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Episode Summary:

As educators of mathematics around the world embrace the ideas shared in Building Thinking Classrooms, more and more students are being positioned to think collaboratively during math class instead of simply mimicking steps, rules, and procedures. 

While this shift in mathematics teaching practice is a massive leap in the right direction, our work does not stop once students solve the thinking task at hand. 

Rather, the real work for the facilitator now begins. 

Join Peter Liljedahl, Jon Orr, and Kyle Pearce from the opening session of the 2023 Make Math Moments Virtual Summit as they discuss one of the most important, yet often overlooked parts of an effective problem based mathematics lesson: the closing.

This is part 1 of the hour-long Keynote session from this past year’s Virtual Summit in November. Look for part 2 in two weeks. 

Resources:

2023 Make Math Moments Virtual Summit

Make Math Moments Academy

Attention District Math Leaders:

How are you ensuring that you support those educators who need a nudge to spark a focus on growing their pedagogical-content knowledge? 

What about opportunities for those who are eager and willing to elevate their practice, but do not have the support? 

Book a call with our District Improvement Program Team to learn how we can not only help you craft, refine and implement your district math learning goals, but also provide all of the professional learning supports your educators need to grow at the speed of their learning. 

Book a short conversation with our team now.

Other Useful Resources and Supports: 

Make Math Moments Framework [Blog Article]

Make Math Moments Problem-Based Lessons & Units

FULL TRANSCRIPT

00:00:00:06 – 00:00:17:22
Peter Liljedahl
Because what happens then is that the students start to realize that all that stuff they do at the beginning, all that inquiry discovery, all that whiteboard work, all that stuff they’re doing is just to kill time. It’s just there until we get to the end of the last thing where the teacher’s going to sit us down and tell us how to do it the right way, the best way.

00:00:17:23 – 00:00:21:19
Peter Liljedahl
We don’t want that either. We’re not going to do the organizing for it.

00:00:21:19 – 00:00:48:01
Jon Orr
As educators of mathematics around the world, embrace the ideas shared in building thing in classrooms, more and more students are being positioned to think collaboratively during math class instead of simply mimicking steps, rules and procedures. Well, this shift in mathematics teaching practices is a massive leap in the right direction. Our work does not stop once students solve the thinking task at hand, rather their work.

00:00:48:01 – 00:01:13:22
Jon Orr
The real work for our facilitators. Those of you who are like us teachers, classroom facilitators, the work now begins. Join Peter Lowell, myself, John Awe and Kyle Pierce from the opening session of the 2023 Make Math Moments Virtual Summit as we discuss one of the most important yet often overlooked parts of an effective, problem based math lesson. The closing.

00:01:13:24 – 00:01:43:24
Jon Orr
This is part one of the hour long keynote session from this year’s virtual summit session in November. And look for part two in two weeks. So stick with us and you’ll learn valuable insights into reevaluating traditional teaching methods and discover why leveling to the top might be holding back student engagement. You’re going to explore the power of the consolidating from the bottom strategy a teaching approach designed to make lesson closure inclusive and effective for all students.

00:01:44:01 – 00:02:08:21
Jon Orr
Finally, you’re going to learn the art of responsive teaching, fostering flexibility in your class lesson plans so that you can create dynamic and student centered learning environments. Here we go. Oh.

00:02:08:23 – 00:02:13:08
Kyle Pearce
Welcome to the Making Math Moments That Matter podcast. I’m Kyle Pearce.

00:02:13:08 – 00:02:16:03
Jon Orr
And I’m Jon Orr we are from makemathmoments.com.

00:02:16:04 – 00:02:26:22
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.

00:02:26:24 – 00:02:32:13
Jon Orr
And we do that by helping you cultivate foster your mathematics program like a strong, healthy and balanced tree.

00:02:32:16 – 00:02:41:01
Kyle Pearce
If you master the six parts of an effective mathematics program, the impact of your math program will grow in reach far and wide.

00:02:41:05 – 00:03:03:12
Jon Orr
Every week you’ll get the insight you need to stop feeling overwhelmed, gain back your confidence, and get back to enjoying planning facilitating of your mathematics program. Students or educators you serve. All right, folks, here we go with part one of the building thing classrooms Opening keynote session with Peter Little on closing your math lesson. So let’s get rolling.

00:03:03:12 – 00:03:12:12
Jon Orr
We’re here to talk about all of your favorite topics, but we also have Peter Little. He is giving away. Peter, he’s here. He’s coming from.

00:03:12:14 – 00:03:14:16
Peter Liljedahl
Halifax, Halifax, Nova Scotia.

00:03:14:16 – 00:03:30:21
Jon Orr
He’s over in Nova Scotia. Halifax. And we are honored to have Peter kick off the session as he did last year. Peter, you said this is your fourth summit of five summits. This is our fifth year in a row of hosting the summit, I think. I don’t know. That might be a record. I don’t know if we’ve had is.

00:03:30:21 – 00:03:40:22
Kyle Pearce
Just going to say it’s like the James Tanton to making Mathematics that Matter podcast to the summit world. Little analogy there but it is awesome to have you.

00:03:40:22 – 00:03:43:10
Peter Liljedahl
And it’s great to be live again this year.

00:03:43:12 – 00:03:59:11
Kyle Pearce
And yes, awesome. So you were obviously or I say obviously someone was in the same room as you today, so I’m guessing that you did a little session today. Tell us a little bit about it and then we’re going to get rolling here, my friend. So, Peter, tell us, how are you doing tonight, my friend?

00:03:59:13 – 00:04:29:16
Peter Liljedahl
I’m doing great today. I was in Port Hawkesbury in Cape Breton, so I was working with a group of teachers up there, mostly high school, but we had teachers all the way from kindergarten and it was sort of a day. One introduction to building thinking classrooms where they got to experience a lot of what thinking classrooms is like doing tasks at the board, starting with non curricular, finishing with curricular tasks and really experiencing those closing elements to a lesson so that they had something they could take away.

00:04:29:22 – 00:04:49:14
Kyle Pearce
That is a great segway because really every time John and I do a session, we always like to let people know upfront what they’re going to get. Sometimes there’s three, sometimes there’s four. Tonight we kept it to one because we want to keep this super, super focused. It’s something that John and I have been talking about for a very long time.

00:04:49:14 – 00:05:04:22
Kyle Pearce
I know it’s something you’ve been talking about for quite some time as well, and you were like geeked when we decided. We said, Hey, are we going to do another session together, just like we did last year. We had a blast. We enjoy every time we come together on the podcast. And we said, What about the end of the lesson?

00:05:04:22 – 00:05:35:03
Kyle Pearce
I’m going to say end because your word we said consolidation. You said, I like the word closing because to you you’re like, there’s other elements there and we don’t want to forget about them. So I really respected that and I thought, I’ve been saying consolidation for a really long time. And really what it comes down to is if I say a word and they’re not exactly sure what that means in their context, one teacher might think they’re doing a consolidation when in reality maybe they’re not, or maybe they’re forgetting something.

00:05:35:03 – 00:05:48:00
Kyle Pearce
So I love this idea. We’re going to dig into the closing. So, Peter, tell us a little bit about why this is on your mind lately and why I guess we’re going to be digging into the closing of the lesson here tonight.

00:05:48:06 – 00:06:11:15
Peter Liljedahl
All right. It’s not that it’s only been on my mind lately, but I finally feel like I’m able to talk more about the closing. Up until very recently, building thinking classrooms has been in sort of its infancy in terms of uptake in the world. I’ve been living and breathing this for 20 years, but most of the world is coming into that space with the book and so on and so forth.

00:06:11:17 – 00:06:37:19
Peter Liljedahl
And so much of that is just the beginning of the lesson we’re launching differently. We’re thinking differently about how we choose tasks. We’re trying to say, Let’s get the students doing more, getting them up on the whiteboards. We’re trying to get there as fast as possible. It’s a such a different thing right now. I feel like we’re at that point where I think most people that are in the room here have at least heard a building thinking classrooms.

00:06:37:21 – 00:06:56:01
Peter Liljedahl
They may have even tried it, they may have read the book. Everyone’s at a different stage. But I think that we can now finally start talking about, okay, what happens after all that exciting time with the whiteboards? Because and I don’t want to diminish that, that is incredibly important that that meaning making time at the whiteboards in their groups.

00:06:56:01 – 00:07:16:11
Peter Liljedahl
But then how do we close that off? How do we finish off the lesson so that the students walk out of the room with retained learning? And that for me has been really, really important. Data indicates that when we close off the lesson, students are more likely to come back the next day. Having remembered things from the previous lesson that we don’t.

00:07:16:11 – 00:07:34:14
Peter Liljedahl
If the kids are just at the whiteboards, they’re having a great time, the teacher is having a good time. Everyone’s feeling good. I feel like the best teacher on the planet. The kids are smiling. We got engagement. Lots of those sorts of things, but their ideas are still kind of floating out here, right? They’re spitballing as they’re working out the whiteboard, they’re clawing their way forward.

00:07:34:16 – 00:07:56:15
Peter Liljedahl
They’re they’re making meaning. But the meaning isn’t really trying is the term that I use. It hasn’t completely solidified in their mind. Their ideas are still floating out here. And if we don’t help them organize, structure and formalize those ideas in the closing of the lesson, those ideas are likely to float away as they walk out of the room.

00:07:56:17 – 00:07:59:11
Peter Liljedahl
So that’s why closing is so important to me.

00:07:59:13 – 00:08:14:11
Jon Orr
For sure. For sure. And for me too. And I think for a long time, I think when we went into putting kids at the boards and getting kids to do group work and this is a common thing we hear from many teachers is that they’re doing exactly what you said. Peter’s is you’re seeing great work at the board.

00:08:14:11 – 00:08:34:02
Jon Orr
You’re like, man, they’re engaged. And you also were going into that lesson with your learning goal in mind and you’re going like, my goal was to get from here to here. And sometimes you see that at the boards, you’re like, okay, I see that happening there. As okay. I get a sense of like, yes, this everyone’s got from here to here or a good chunk has got from here to here.

00:08:34:02 – 00:08:45:21
Jon Orr
And then you’re giving yourself permission to go, All right, everybody, when we’re done, we’re done and we’ll just keep on rolling or you can head back to your seat and start the practice, or you can stay in the boards and do the practice or do your practice up there.

00:08:45:22 – 00:08:46:22
Peter Liljedahl
Or the bell rings.

00:08:46:22 – 00:09:02:13
Jon Orr
Exactly in the bell rings. And everyone’s like, the kids will say, the thing that you want to hear the most, right? It’s like the bell will ring and the kids will say, Done already. And you’re like, Yes, that’s exactly what I hear. But they’re walking out the door going, Wait a minute, did they solidify what we really needed to solidify?

00:09:02:16 – 00:09:21:13
Peter Liljedahl
And I saw them do it on the whiteboard. That was one of the things that kept happening in my research was I’d be in these classrooms and I’d be watching that group. And I’m like, They definitely got it. Everyone in this room has it. Nobody’s riding coattails. They are all getting it. They can demonstrate it, they can explain it.

00:09:21:19 – 00:09:46:20
Peter Liljedahl
They’re all doing it at the whiteboards in their groups. And then we give them a quiz two days later, and 30% don’t have it. And I’m like, okay, where did that go? And it wasn’t that they weren’t able to do it. They did have it. And it’s more than learning loss. It’s not that simple. It’s not just that they forgot it, it’s that they never really reified or personalized or organized, structured or formalized those ideas that they were demonstrating at the boards.

00:09:46:22 – 00:09:56:09
Peter Liljedahl
So, yeah, we got to that closing is so important because it helps bring those ideas into better structures that they can hold with them as they leave the room.

00:09:56:15 – 00:10:14:13
Kyle Pearce
Well, you know what? I think it’s easy for us to fall into that trap, right? We want kids thinking. We don’t want to be doing too much hand-holding early on. And then I think, again, we carry that thought process through the entirety of the lesson. So it’s almost like we’re like, we’re just going to they did great thinking everything’s great.

00:10:14:13 – 00:10:36:04
Kyle Pearce
But the reality is, is how many students recognize what it is they really did, right? Like, we’re walking around, I’ve got a math degree and I’ve taught math for 15 years and I walk around and I go, Holy smokes, that student just did blank without even recognizing had never been taught this algorithm. But they’ve essentially done the same thing.

00:10:36:04 – 00:11:03:08
Kyle Pearce
Well, guess what that students like. Well, all I did was I was thinking, and I don’t recognize that there’s actually a behavior here. The word behavior is something I always steal that from Kathy Fasano, where this idea of the behavior of the mathematics is like something we want students to latch on to so that when they go, you know, the last time I saw something like this, this idea was really helpful.

00:11:03:12 – 00:11:25:12
Kyle Pearce
And then my thinking the next time I do a thinking lesson, now I can use that and I can actually use it as a tool instead of always sort of starting from scratch, right? It’s like you built a bit of that foundation and you’re kind of growing and building from there and then you’re actually going, Huh? I can start with what I built last time to apply to this time.

00:11:25:12 – 00:11:46:06
Kyle Pearce
Whereas if a student walks away and they don’t have anything sort of structured, if they don’t have any assistance, and it doesn’t mean we can’t and I don’t want to put words in your mouth, but I believe you agree. From all we know of your work, that giving students the opportunity to formalize and structure some of that learning is something we can also do.

00:11:46:06 – 00:12:06:13
Kyle Pearce
But the one thing we don’t want to do is leave. John likes to call it like leave the bow on tied, right? If the students are trying to wrap the present here and they’re doing a great job, we’re not going to throw it under the tree without actually finally taping that final flap and then tying that bow and then placing it nicely under the tree.

00:12:06:13 – 00:12:11:08
Kyle Pearce
Otherwise, by the time you wake up the next morning, it’s under the tree and.

00:12:11:10 – 00:12:25:23
Peter Liljedahl
The cat got into it. But at the same time, we also have to temper this with this care to not now take over the lesson and just take all that lecture that we used to do at the beginning of the lesson and I’ll do it at the end of the lesson.

00:12:25:23 – 00:12:27:03
Kyle Pearce
Well, backfill, right?

00:12:27:06 – 00:12:46:10
Peter Liljedahl
Yeah. And we’re just okay. Because what happens then is that the students start to realize that all that stuff they do at the beginning, all that inquiry discovery, all that whiteboard work, all that stuff they’re doing is just to kill time. It’s just there until we get to the end of the last thing where the teacher is going to sit us down and tell us how to do it the right way, the best way.

00:12:46:11 – 00:13:06:06
Peter Liljedahl
We don’t want that either. We’re not going to do the organizing for them. We’re not going to do the structuring. We’re going to help them organize. We’re going to help them structure right. And this hearkens back to Kathy four note as well. It doesn’t matter how organized I am, I can’t understand for them, right? I have to help them make the connections.

00:13:06:06 – 00:13:25:23
Peter Liljedahl
I have to help them do the organizing, the structuring, the I can help with the formalizing. I can give vocabulary, I can name the learning intention or the learning outcome. It is at this point, but it’s how do I help them take their thoughts and organize them as opposed to giving them my already organized thoughts.

00:13:26:00 – 00:13:44:21
Jon Orr
So, Peter, I know that we drove into this on the podcast, but I want to do that here with everybody who’s joined us too, is to talk about what are the three you’ve talked about before. Like there are three big ways to consolidate that lesson. Three pieces. Oftentimes we kind of think about it’s one thing and I quickly said it consolidating, but there’s more to it than that.

00:13:44:21 – 00:13:47:11
Jon Orr
Walk us through the three things and we’ll unpack each one.

00:13:47:13 – 00:14:11:13
Peter Liljedahl
Okay. So there are three things that close adolescence. It’s consolidation, which you talk about and I talk about, and I think we mean the same thing when we say consolidation, right? It’s how do I bring order, structure and formalism to the learning that happened in the lesson? But then there’s also opportunities for the students to make meaningful notes, which happens at the end of the lesson, right in normative classrooms.

00:14:11:13 – 00:14:31:06
Peter Liljedahl
When I would sit in those normative classrooms, notes was always something that came at the front end. It was sort of that here’s a prerequisite knowledge here, let me teach you some stuff so that you can do that. Now you try one. So notes always sort of replaced or was a precursor. It was learning. It was what we conceived as learning.

00:14:31:08 – 00:14:52:04
Peter Liljedahl
And then it was like practice came after notes have shifted from being at the front of the lesson to the back to the lesson. It’s now not what replaces learning, it’s what is a summary of the learning. It’s what is the takeaway for the students when they walk out of the room? How do they actually organize this into an object, into a thing that they can take with them?

00:14:52:04 – 00:15:17:09
Peter Liljedahl
Because it’s great that I consolidated the lesson for them, but consolidation doesn’t actually take on a form. It’s still discourse. It lives in the ether. It lives in our thoughts and our conversations. But notes is actually something that’s really concrete. Then comes the check your understanding questions, which we used to call homework, which is now in some people called practice, but it’s let’s name it what it is, right?

00:15:17:11 – 00:15:38:14
Peter Liljedahl
It’s not that I want them to practice. I actually wanted to self-assess. I wanted to check their understanding. Right. And so that’s the only thinking classroom around that We need to name it. We need to call it what we want it to be. And that’s that third part of the closing. There is a what I can call a meta closing, which is the one highlighted in blue, which is that formative assessment.

00:15:38:14 – 00:16:02:01
Peter Liljedahl
And it’s not how we close out a lesson. It’s often how we close out a unit or how we close off a part of a unit. It operates in a larger structure in transit. It ends the lesson space, but it’s still a form of closing. It’s still a form of helping students organize and structure their thoughts, and in particular, helping them understand where they are.

00:16:02:07 – 00:16:04:18
Peter Liljedahl
But I highlighted in blue because it’s a little different.

00:16:04:24 – 00:16:25:05
Kyle Pearce
I love this as you see it in the visual up on the screen. I think it’s really helpful for people, as you’re describing this idea in something when you really think about it, you had highlighted sort of the normative classroom structure. And John and I, we openly discussed this often about how we used to do that gradual release sort of mode, right?

00:16:25:05 – 00:16:41:05
Kyle Pearce
Like we’re going to like we’re going to do all this teaching the gatekeeper of knowledge. We’re going to share all this information. You’re going to copy it all down because of course, you don’t know this stuff. So you’re going to have to write it down because I’m going to give you problems later and you’re going to need those notes in order to help you do them.

00:16:41:07 – 00:16:48:03
Peter Liljedahl
Or as I used to say, don’t do any thinking right now. Write this down so you have it in case you want to do some thinking later.

00:16:48:05 – 00:17:02:21
Kyle Pearce
And it’s always fun when we give the, you know, a student goes, But wait a second, I’m not sure I understand. You go, Don’t worry, don’t worry, don’t worry. You’ll get it later. You’ll get it after. We’ll do some examples and everything will be fine. Meanwhile, that student’s anxiety is just pinned to the ceiling, right? And they’re.

00:17:02:21 – 00:17:03:24
Peter Liljedahl
Just, Oh, my gosh.

00:17:03:24 – 00:17:27:20
Kyle Pearce
I can’t stand this stress that I’m feeling. But as we do this, so as students are having this opportunity to think and we’re looking around the room and we’re engaging in, we’ll call it during the lesson, the formative assessment that we’re getting to kind of look around. It really informs the opportunity for us to lead a true responsive consolidation.

00:17:27:20 – 00:17:49:01
Kyle Pearce
Right? So, you know, we have normative classes room and I would go, well, here’s the note dusted off from the last time I did it last year, and then we’re going to do the same exact thing when there is a different group of students here that may already be excelling in that area. Or maybe they’re nowhere near ready to be tackling exactly what I had.

00:17:49:01 – 00:18:08:18
Kyle Pearce
On that note, what we’re going to do it anyway. And really what we’ve got is this opportunity to essentially assess and go, okay, where we at, who heard what, who has what and where do we go? And the question that I’m moving towards here is I want you to dig in a little bit more about this idea of consolidating from the bottom.

00:18:08:20 – 00:18:33:05
Kyle Pearce
I’m wondering how many people out there I mean, there’s only oh, I don’t know. Let’s do a place value finding here. It’s 1.1 thousand people in the room or so rounded to the nearest 10th of a thousand. But anyway, tell us for those who are in the room and they’re looking in, they’re saying, okay, I’ve heard of consolidation, but I know me personally, I’ve never used the term consolidate from the bottom.

00:18:33:05 – 00:18:39:11
Kyle Pearce
So I want to hear what does that look like in sound like to you? I have a feeling I know. Yeah, but let’s hear from you.

00:18:39:15 – 00:19:00:14
Peter Liljedahl
Okay. You said something really, really important there. This idea of we have an opportunity to be responsive to the room, and I think this is what’s really important in the closing. And it’s shot through in all parts of this closing, which is how am I going to respond to what happened in the lesson, rather than forcing the lesson to respond to what I wanted to happen.

00:19:00:16 – 00:19:21:18
Peter Liljedahl
Right. So of course, we go in with an intention. Of course we have a plan, but we have to be responsive to what the group is actually giving us. And you gave the example that maybe the group goes here when I was planning here. Maybe they don’t get that far, but I have to be responsive and the closing should be a response to the learning and meaning, making that happen in the room.

00:19:21:20 – 00:19:54:14
Peter Liljedahl
Right. Okay. So what is consolidation from the bottom? To understand this, I got to go back in history to 1985. Alan Schonfeld, who is a mathematician and a math education researcher out of California, wrote a really seminal book called Problem Solver. And it was really an empirical study into George Poorly, his work on how to solve. And as part of this work, he sat in classrooms and he was trying to understand the difference between expert problem solvers and novice problem solvers, right?

00:19:54:16 – 00:20:16:02
Peter Liljedahl
Experts being mathematicians, novices being students in a classroom. Right. And one of the things he noticed tangentially when he was doing this was he noticed teachers were doing something he came to call leveling to the top. So leveling to the top is, is this idea that, okay, I’m going to teach, teach, teach notes, notes, notes, lecture, lecture material.

00:20:16:02 – 00:20:37:20
Peter Liljedahl
Remember, it’s 1984, right? Or it’s prior to this. The book came out in 85. And then at some point the teacher turns and says, I’ve done three examples. Now you try one. Okay, So that’s sort of I do we do you do type of situation. Now we’re in that you do it. So the teacher gives a task and then as soon as you’ve given that task, what happens in the room?

00:20:37:22 – 00:21:00:23
Peter Liljedahl
All right. Well, so. Student A Well, they get to work and they try really hard and they actually solve. Student B Goes to the bathroom. Students see starts and works for a bit, but happens to be sitting next to Kyle and they start talking about what they’re doing this week and they only get so far right? Student E tries really hard, almost gets done before time runs out, but it’s really close.

00:21:01:00 – 00:21:06:12
Peter Liljedahl
F and G get it all the way. H decides it’s a good time to sharpen all my pencil, crayons and so on.

00:21:06:12 – 00:21:11:14
Kyle Pearce
And so the Canadian term, by the way, our U.S. friends pencil crayons. Keep going. Oh.

00:21:11:16 – 00:21:14:04
Peter Liljedahl
Okay. Yeah. So what would they call it in the U.S.?

00:21:14:06 – 00:21:16:21
Jon Orr
Yeah. Now you got them all sidetracked. It’s like.

00:21:16:23 – 00:21:43:11
Peter Liljedahl
Okay, I’ll get back on track. So anyway, my research shows in situations where the teacher does this, now you try one that we give them on average, 4 minutes and 20 seconds, and that’s just an average of something interesting. What’s more interesting is how many students are finished. By the time we do the consolidation in a now you try one situation and the number somewhere between 20 and 30%, which is a woefully low amount.

00:21:43:13 – 00:22:05:08
Peter Liljedahl
What happens is you think about it as a teacher, okay, you’ve got all the kids working at their desks. The first students finished. They’re going to sit quietly. The second students finished, they’re sitting quietly. Not a think about the sixth student that’s finished, right? Odds are they’re sitting next to someone else who’s finished. They’re not sitting quietly. They start talking and they start recruiting.

00:22:05:10 – 00:22:23:06
Peter Liljedahl
Right? They start pulling other people off task and so on. And the teacher goes, okay, that’s it. All right. We’re going over this thing, right? Because now the class has tipped from being mostly engaged to being mostly disengaged. So now I’m going to consolidate two from the top. And because I gave this task and this is only the first.

00:22:23:06 – 00:22:39:06
Peter Liljedahl
So that’s going to be hard for I need everyone to understand this one first. So I wanted everyone to get out here, so that’s why I’m going to go over it. Okay, let’s go over let’s talk about how to do this one. You know that student who almost got it, That’s who sharpened their pencils, that their coloring sticks out, look like pencils.

00:22:39:06 – 00:22:58:08
Peter Liljedahl
Better not pencil crayons in the U.S. Is it working for them or that John and Kyle, in the middle of the room, we started talking about what kind of sandwiches they like best. It’s not working for them. This consolidation to the top. Well, he called it leveling to the top doesn’t work because it’s outside of the students zone of proximal development.

00:22:58:10 – 00:23:23:11
Peter Liljedahl
And in fact, it doesn’t even work for the students who actually got it because this is redundant. So it works for such a small percentage of the students. And he named it. He identified the problem with it. When we started doing our research, we saw the same problem. So what’s the counter to that? What’s the opposite? The opposite is rather than start at the place I wanted them to finish, let’s start at the place where I wanted them to start.

00:23:23:13 – 00:23:43:21
Peter Liljedahl
Let’s do the consolidation there. So if I’m doing a rich task, write one of your three act tasks. Maybe we’re not going to talk about the most sophisticated part of the pattern or the number sequence or the ratios that you’re noticing. We’re going to talk about did Heinlein count how many chocolates I ate? We’re starting down there. Everyone can participate in that.

00:23:43:21 – 00:24:09:15
Peter Liljedahl
Every one is part of that conversation. They want to count how many correlates, right? And then we’re working our way up in more of sophistication into the task. So we’re starting at the bottom. We’re consolidating from the bottom and working our way up. And in doing that, we move through every one zone of proximal development, which means that in theory I can lift everyone up a bit.

00:24:09:15 – 00:24:20:05
Peter Liljedahl
Does that mean I can lift everyone to the top, but everyone can get something out of the consolidation rather than being left behind because they’re here and I started it here. So there it is in a nutshell.

00:24:20:10 – 00:24:42:14
Jon Orr
Well, what I love about you summarize it so nicely about leveling from the bottom, because Kyle and I used to we’ve toured all over giving workshops, and for a long time we were giving workshops on the five practices for creating productive mathematical discussions. And we were talking about the console or the connecting stage of that practice. There is lots of different ways from Peggy Smith and Mary Kate Stein’s book about how to do that.

00:24:42:14 – 00:25:03:23
Jon Orr
And a lot of times we used to say, you could go with most common. What does everyone doing around the room? Like Let’s start with the most common. But then it was like, Well, why don’t we start with the one that we everyone can access? The most accessible areas are solutions, and then you can start moving up. It just place so nice into some of the other work that people have been doing for a while too, and that connections make a lot of sense.

00:25:04:00 – 00:25:19:05
Jon Orr
I’m curious about, as I’m sure lots of teachers here are wondering This too, is when you consolidate to the bottom and you have your learning goal in mind, are you consolidating from the bottom all the way to the top? And if so, or are you pausing where everyone got to? I think this is where a lot of people are.

00:25:19:05 – 00:25:35:08
Jon Orr
Like everyone’s going to say time is a factor. Well, I got to get to my learning goal. We want to make sure that that learning goal came out. Do I consolidate all the way to the top if I need to? Because we didn’t get there and I get to step in and do it and show them or do we just consolidate to where we are.

00:25:35:10 – 00:25:39:11
Jon Orr
And then I have to recraft what we do tomorrow so that we can get there on our own.

00:25:39:17 – 00:26:01:04
Peter Liljedahl
Okay, So I’m going to say yes, but okay, so yes, we consolidate all the way to the top, but the top is set by that responsive reaction to what happened in the lesson. The top is what is the furthest anyone in the room got to rather than where did I hope we would get that day to that day?

00:26:01:06 – 00:26:17:24
Peter Liljedahl
So maybe three groups got this far, but maybe I was hoping to get this far. I’m not going to consolidate all the way up to there because I’m like, I’m speaking Greek by the time they get up there. Right? And I left everyone in the dust. But if I got three groups getting two here, then I had two or three groups that were really close.

00:26:18:01 – 00:26:34:18
Peter Liljedahl
And a consolidation to that point is going to lift a whole bunch of people up to that level. But it can go the other way too, which is that my plan was to get this far today and we actually got this far and that happened a lot of times before I got better at anticipate what was possible in a thinking classroom.

00:26:34:20 – 00:26:53:16
Peter Liljedahl
I’ve so often went into a room with Section 6.1. We’re going to do 6.1 right? Or what I normally would have done in my normative classroom as a Thursday lesson. And we shoot through that minutes and now making up tasks on the fly because we’re getting all the way into the more complex stuff. We’re three lessons ahead now, right?

00:26:53:16 – 00:27:04:15
Peter Liljedahl
I’m not going to just consolidate to where I was hoping we were going to get to that day because everyone’s staring at me going like, Duh. Yeah, right, exactly. That’s it. That was all you were going for reading. Yeah.

00:27:04:17 – 00:27:25:11
Kyle Pearce
Well, and I want to be super explicit here because what I just heard you say, John, you mentioned about this idea in the chat. Let us know if time’s ever been a concern for you, because what I just heard you say, Peter, and what John and I have experience in doing this process where we’re actually letting students do the thinking first and actually being responsive.

00:27:25:11 – 00:27:47:08
Kyle Pearce
When we talk about differentiation and we talk about all of these things in the math classroom, everyone’s thinking it’s like, I’ve still got to do all the same things I normally do, plus I need to do all these other things. And in reality it’s like, No, no, no, because there’s so many times where you go, They don’t want to do that because they’ve got that part right.

00:27:47:10 – 00:28:06:17
Kyle Pearce
Like, so sometimes we spend too much time in something that isn’t necessary. Other times we spend too little time in an area that we should be spending more time in. And in reality, if we’re actually being dynamic. I like this idea where you set the top. I was picturing that top really is very dynamic. It’s going to be moving.

00:28:06:17 – 00:28:08:20
Peter Liljedahl
I know where we’re starting that. I know.

00:28:08:24 – 00:28:33:10
Kyle Pearce
Exactly. And from there you might learn something, right? We might learn something. And I think the other thing that I heard you saying earlier, and I lost it because there was so many other good nuggets that sort of took my attention away. But just this idea of starting, oh, this is what I was consolidating from the bottom, this idea, we always talk about it from this idea of accessible, the most accessible solution so that everyone can see it.

00:28:33:10 – 00:28:51:02
Kyle Pearce
And the thing that’s valuable about that that’s easy to miss is that when we start at the bottom, people think that the kids who are maybe up near the top, whatever the top is for that day, are not going to be able to relate or they’re not going to be interested or they’re not going to get to see the math.

00:28:51:03 – 00:29:24:05
Kyle Pearce
In reality, what I see a lot, and I’m sure you see it all the time, is that when we do this, those students who have actually been mimicking algorithms and they’re just like way ahead of everyone in terms of their memorization skills, they’re actually starting to see why what they did actually works. So it’s almost a way of helping that student that’s like off to the races maybe is doing the after school program or whatever it is, and they’re doing all kinds of stuff and in reality they actually aren’t able to model.

00:29:24:05 – 00:29:46:22
Kyle Pearce
And then you have a student that actually did something that’s very, very visual in terms of their solution strategy and almost makes it obvious. And then everyone looks and goes, Yeah, I can totally see that. And this is sort of that assessing piece for us where we go, Okay, this is great when this happens because we got one student way over there, we got one student over here.

00:29:47:02 – 00:30:02:16
Kyle Pearce
And I love when we have this opportunity go, you know what? Let’s spend some time at this board. I want you to have a look, chat with your partner about things that you notice, things that maybe you’re still wondering. And then I want you to go walk over to that board over there. And I’m wondering, what do you notice?

00:30:02:16 – 00:30:26:01
Kyle Pearce
What do you wonder here? You see anything similar? Do you see how it’s connected? Where are you stuck? How is it possible that they got the same answer and you’ve got the same answer, and yet you’re solutions appear to be very, very different. Your approaches appear to be very different. And that connectedness, it’s almost like you build in a sunk cost of the students, right?

00:30:26:02 – 00:30:32:11
Kyle Pearce
It’s like they start thinking and then now they’re like, well, now I’ve been thinking about this and now I don’t want to give up on it.

00:30:32:13 – 00:30:50:19
Jon Orr
There you have it, folks. This was part one of two from the opening keynote session from the 2023 Math Moments Virtual Summit, where we spoke and led that keynote session with Peter Little. All you heard him just now. And when I reflect on that first half of the episode, second half is going to come out in two weeks.

00:30:50:19 – 00:31:06:18
Jon Orr
So look for that episode in two weeks, not the next week, but the week after that. And if you’re listening to this, after a bunch of episodes have already come up, you can flip ahead and go to it right now. But when we talk about the closing of the lesson, we’re talking about a few different pieces of our classroom tree.

00:31:06:18 – 00:31:26:03
Jon Orr
We’re talking about the branches of the tree. Obviously, when we talk about pedagogical moves in, pedagogical content moves, when we think about what some of those leveling to the top looks like and what leveling from the bottom looks like. These are teacher moves that we do in the classroom, in our branches are those teacher moves that we need to implement to create student thinking.

00:31:26:05 – 00:31:53:22
Jon Orr
That’s what we are hitting on here. Mostly today in this first half of the lesson. But I think we always touch on when we start to change our pedagogical practice, we always touch on the soil and the water and the sunlight of our trees, which is our mindset in our belief system about what real mathematics is. And for us teachers who are pivoting into a problem based math classroom or a building thinking classrooms or a thinking classroom, this is a mindset we shifted to already.

00:31:53:22 – 00:32:21:23
Jon Orr
This is our fundamental belief that mathematics is getting our students to think mathematically and see themselves as mathematicians. This is our soil water. Suddenly our mindset of what we believe to be true about mathematics and how do we see ourselves in mathematics. So those are two pieces of the tree we’re growing in this episode. And if you were with us on the 2023 Make Math moments virtual Summit, you may have been in on this particular session and it’s a good to get a refresher in here.

00:32:21:23 – 00:32:46:18
Jon Orr
It again, all replays from the session have been placed in the Make Math Moments Academy. You can get over there and join the academy right now so they can capture all the replays, all 30 sessions that we gave out in that summit. You can head to make math moments dot com forward slash academy. Join the academy and you can binge watch all the sessions, plus all of our courses and our tasks are all included in that membership.

00:32:46:18 – 00:33:06:24
Jon Orr
Plus, I actually just finished recording with Kyle and our members, our live Q&A session for this month. Every month we get together with teachers. We have a live learning session, go over big questions and big struggles. Pebbles in shoes for a group, the Academy group. So we just finished recording that every member of the academy gets participate in those as well.

00:33:06:24 – 00:33:25:19
Jon Orr
So head on over to make math moments dot com for such academy. Join the academy. You can get all those sessions and that will see you in those live Q&A sessions as well show notes and links for this particular episode folks is found over at make math moments dot com for episode 270 that’s make math moms dot com for episode 270.

00:33:25:21 – 00:33:34:24
Kyle Pearce
Well until next time my math moment maker friends I’m Kyle Pearce and I’m Jon Orr high fives for us.

00:33:35:01 – 00:33:49:01
Jon Orr
And a high five for you

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