning and facilitating of your mathematics program for the students or the educators you serve.
Kyle Pearce: All right, my friends, keep your ears open during today's episode because while we'll be exploring all six parts of our metaphorical tree, we're going to focus in on the branches of the tree that represent an effective mathematics program. That means our pedagogical content knowledge, looking at the teacher moves and how we actually facilitate mathematics with our students.
Jon Orr: So this is the 2022 virtual summit opening session with Peter, and we chatted all about the intermingling of Peter's work in his 14 elements of a thinking classroom and in the work that we're doing here with our frameworks, and we talk a lot about the teacher moves that you should be kind of thinking about as you're starting tasks, teaching through tasks. We get Peter's take on that. We get our take specifically remember us talking about using non-curricular tasks and curricular tasks and when we should use some of those and when we shouldn't use some of those. So stick around here and let's dive into the branches.
Kyle Pearce: All right, here we go.
Peter, great to see you. How are you doing tonight as we get ready to kick off the fourth annual virtual summit?
Peter Liljedahl: I'm doing great. It's sort of ironic that I'm actually here in Ontario as we're doing this and we're probably sitting two hours apart from each other, which is-
Kyle Pearce: We are.
Peter Liljedahl: I don't think we've ever recorded anything together where we've been this close.
Kyle Pearce: That's true. If we actually all get in the car and start driving now by the end of the session, we'll maybe be able to, we can meet-
Peter Liljedahl: In Hamilton.
Kyle Pearce: Yeah, I was thinking Hamilton, London, somewhere around there. Why don't we dive into what we're hoping to achieve here in this session and then we'll dig right into the conversation.
Jon Orr: We'd like to start one of our sessions off with kind of giving you the outline of what you can look forward to by being here, by being with us, by participating and engaging. So the first thing we're going to give you in this session is where you can begin to craft a task to ensure that students' curiosity is sparked right at the beginning of that task and then all students can enter. That's something that we always are striving for in our lessons. What in our teaching and what we do over at makemathmoments.com is how do we get that curiosity sparked and get students to enter into this low floor environment? Kyle, what else are we going to do here?
Kyle Pearce: So we're going to get thinking how are we going to generate thinking? And finally, Jon, what's the last piece before we introduce our special guest for the evening?
Jon Orr: Yeah, we're going to talk about teacher moves, intentional teacher moves on during the planning and the facilitating process that maximize effectiveness when implementing either building thing in classroom ideas from Peter's book or make math moments ideas and the tasks that we share over on our website. So we're hoping to give you those teacher moves so that you can make the most impact when you're using these ideas in the classroom.
Kyle Pearce: Awesome stuff. And our special guest that we are so honored to be able to chat with again is Dr. Peter Liljedahl. We had him back on for episode number 98 and continued that conversation and dug deeper and he's also been a part of the virtual session too, of the three previous years. So it's great to bring Peter back for the conversation. So Peter, we are here to chat with you about what actually matters in order to make some math moments and promote thinking, and maybe it's not even an, and it's like if you want to make math moments, you have to get kids thinking, I think is more of the caveat there. What are your thoughts?
Peter Liljedahl: Well, first of all, thanks for having me and thanks for everybody who's decided to give up their Friday evening, and I know you're coming from all over the world, so Friday evening is probably only a local descriptor. Some of you're coming from yesterday I think. And some of you are coming in from tomorrow.
Jon Orr: That is wild.
Peter Liljedahl: So welcome everybody. Like I say, always, I think this is a height of professionalism when teachers are willing to engage in this kind of professional learning outside of the confines of their contract. It just shows how dedicated we all are and how you all are to your profession.
Yeah, let's talk. So I think that it's sort of ironic that you had that heading make math moments and/or thinking as if they're mutually exclusive events, but I think you and I have always, we've been on the same journey for a long time. We've been trying to find ways to engage students to captivate their interest, to capture that curiosity, to vector that curiosity towards something that is productive and satisfying and rewarding for the students and so that they can start to learn, think, attain and start to feel like that they are capable of doing mathematics. So I think we come at it from different angles, but I think we're all trying to achieve the same thing.
Jon Orr: Awesome. Awesome. Yeah. Peter, why don't we start here? When we work with districts and we work with teachers and what we primarily start with is when we help these people along a pathway of learning to change some of what's happening in their classrooms, often we start with what type of lesson do we pick? How do we pick a task to use with our classroom? You saw everyone here say yes, they know about building thinking classrooms. They probably know about make math moments. They're here with us. They kind of know that the tasks are there or they know about elements of how to build a thinking classroom. But a lot of questions we still get a lot of questions I think you probably still get is how do you pick that first task to start with or a task just in your classroom? What do you look for when you think about a good task to use in your classroom and what motivates you to pick that task?
Peter Liljedahl: Okay, first of all, so when I work with a school district, I work in different capacity. So in one capacity I'm going into a session, maybe I'm working with teachers that day, maybe that's what I'm doing and I want them to live and breathe and feel a thinking classroom. So I'm picking a task for them, and this is sometimes I actually have to say explicitly, do not use this task with your students. This is a task I pick for you as learners to experience a thinking classroom. I'm not going to come into a professional development session with a really exciting long division question. I got to come in with something that is clearly outside of the curriculum but also hits them as a learner, much more student as a teacher.
Jon Orr: Is that because you want to get away from them steering their math learning at that point and getting them to experience something that they don't know at that point? Our students, when we do a math task with them right at the beginning, they don't know exactly what math is going to happen that day or they might not even, we're trying to develop and connect some math concepts that day. So we're not expecting them to know the math when we start with a task task is that why you're picking a task for teachers? It's like, Hey, I'm going to use this task because I know you don't know where we're going here.
Peter Liljedahl: I want their first experience with me and anybody who's in this room who has done in-person or even a virtual workshop with me knows that I want your first experience with me to be as a learner. So I need to immerse you into that space. I need to give you something to think about, and because I'm giving you something to think about, it can't be something you already know about, right? So this is why I try to get as far away from the curriculum as possible, but it also fits into that research that we found that we got to start with a non curriculum task. I want to capture curiosity without that forward seeking radar going, beep, beep, beep, beep, beep. Where can I use this in my curriculum? Where does this fit in? I just want you to be a learner. And I think that's the same thing we want with our students as well when we first start trying to expose them to a different way of being in the classroom is we want to shed those expectations that we're trying to hit an outcome or a standard that it's No, we're going to strip away the sort of outcomes here and we're just building an experience and I want you to just be present in the experience.
Kyle Pearce: I love it. The way we look at planning and I see a very, very similar thinking process here is what is it that we want our audience to land on? What are we hoping that they'll walk away with? And of course, as you had mentioned, for educators, what you want them to walk away from is probably very different than say what we're after our students to walk away from. I have a curriculum I need to teach and that I want to make sure I hit this standard or that standard. But for educators, it's almost like you want them to experience what you want students to experience. That thinking, the experience that Jon and I didn't give our students, when we look at this gradual release of responsibility classroom, I realized that my students actually didn't do a whole lot of thinking and it was almost as if I told them exactly what we were going to do right up front and the cat was out of the bag.
So I'm wondering why don't we get started with that in order to this quote, we were chatting the other night as we were preparing, and you brought up this John Dewey quote, which I thought was really helpful for where we want to take the discussion now, which is around this idea of how do we capture the attention, how do we capture and engage our students? Now the same is true for adults. Right now we're trying to capture the attention of everyone that's in this room. There's over 850 people who are joining us live right now, and we're trying to make sure that they want to look this way and we need to do the same thing for our students.
I know for myself, I don't know if you fell into this, Peter, I'd love to hear your perspective on this, but for me, I used what I call attention gimmicks at first. I tried to appeal to whatever the students are interested in, but it had nothing to do with what my goal was for the lesson that day. And then there was this massive gap in between. I'm not sure if you've been there or how you helped teachers overcome that. What are your thoughts on that?
Peter Liljedahl: Well, it's like, okay, everybody look over here. Everyone get really interested over here, get really focused on this. This is really exciting. Now we're going to do this thing over here. And it was like, which is less exciting? It's sort of like I'm trying to entice you with a little bit of reward upfront and so that I can ... And you're right, John Dewey has this beautiful quote about we want to capture students attention in the direction we want to hold it. Capturing students attention isn't difficult. All we need is a wig and a rubber chicken. You got their attention, but it's not in the direction you want to hold them. If we're trying to get them engaging in patterning or looking for constructing number or constructing shape or we want to get their attention on extrapolation, we got to capture their attention in that direction.
It's not trivial, but it's not impossible. And one of the ways that I've come to reconcile with this is that the thinking classroom research really showed me that my goal is not exclusively to find engaging tasks, it's also to build engaged students. And I think we're alike in this regard, which is that to kickstart the process, the task needs to really hold all the attention. It's got to be where all the engagement is and it's got to hold that attention in the direction we want to hold the students. But as we get better at this, we still want to use good tasks, but the students are now, they're walking alongside us. They have some inaudible engagement as they come into the activity.
But getting back to your first question, which is, okay, so I'm going into a classroom and what am I going to pick as a task? Well, I often start with a conversation with a teacher about, there's always two questions. One, where are you in your building thinking classroom journey? Because if you haven't gotten off the mark, guess what? We're doing non curriculum. But if you are off the mark now, what is it that you're teaching? What is it that you're trying to get to? Okay, you want to do linear relations, let's go in that direction. You want to do Pythagoras, go in that direction. You want to factor quadratics, complete the square, whatever it is, let's go in that direction. And then we start trying to think about how do we pick a task? And I always keep a couple of things in mind when I'm trying to bring curriculum into the classroom and I'm trying to do a task that's curriculum focused.
One, is what's my first question? What is the context, the setting, what is the problematic situation in which I want to engage them in? So sometimes it's something as mundane as, let me show you a really cool relationship in a right angle triangle. But sometimes it's what do you notice if I do this? And so what do you notice? If I take a ream paper and drop it on a desk and I take another one, I drop it on a desk and I drop another one. What do you notice? Now I've got their attention. But then the next question is, where do we go with that? And now we're talking about what is my goal? What is the outcomes I want to hit? Where am I willing to zig and zag? But what is always on the forefront of my mind is what am I going to ask them second and what am I going to ask them third, and what am I going to ask them fourth?
Because I can't ... one of the hard and fast rules in a thinking classroom is nobody ever gets to be done. So I always have to think about, it's not just like we used to do in the old days. Here's the activity, here's the worksheet that you're going to fill in. It's already drawn the table of values for you to fill in, and the graph is there for you to graph it. And then the four questions I want you to answer, no. It's this sort of slow reveal.
Kyle Pearce: Peter, I feel like you were looking in some of my old planning materials when you were talking about that because that's exactly what we would do. And I brought this quote up. I'm sort of for those who are giving context to everyone who's joining us, I've got a massive slide deck with all kinds of ideas that may or may not come up in this conversation, but this quote sort of jumped into my mind. We use it a lot when we're sharing and presenting with educators that we often do a lot of these things ahead of time. And what Jon and I have realized over time is that if we actually stop pre-teaching everything and actually allow students to do some of the thinking, if we actually allow them to engage in a task first, then we can actually keep their attention much longer. So again, planning with that intentionality as you mentioned, I think is really important.
And something for us. I know Jon, maybe you can speak to this a little bit, and I know Peter for you, it's much the same is that we have to be thinking about what do students actually know? What is it that we want them to now know and how high can that ceiling be? And as you mentioned Peter, it's almost like it's a limitless ceiling. We need to be thinking far enough down the road that students aren't just standing there saying, okay, now what? Or I'll just go sit down and wait for the rest of the group. We sort of have to have this nice long runway of ideas here. Jon, what are you thinking about when you are planning your tasks and trying to figure out which one we need to select to try to engage students, but then also get them to that thinking piece that we want?
Jon Orr: Totally, totally. And I think we're along the lines here with Peter. What Peter is thinking from what I'm hearing in what I've experienced when viewing your live sessions and being in your live presentations is thinking about those questions. And I love that you're thinking about all those questions down the line because you know exactly where you're going to go. So that intentionality is key. And I think when we start to create our tasks in our classrooms and the tasks that we have over on our websites, we start with that fundamental idea on context. We'll take a context and we'll think what is a truth I can reveal from this context that kind of keep us going down that line. So a lot of times we try to pick a task. We might go to the textbook, we might go and find a common task that we're going to tackle.
But what we like to do is we like to put in what we call the curiosity path. So we strip away a lot of information. And Peter, you're saying the same thing when you said you're thinking about those four questions or five questions down the line, and it's a slow reveal. We use that in our curiosity path. On the screen here. We're looking at this common problem that's from the textbook, but what we like to do is strip as much from that away as we can so that we can do the slow reveal. So it's kind of like, let's pull all that back. Let's see if we can spark a little bit of curiosity here because we know that if our kids are leaning in for curiosity, then we can take that attention we're getting and steer them in the right way. And then we have them focused in that right direction like that quote that we had up on the screen earlier.
So the curiosity path, we tossed a name on that, but it starts with withholding as much information as you can. If you think you pulled a couple questions back, you gave them A and B, and then you're like, you know what? I'll hold C and D for later. You probably still didn't strip enough away. It's like we got to go right back to the beginning and go let's start with nothing and let's see if I add something in here that can give them a little bit of a nugget to get started, and then we'll keep giving them nuggets because that will create anticipation. And that anticipation is what kids are going to be like. I want to know more. Because if we can just start a problem by showing them some piece and then asking them to say, what would you notice? What do you wonder here? Let's get to an estimation and then go, "Okay, if I wanted to take this estimation and make it more accurate, what do I need?" Because that part for us is our game changer question.
Because if students are starting to tell us what they need to make that estimate more accurate, they're already on this thinking pathway, right? They're already thinking about strategies. They're already thinking about what they're going to do with the numbers and hey, and they might even be going down a pathway that we didn't anticipate. That's okay. We can say, you know what? Okay, you wanted this, this, and this. You wanted to know the height of this paper stack. Maybe you wanted 10 stacks. Well, I don't have 10 stacks. I have six packs of paper for you only. I'll help you with the height here. So we like to take that information and give them what we've got in mind, but listening to what they have in mind is the problem solving practice. And then think how much information you get as a teacher to know where that student is by just listening to their strategies.
Peter Liljedahl: So you have this nice visual here. Let me tell a story about when I use a derivative of this task with a group of grade eights.
Kyle Pearce: So that was a inaudible little math term there too. I like that.
Peter Liljedahl: So I'm in a grade eight classroom and I'm working with a teacher and she wants to do linear relations. And I thought, "Hey, why don't we do this three math task," as the paper stacking one from you. But I deliver it a little bit differently. So it was the kids don't know me at all. They've never met me before, but they know I'm coming in. I come in, the teacher launches, and then I call everybody over to me and we're standing around this table and I'm standing sort of near the edge of the room and there's a counter behind me, and I reach behind me and I grab one of these reams of paper and I drop it onto the desk. And when you drop a ream of paper under the desk, it makes a big bang right now I got their attention, but not in the direction I want to hold it, but I got their attention. Then I grab another one, drop it, bang. They are hyper focused on what I'm doing now because obviously something's going to happen here. I grab another one, bang, I grab another one, bang, and then all I do is I look up at the ceiling, I look down at the pile of paper, I look up at the ceiling, the kids are all looking up at the ceiling and-
Jon Orr: What's up there?
Peter Liljedahl: And then I just said, I wonder how tall it would be if I have 50 of these. And then I said, data's around the room. Go. And we put them into groups and off they went. And we had photographs of the paper stacks on the table spread out around the room. And on each photograph there was a height. And so they had to run around the room and gather this information, what these different data points. So there was one for two high, there was one for three high, there was a five, there was a eight, and there was a 10. Okay? And they had these data points and those numbers were chosen very deliberately because one of the things I'm always thinking about is not just what is it, now I've sparked their curiosity. Now I've got them going in the direction I want, but where is the cognitive dissonance that I want to occur?
Where is that, when they think that everything is going exactly as they're expecting that something's going to happen. And the place I place that in that one was I have a data point for five, and I have a data point for 10, and I'm asking how tall 50 would be. What do you think students do, right? Well, one group grabs a five and says, we're just going to multiply this by 10. There's our answer. There's there with a big grin on their face. We got it. Another group grabs a 10 multiplies by five, we're good. Then I put those two groups together and I say, how come you have different answers? Because one of the things that I never said anything about was the table.
Jon Orr: And now just to be clear here, Peter's describing the same task, but he said he delivered it differently. So this picture didn't match. You didn't give them this picture here on the screen. They had no idea that this was on a table. You were stacking them on the table right in front of them, but you didn't clearly say they're on a table.
Peter Liljedahl: I didn't say that the measurement's from the floor.
Jon Orr: And so now the pictures, the data around the room, Peter fill us in was the table in the picture.
Peter Liljedahl: Yeah.
Jon Orr: And so then I'm picturing the number, the height is around that many stacks, right?
Peter Liljedahl: Yeah. It says if there's three stacks, it says 90 centimeters.
Jon Orr: Beautiful, beautiful. I love it.
Peter Liljedahl: And the kids, they don't really bat an eye about that. They don't interrogate that data at first, right?
Jon Orr: Yeah. So what did your students do? Because I'm imagining that flow is that students are multiplying and they're saying, "Well, how did they get different answers?" You've obviously done your four question, five question plan. You knew they were going to try to do this. That's an anticipation stage of you anticipating their thinking in advance. And then what was your next question to them after you're saying, "Well, why do you have different answers?" And then I'm imagining there's this key move next for you.
Peter Liljedahl: So then the next one is, okay, whatever they've done is if they've done something where they've made too hasty an assumption, you got to slide in there. And my favorite move is to pair them with a group that has a different answer.
Jon Orr: Right, right, right,
Peter Liljedahl: Right.
Jon Orr: Yeah, convince each other who's right?
Peter Liljedahl: Yeah. My statement is always, huh, you have 140 meters and you have 32 meters. I can guarantee you that at least one of you is wrong. And then they start talking because I've created that cognitive dissonance now. And then what happens is now they start interrogating the data a little bit closer and they're like, wait a minute, "Two stacks of paper? There's no way that 78 centimeters."
Jon Orr: Right.
Peter Liljedahl: "How can two stacks of paper be 78 centimeters when three stacks is 81? How is that possible?" So they start interrogating the data. Once they start interrogating the data, we're off to the races. Now they're starting to look at calculating how thick one is, or some will calculate how thick two is, and then they'll extrapolate that upwards to 50, but then add on the original condition that they have. So we're off to the races. Now they've created a table of values. The table may or name may not be ordered. So here comes the next question, nice table of values. What do we normally do with tables of values? And they're like, oh, we graph it. Now grade eights would rather chew off their arm than draw a graph. So it's like most-
Jon Orr: inaudible to be honest. Let's be honest here.
Peter Liljedahl: But the questions I ask is also like, well, how high would it be for a hundred? How many stacks of paper do I need to get to a height of six meters? So now I'm flipping the question, which is not if we're thinking about graphs or tables of values, I'm flipping between the X and the Y here. And what's nice. So you always pick a number that has that. It doesn't go perfect that they have have a decimal. They have to have a discussion about whether they round up or round down. And then we have this discussion, and then here comes the hard part, and I've made a modification to this. Walking around and trying to convince them. So then I walk around and I hand him a ream of paper. I said, this ream is yellow. Now, I don't know if you know this, but yellow paper is thicker than white paper. That's a hard sell. Yeah. Trying to convince them that this ream paper is thicker. What's going to happen to your answers now.
Kyle Pearce: Card stock.
Peter Liljedahl: Right? So now what happens, because now we've been pushing into, we may have been table of values. We may have, depending on what the outcome is, we may have pushed to a graph. We may have pushed to an equation with a slope and a constant. And this constant now takes on a real meaning. It's not just a y intercept. That constant is the height of the table. It's a real meaningful thing. So then I'm trying to convince them that yellow is thicker and blue is thinner, and I'm asking more conceptual questions. So what happens to your graph or your line or your equation or your table of values or your answers? If we are now doing it for yellow paper, and this is why I've made a modification, I don't do reams of paper anymore, I do books.
Jon Orr: You can get different book sizes.
Peter Liljedahl: I can get different book sizes. So I can go up to a group and I just hand them a book and I say, what do you think will happen? If the question was about this book.
Kyle Pearce: Our consolidation prompt for that particular task is about books. So it's interesting, we've never had this discussion before, but our heads went in a very similar spot. Something I wanted to mention as you were describing that scenario was again, I'm hearing when you described it's like we're using the same context, and I would argue that some of the same big ideas are being revealed, but you are going for a certain part of that big idea. And we tend to use that particular task to try to explicitly get students to calculate or to kind of bump into this idea of the initial value or the why intercept. Now mind you, same thing as causing your struggle for your learners, but you're trying to almost have them bump into this non-proportional relationship or this partial variation over direct variation. So again, I wanted to bring this back up about this idea of planning with intention because it's exactly that. It's not what task will engage my students today and sort of keep everyone thinking just to keep them thinking.
Peter Liljedahl: It's not a set it and forget it, that's for sure.
Kyle Pearce: Exactly, exactly. There has to be that intentionality at its core. And I would argue that the more intentional we are about what we want to do, the easier it is because then you know what not to do and what you do want to do, it allows you to say no to more things. So when I say yes to this thing, it means I'm saying no to something else. And if I can actually focus in on the things that I need to say yes for today's lesson, I think it makes that decision making process a whole lot easier.
This last piece I wanted to kind of articulate, which I'm hearing from is this idea of emerging strategies. You had already thought this through, you pre-planned that students are likely going to bump into this idea. I saw someone in the chat saying, no students knew it was the table. There may be some students that might have bumped into that idea, but the reality is that we often rush to tell kids these things before we give them the opportunity to think it through themselves. So it's not something they'll just immediately know because they don't have experience playing with the context. And then obviously the mathematics that are underpinned that context as well.
Jon Orr: And I wanted to add something here too about the way that this was unfolded because I think sometimes this whole set it and forget it thing can come into play. It creeps into our lessons, it just shows up. And what I mean by that is a lot of the tasks that we share on our website start with a video prompt, and that is done to capture some curiosity. It's allowed us to strip away information and present it visually. That helps us kind of tell a story, wraps it around in some context. And what Peter was doing this at the tables, just doing this at the tables, it doesn't have to be a video context, it doesn't have to be at the tables. And I think a lot of the times teachers, they misconstrue engagement or the beauty of this task comes from this video like this one you're watching here of Kyle and I eating some chocolate or from the storytelling or I think Peter, maybe you can admit to this as well, I think a lot of teachers grasp building thinking classrooms in the work that you've been doing.
One element that we hear so much is the vertical non-permanent services, right? It's like everyone's up, the boards are working, and as a teacher who's had kids at their desks for years and mimicking what's happening at the board, and as soon as we get them up working, we're like, we've got engagement. We've got engagement. And so I think that one move one thinks "I'm doing a thinking classroom because I have students at the boards," and I think that that part is the same idea that Kyle and I are experiencing with teachers who are like, "I'm making math moments that matter because I'm playing a video." It's not the essence of what makes a building thing in classrooms or a math moments. It's the questioning in the way that you handle that task that actually does it.
Peter Liljedahl: Yeah. And by coming back to this, the planning, so there's planning with intentionality. Now does that mean it always goes well? No, but we still have to be intentional. We have to try to anticipate what are the things that the students are going to do? What are they going to do wrong? Where are they going to bump into things? And then can I cause them to bump into something? There's this wonderful word out of complexity theory called occasioning, which is I can occasion something to happen, that means it might happen. I can set the environment so that the environment is ripe for this thing to happen. It doesn't mean it's going to happen, which means now I've done all the planning. Now it meets with first contact, right? The kids are now on the boards, they're thinking their curiosity is hooked, but now what? Right?
Because that's going to run out real fast. Now comes the on your feet dynamic. I have a plan and now I have to somehow delicately see if I can help that plan come to fruition by asking the next question, by giving the next prompt, not synchronously, asynchronously. When a group is going this way, what can I say to them? Yeah, when a group is going this way, how can I enter that conversation and enter their conversation, not just say, "Okay, you're onto part B now and part B, you have to do this." But rather, okay, what is their particular A look like and how can I feed into that? And there's this delicate dance between trying to bring my vision to life and trying to avoid them sensing that I'm sheep dogging it too much, is what I call it where I'm they start to feel like I'm starting to nudge them too much, that I want them to live in their own curiosity, in their own engagement as we're working.
And it doesn't always go well. And sometimes it goes better than expected. I'll give you an example. We were doing, I was in a grade nine class. We were doing one of Alita Classen tasks about jobs. It was linear relations. Here's a job. So she has all four jobs at once, but we didn't do that. We said, "Here's your first job. Your parents have said you can get a job. Here's your first job." And it's a graph. So put this graph up on the board and it's a job that starts in negative $60 and it's a linear slope. It's a slope, it goes up, I can't remember what it goes up by, let's say $20 an hour, but it starts in negative $60. And the kids are like, grade nines are going, "What the heck kind of a job starts starts at negative $60?" We haven't said anything, but they're trying to figure out what kind of job starts at negative $60.
And we didn't anticipate they would have this conversation, but they had amazing ideas. Well, maybe you got to pay for gas to get to work, or maybe you got to buy a weed whacker because you're going to be mowing lawns. Or maybe you had to buy some clothing or a uniform. They were amazing. And then, okay, so here comes your next job. And the next job was an equation. So they graphed this equation onto the same graph and it started at zero, thank goodness, but it had a lower slope. And now the kids are going, and we expected them to say just sort of like, "Okay, well this one starts higher, this one will be better." Or they'll say, "This one's better eventually because it's steeper." No, every single group without us expecting it spent literally 20 minutes trying to figure out where these two lines intersected.
And we hadn't asked them to do that. And it's just like we had piqued to curiosity and this thing was going to run its course, and now we're in the thick of it. Now we have two choices. We can get them on to C. C is a table of values. You got to plot that one. Or we can let this run. Right now, they're going in a direction we hadn't expected, and they're going in a direction that in the end might not actually make a difference when they get to Part E. But they're curious. They're thinking, they're engaged, they're chasing their own passion and they're doing a ton of math. As much as we plan. We also have to learn to be responsive and responsive to where they are in their conversation not where we are in our conversation.
Kyle Pearce: I love the idea as well that there are going to be, and I think it's really important for everyone to know that you could spend a ton of time prethinking and planning your lesson, and sometimes it doesn't go anywhere near what you're expecting, but the thing that you should anticipate is what you are going to do regardless, which is we need to tie those loose ends at the end. So I'm showing some sample student work here, and it's like I want to be able to look at the student work and try to think how am I going to take some of those ideas and then weave them into the new learning from today. So if it is a new model, for example, let's say it's a double number line that you're trying to help students understand or see, I could take some of that student work in order to help them see it maybe in a new way.
Is it great if the students brought up some student came up with this solution? That would be perfect, that would be ideal. But we know that we don't live in an ideal world. We still have to get to that new learning. But the thing that we did do that I never did for a long time in my career is I actually let the students think I got to. Actually, the best part is like you were saying, Peter, you're walking around listening to what students are saying. And I think that's part that oftentimes we miss out. It's like, okay, the kids are doing work, they're engaged, but you're learning so much about where students are, right? Are they on board with the context? Do they understand what's happening or are they lost? If they're lost, that means that next prompt has to come where, okay, I need to help redirect them a little bit to try to get them back on track. But ultimately at the end of the day, we don't want to just sort of go, oh, it didn't work out. We still need to ensure that they took something new away from the experience, hopefully building on the work that they've done so far.
Peter Liljedahl: And that this is in my book in chapter nine, I talk about seeding, S-E-E-D-I-N-G, this idea that, for example, I drop that thing about, so what do we normally do with table of values? Oh, we draw a graph. Maybe I really want this lesson to culminate in a graph. A graph that shows all the data and how we can extrapolate and that we can look at, well, look, if we want to understand how tall 100 reams of paper is, we just have to go out on the X axis to a hundred and look at what the Y value is. Or if I say how many reams of paper is going to take to get to six meters, I can go to the six meter mark on the Y axis and go across, and I really want that to happen.
So I may plant that idea. Wow. But like I said, still want to draw that graph. So I may have to circle back around, go, how's that graph going? And like, Hey, would you like some extra colors to draw that graph? When you plant the seed, it doesn't always want to grow. And sometimes the ground is not fertile and you have to maybe plant it in multiple places, but sometimes it just isn't going to grow. And now you have to ask yourself, what do we do in that situation? Bait, do we start pushing too hard so the students start feeling like we're trying to guide them too strongly? One of the little tricks that I've done, if no one's drawing the graph and I really want the graph, I just grab a blank piece of whiteboard and I draw a graph.
Jon Orr: Exactly.
Peter Liljedahl: And then I put a red box around it, and then I put a number on it that nobody had. And then when we're doing the consolidation, I go, so can someone not in this group tell me what this group was thinking? And nobody has a clue that I drew that graph.
Jon Orr: Because they were too busy at the boards. This is good because I think we also get questions like that. What do we do at the end of a lesson? What happens if the students didn't get to what I thought they were going to get to? I plan these questions out and they didn't get here. And I think we always said the same thing. It's like we step in and we can say like, "Look, we're going to tie a bow on this. We want to walk out the door today with a learning goal, going out with them" going, and the student knew exactly what the learning goal was that day and what they can do going forward. We got them thinking, we don't want them leaving, going, I did some thinking today, but I'm like, how did it tie into where we did two days ago or what we're going to do next? I did some stuff today, but I don't know how it fit in with the grand scheme of things.
So we like to say, tie a big bow on it. And so at the end of that lesson, it's like we have to restate, and Kyle, we were chatting about this on the last podcast episode about the oblivious viewer effect. And Peter, we were talking about it with you on that chat we had last week about it. And you're watching a show and it's a mystery show. And then it was subtly revealed what the big reveal was of this mystery. And you're sitting there going, oh my gosh, I got it. I can't believe that. We had used the Sixth Sense as an example of catching on.
Peter Liljedahl: In hindsight I couldn't believe, so clear.
Jon Orr: It's so clear now. And then where I was talking with a friend of mine who had seen that same movie or show and they're like, I missed that part. Right?
Peter Liljedahl: Or the one who got it right away, right?
Jon Orr: So some of us are getting those things in the movies and some of us aren't. And that's also what's happening with us in our classroom. If we were not making the connection to the learning goal they needed that day, what is the success criteria here? We had to make those connections manually in that connect stage. So it's like we had to tie that bow on it so that they could walk out going, "Okay, I know exactly what I need to do going into tomorrow's class."
Kyle Pearce: Isaac said, don't forget a beautiful mind has a similar effect.
Peter Liljedahl: Yeah, absolutely. So I want to just come back to this idea. You plan with intentionality, then you got to be able to zig when they zag. So you have all your contingency plans, and sometimes you got to think of those on your feet. And then like you were saying, we got to tie a bow around it, but we also have to decide where we're going to tie the bow.
Jon Orr: True.
Peter Liljedahl: If our goal was for them to get this far, sometimes they only get this far. If we tie the bow up here, that's not helping. We tie the bow here, we revisit the next day, we do some introspection. But the other thing happens too, which is this is how far I was expecting to get to. And they went this far and now it's like, okay. I'll give you an example of that. I was observing a lesson by Jamie Depepo in Ottawa, Ottawa, Catholic, and she was doing another three act task with a candle burning. And now she had the same thing, the data around the room. And she's an amazing storyteller. It's performative, and she's talking about that she's cooking a meal and for her in-laws, and last time the candles burnt out too soon and she needs to have ... And then so now the data's around the room and the kids start gathering this data and they get it into a table of values and then this conversation start. These were grade eights.
There is no way this is linear. This is the students. There's absolutely no way this is linear. And we're listening to this. Every group is saying this, this can't be linear, this is not linear. And we're just listening to have them, having them have this conversation. And then they draw the graph and they're like, "Huh, look at that. It's linear." So they overshot what we thought they were going to go to and we're like, okay, so we need to have a conversation about this. What made us think it wasn't linear? What does it mean to be linear? And what were you attending to? And now we're having to think about how they think and pull that out of them. And of course, the reason they thought it wasn't linear, because every linear function they'd ever seen, the X values go up by regular intervals. And this was a candle burning. It was all over the map.
Kyle Pearce: I love it. I love it. And I really like how you stated that going back to if you don't make it as far as you want to, oftentimes people want to rush get there. And then it's almost like all that hard work you did earlier is almost wasted because now there's confusion at the end. So take what you've got. But then on the other end, if students are pushing it further than you thought, ride with that as well. You want to make sure that you really take it to that next place. I'm looking at the time here. I see we've got about seven minutes left, so I want to make sure that we have an opportunity to kind of tie a nice bow around on this idea. I'm hoping friends, as you are thinking about this conversation thus far this evening, let us know in the chat if you have any questions, any wonders that you'd like us to address here and ask with while Peter's with us.
One piece for me, I think that's really important as well, and I have it up on the screen at the bottom here. So we talked about tying that bow with consolidation, but also giving students the opportunity to do practice as well. We do need to give students the opportunity to engage in more than one problem about an idea. So for example, if we go back to the stacking paper opportunity, we're stacking it on the table. Students finally understand that, oh my gosh, this isn't a proportional relationship. There's something else going on here. We do need to ensure that students do have an opportunity to play with another context possibly, another idea where they're engaging with this idea also, so that they actually can build a little bit of that fluency and flexibility.
I know that's something that I really struggled with for a long time. I had just like problem after problem after problem, but they were all kind of disconnected and all over the place. And students will still do great thinking and they'll walk away better thinkers, better problem solvers. But if at the end of the year you're trying to say, "Oh my gosh, my students didn't do any better on this exam or test" or whatever. We have to make sure though students also have the opportunity to kind of reiterate some of these ideas that we're sharing with them. Do you have any thoughts on that as well, Peter? What would you be recommending for educators on that piece? Because I think that's kind of a trickier one. It's not a sexy idea or topic to chat about or to keynote on. So I know it doesn't come up as often for us when we're presenting, and I'm sure it's the same for you as well. So I'm just curious.
Peter Liljedahl: Well, ironically, I did a keynote of about a month ago at the VC Association of Math Teachers Conference, it wasn't a keynote, it was just a parallel talk, but there was a lot of people there. But it was on exactly this idea. The idea of students writing notes, which is a form of reflection, note making, not note taking, and then what we call check your understanding questions and why are these so important? Because when students are in the thick of it, so we like to assume that the students are being logical and they're using deduction, they're making connections, they're seeing the big picture, they're seeing the minutia, that they see the logic of the whole thing. They see the arc of the problem from beginning to end, that it's all buttoned up and neat and tidy. It's not because they've been engaged the entire time in that activity through informal doing, right? They're spitballing, they're trying this, let's try that. Oh, that's a good idea. Let's try that. Okay, let's do it. Let's draw the line. All right?
And they haven't had a chance to connect all those pieces together and actually build that narrative, that narrative that goes the narrative that we see. So clearly we see this narrative of how this problem unfolds. They've been in the thick of it. They don't see that narrative yet. That's what the consolidation is for, is to help them, is to find that narrative inside of their work. And it's important that it's inside of their work because otherwise it's just do a bunch of thinking for 40 minutes and now ignore everything you did and just listen to me, right? Because then we're not honoring their work. So it's helping them find that thread, that arc within their work. Then they have to reflect on it. That's that meaningful notes.
And then it's what we call the check your understanding questions, which is, now, can you actually do this on your own now? And that's not a meaning entirely on your own, but were you able to take something out of this? Have you encapsulated it, reified it, right? Have you turned it into a construct, a schema that's inside your head now that you can take with you out of the lesson, but then use in a different context, what if they were dictionaries? What if we started stacking them top of the roof of the school? How does this change things? What if we used a chair instead of a table? What if? And being able to flexibly think about these things. So that purposeful practice, although I don't use the word practice, but that purposeful check your understanding question, that opportunity to actually test run your new construct that you have walked out of the lesson with.
And it's so important. And if I was to put a little bow on this whole thing, it's plan with purpose. It's not going to go perfect. The intentional in the lesson zigzag, it's not going to go perfect. Try to close it off where you were hoping it was going to close off. It may or may not. It's not going to go perfect. But we never learned from anything that goes perfect. When it goes perfect, there is absolutely no learning to be had by us. The first time we did that task with a paper stacking, I had a 10, I didn't have a five, and then a group did the 10 thing. They just multiplied by five. And I'm like, "Huh, what am I going to do now?" And then I learn, right? I need to have something that entices students, that creates that cognitive dissonance so that I can put them together. So I better throw a five in there as well. If I'm going to entice them with a 10, I got to entice them with a five so that we can get some cognitive dissonance going.
Kyle Pearce: The piece I really like about that as well, Peter, is I noticed in my shift in going from this, we'll call it pretty traditional, or at least the way I was taught form of teaching math to this way where I want students thinking, the thing that you also get, whether you like it or not, is your thinking. And to me, I like that it really made me enjoy teaching again. Whereas when you feel like you're reciting a lesson where it's like everything is exactly as it should be or as it was planned, it's not that engaging for you as the educator as well. So this really keeps you on your toes, but in a good way. You have to be open to it and just know that things aren't going to always go perfectly, especially the first time. You should not expect it to go well the first time it's going to be a lot of learning.
Peter Liljedahl: Don't force it. The got to work with teachers who will their plan to come to fruition. And it's just painful. It's a plan. It's taught a script. It's a plan. All plans are subject to change.
Kyle Pearce: And I love that. And one of the biggest pieces is reflecting, right? I mean, without reflection, you aren't going to be able to get yourself to do it differently or better the next time. And the next lesson plan, you'll learn something from another unrelated lesson. And with that, we're going to turn it to the group here. We have 924 people in the room right now, and we want you, my friends, to do a little bit of reflecting yourself. We want you to share what is something that resonated with you here today. Maybe it's something that you want to try next week. Maybe it's something that you want to think about and try next month. Maybe it's something more big, more lofty that you feel like might take the next 12 months, that next year to be working on whatever it is. Let us know in the chat.
And I'm going to nominate Jon. I was going to nominate Peter, but I want him to come back again, so I'm not going to give him this job. I'm going to nominate Jon to have to randomly select someone from the chat, which is a hard job with a couple hundred people, with almost a thousand it is incredibly difficult. So we're going to give you a second to do that. And while we do, I want to thank Peter again. Friends, I hope that you feel that through this discussion you are seeing how the beginnings of a task selection process or how a task creation process might look in order to ensure we get students focused on the things we want them keep their attention directed at the piece of curiosity that we had sparked with them so they can enter that task. I'm hoping that you're also feeling more confident in how you design your lessons for fueling sense making to make them think.
And then finally, those intentional moves we've discussed in order to make math moments through the making of students and thinking in your math classroom. So there was a lot of key items here. Jon, how are you feeling about selecting someone?
Jon Orr: Yeah, I'm going to scroll here-
Kyle Pearce: Just an academy membership too. I think it's scrolling for you.
In today's episode, we were focusing in on the branches of our effective mathematics program, keeping in mind that the branches of your tree represent the development of educator pedagogical content knowledge. This includes effective teaching and equity-based teaching practices. If you're spending valuable professional learning time on too many things that don't make an impact, the trees canopy will get heavy and begin to sag hindering growth. And as you heard here, the work that Peter does through the Thinking Classroom and the three part framework from Make Math Moments, they work so well together in order to draw in students, but then also draw out the pedagogical moves that we need to be doing in our math classes to ensure that students aren't just engaged, but that they actually make sense of the math that they're engaging with.
Jon Orr: So here's your action plan. What are you going to pull from this episode? What are you going to think about putting in place to support your branches and just create that strength. What pedagogical moves that you can make tomorrow, we're going to challenge you to pick one to try in the classroom if you have not yet done so already from Peter's work. And if you have not yet picked up Peter's book, make sure you go and do that so that you can learn all about his 14 elements of a thinking classroom and then how that jives with the Make Math Moments framework.
Kyle Pearce: I love it. I love it, Jon. Hey friends. Listen, if you're curious about the six parts of an effective mathematics program, you might be in the classroom where we call the trenches doing the hard work with students, or maybe you're at the district level. Hey, don't get me wrong, that's hard work to do as well, trying to help your educators across maybe a bunch of schools are across a very far distance. Either way, you can head to makemathmoments.com/report, and that'll give you an opportunity to take our 12 minute assessment, which is going to not only ask you some questions that might get you thinking and reflecting maybe about some of the things that you haven't even thought about before, but it's also going to send you a customized report right to your email that will highlight your most urgent next step in order to make the biggest impact in your classroom.
I don't know if you've been there before, but I was constantly making little changes in my classroom and putting a ton of effort into doing some of those things. And sometimes they came up flat, sometimes I had worse results, and then sometimes it was a massive impact. Well, this report is designed to help you decide what should you do next? What area do you want to focus on next? And then in that report, we'll even give you some next steps that you can take and some resources, some links, all kinds of goodies for you. So head on over to make math moments.com/report and you can take that 12 minute assessment regardless of whether you're a classroom teacher or a district leader. You're going to have the option to take the assessment that matches your role and you'll get yourself that customized report.
Jon Orr: Hey folks, don't forget if this is the first time you've listened to our episodes here. Hey, awesome, welcome. If not, you're a few in, you're a few deep, then no matter what, we'd love for you to hit subscribe and give us a rating and review on your favorite podcast platform. We put episodes out every single week have been doing so. This is episode 225, Kyle, so 225 episodes weekly. If you have not yet left that rating and review, please do so. We read them and it fills our hearts every time we read those.
Kyle Pearce: Awesome stuff. Friends on our website makemathmoments.com/episode225 is where you're going to find the links to resources, including a link to Peter's book, a link to some of his other useful resources as well, the Make Math Moments Framework, and complete transcripts from this particular episode. Once again, head over to makemathmoments.com/episode 225 and you'll be able to grab those.
Jon Orr: Thanks for listening to the Make Math Moments That Matter podcast, where we help you grow your mathematics program like a tree so your impact can reach far and wide.
Kyle Pearce: Well, until next time, my math moment maker friends. I'm Kyle Pearce.
Jon Orr: And I'm Jon Orr.
Kyle Pearce: High fives for us.
Jon Orr: And high five for you.
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