How we teach and what we teach are both really important. In this episode Pam and Kim wrap up their discussion on Peter Liljedahl's book Building Thinking Classrooms and how it meshes with the Math Is Figure-Out-Able movement.
See Ep 128 for more about students can do more than they can say and say more than they can represent
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
And I'm Kim.
And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using that mathematical relationships. We can actually mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.
So, today, we are wrapping up a four part series all about Peter Liljedahl's Building Thinking Classroom, and how it meshes with Math is Figure-Out-Able movement.
Yeah, this has been a lot of fun to really focus and point out some places where we totally align, and just maybe a few things that we might kind of interpret a little bit differently. Ya'll, we just filmed the two-day, live workshop for Building Powerful Fractions that will be coming out sometime in 2024. We're super excited about that. We just filmed the live portion of it. It was a lot of fun. I have to tell you. We had fantastic participants. We had such a good time. That might have been the best two-day live that we've ever filmed.
Well, you know, I love the fraction, so I thought it's fantastic. Yeah.
It was super good. So much thinking. And I'll never forget, at the very end of the second day Krystal. Right on the front row. She was right in front of me smiled. And she said, "I came to this workshop for some tips and tricks to do with my students, but these two days have actually completely changed the way I think about fractions. I've learned a ton of math." And I just smiled and I was like, "Fantastic!" And I was kind of like, "Was that okay? Did you also learn things to do with your students?" "Yeah, yeah, yeah. Yeah. Like, no, but I'm leaving... Like, I didn't expect to learn this much." It was so amazing. But that's kind of our goal.
Yeah, and I don't think she was alone.
I think there were a lot of people there who walked away thinking about fractions more deeply, with more understanding. Yeah, that's super cool.
Yeah, (unclear) for everybody to walk away. Yeah. Nice.
Okay. So today, we're going to mention four things from Building Thinking Classrooms in part two and three. It's a fantastic book. There's so much. We can't dive into everything. So, we're going to do just a quick mention for the first couple, and then there's one that we're going to dive more deeply into. The last one we'll talk about, so (unclear).
Hey, before we do that, let me. Sorry, before we do that, I'll just mention. He has 14 teaching practices in four sections, and we're not attempting to address them all. Just kind of the ones that we find really noteworthy. And so we're going to finish today with those last few noteworthy ones. At least that are on our minds today. We might come back and talk about the rest. But yeah, here we go.
Oh, I'm sure we will. Okay. So, Pam, defronting the classroom. We don't do that all the time.
Yeah. So, defront the classroom. When he talks about that, it's the idea that if you look at typical classrooms, you've got all the desks in rows, and they're facing the teacher at the board. And there's this obvious front. And the teachers obviously in charge and knows it all. And the students are supposed to sit and quietly just consume. And so, a way to shake things up and to help students feel that things are different is to defront the classroom, so that there isn't an obvious front. I think, Kim, we're okay with that idea. We're okay that there doesn't have to be one obvious front. In fact, when we just filmed Building Powerful Fractions, we had vertical, nonpermanent surfaces. Thank you Wipebook for donating one of their flip charts. We had those all around the room. And we had our participants up at those spaces for various times doing various things. We also had them at their tables in chairs where I was at a front of the room doing some things altogether. Especially Problem Strings. Now, when we were filming because of cameras and stuff and such, we always did it at the same place. But we wouldn't be okay in a classroom that I could have gone to any of those vertical non-permanent surfaces and said, "Hey, today, we're going to do the Problem String here. Everybody orient your desks this direction, and we're going to do the Problem String with me here." And so, in that way, in a classroom, there wouldn't have to be a front. It could move around the room to wherever the surface is, as you are facilitating something like a Problem String, right?
Yeah, so I think...
Oh, go ahead. So, I think we would say, "Sure, go ahead and defront your classroom," but that doesn't mean that there aren't times when it's a whole class task, like a Problem String, where there needs to be a place where the teacher is facilitating, and the teacher is representing thinking, and bringing together all that thinking in one common place, so that there is that pre-planned display of thinking that now is all kind of going together in this very thoughtful, purposeful way to give the most possibility for learning to occur. That that is a thing we do in Problem Strings, and for that we need to have kind of a front, or at least a focus place of the room. So, I think the second one we wanted to talk about was he says, "Give the thinking task early, standing and verbally." So we've already kind of talked about how we're not going to only be giving thinking tasks, right. So, when you're giving a thinking task...so, we would maybe say a Rich Task or an investigation...students are going to be at vertical, nonpermanent surfaces in the randomly chosen groups. When you are doing that, then sure. I think it's fine to give the task early.
Get going. Yep.
Get her going. Yep. Don't have students walk in the room, sit down at their desks, facing the front of the room, and then give the... Nope. Get them up. Get them up with their groups. They're standing. Give them the task. And then, I think I'd like to focus on the "verbally" part.
Yeah, me too.
I have a feeling you might have something to say about that. So, I agree with a lot of things that Dan Meyer has said about let's not have the... And I'm going to kind of... I'm not trying to quote you, Dan, I hope I do you justice here. That what we don't want to do is have too many words get in the way of students actually learning the math, doing and learning the math. So, if we can have these Rich Tasks, be less a bunch of words and more an image, then do that. So, work towards having more images, more graphics, more things that kids can read, even if they don't speak the language natively. Or if they... Yeah. Something that kids can dive into the math and not have the language hamper them. And I think there might be a small "but" coming from you.
Well, yeah. So, you know, it's interesting to me because maybe this is just like a personal experience from students that I've taught and my own personal kids. That I think there's absolutely such value in the verbal part of it. But I also think that a bulleted, recorded, highlighted image that kids can have in hand is incredibly helpful. And I think that it's, you know, we've talked about "proximity" questions. Some of those like, "Am I doing it right? Did I get this?" I am huge fan...
We could also include, "Did I hear it right? Did I..." Yeah.
I am such a fan of sharing information in multiple ways because I recognize that there are lots of kids who are in and out of attention, who, you know even in that small portion of time where you're just posing the task, you got kids who have heard bits and pieces of it. And I don't want to miss an opportunity for them to go, "Oh, I was in and out. And here I'm going to confirm what I think I heard with something on paper."
Yeah, absolutely. So, I think we would both agree that what we don't want is for the (unclear) to get in the way. So, like this long, unwieldy paragraph that kids have to wade through, where the the reading level is above their reading level. We don't want that to get in the way of the math, of them doing the math, understanding the math, being able to dive into the task. But we also don't want students who have stuff, right? They're going to come to our class, who knows what was happening in the hour, or the day, or the week before. And/or they might have working memory that doesn't necessarily hold on to as much as we would want them to, to be able to do a Rich Task, right? So, I can absolutely have a way to have them have something to refer to, so it doesn't stop there thinking. I think we're both there. I think we would also agree, though, that we're not going to take all the language out of the experience, that when we talk about... Golly, we'll put the episode number in the show notes, but we've talked about that what you think about is more than what you can say clearly, and what you can say clearly is more than what you do. That brilliant thing that you came up with. And then, I kind of added to that that's kind of...
(unclear) You said "do" It's okay.
Sorry? Oh, did I say that wrong? Yeah, sorry. Do. You can do more than you can say, and you can say more than you can record. Thank you. Or represent. And then, I kind of added to that to say that's actually a great way of teaching is give some give kids something to do. My hands are kind of big. It's like big Venn diagram. And then, pull out of them language to help clarify what was happening in their heads. And so, that piece about language is super important because as we help students verbalize the relationships and connections they are using. They actually gain clarity, and then as we, the teachers, represent their thinking, and help them, "Oh, when your brain does that, when you say those words, it could look like this." And we use a visual model to represent what they're doing, and then help them transition to where they can use that visual model. All of that is super important. So, we don't want to take the... What we're not saying is take the language out of it either. Yeah, that's probably good enough. Cool. Alright. Let's do another one.
Yeah, so we did defront the classroom, mentioned giving thinking task early, standing verbally. How about consolidating from the bottom?
Oh, yeah. So, this might be one of the things I'm going to maybe push back on a little bit more strongly than maybe most of the things in the book. May I respectfully suggest that consolidating from the bottom every time is not the most advantageous way to finish a Rich Task? I think... So, Dr. Liljedahl will use the word "consolidate" to talk about what happens at the end of the Rich Task or the thinking tasks that students are doing at their vertical, nonpermament surfaces in their randomly chosen groups. That that's kind of the way of sort of finishing it out. He'll call that consolidation. We like to call it a Math Congress. And thank you, Cathy Fosnot for that term. We like to pull kids together and have a congress just like the International Math Congress where mathematicians bring their work to their peers for review and comment. We like to have that same idea here where, "Hey, mathematicians in the room, let's bring our work to our peers for review and comment." And in that, then we like to be very purposeful, to have a purpose to that conversation. If it's really a Rich Task, then there should have been lots of, or at least a few, different things kind of happening, bubbling, percolating, bubbling up and percolating in the room. And we want to focus on one, or two, or maybe three of those things. More often, one of those big ideas, models, or strategies that's coming out in that Rich Task, in that Math Congress. And so, in that Math Congress, we might do things...I'm going to refer to Smith and Stein's Five Practices for Orchestrating Productive Mathematics Discussions... where you actually want to select and sequence the group work that you're going to toward a particular goal, that it isn't always from the bottom up. It isn't always from the less sophisticated, or what he means is like from the first problem. May I respectfully suggest, that if everyone successfully solved the first problem, then there might be a time where we could gain from sharing strategies for that first problem. But maybe most of the time, there isn't anything to gain because everybody successfully did that first problem well enough, that we don't need it as a jumping off point. There might be occasions where we do, where we want to pull it up maybe to compare it to something else, or as "we know everybody has access to, and we can jump off from there". But if everyone has access to it, then we might want to do the next thing that everyone just barely has access to. We might want to start with, as we consolidate, as we have this Math Congress, we might want to start with the thing that everybody just barely has access to because now we've helped them learn and grow in that sharing, in that congress time, in that time where they're bringing their work in front of each other. Does that makes sense?
Yeah. And sometimes, you know, we might start with a less sophisticated and move towards a more sophisticated if we're talking about building a particular strategy. But there are also other congress purposes that don't have anything to do with least to most.
Absolutely. We might choose either a problem and another problem. We might choose a strategy and another strategy. And then, compare them, right? It might be a comparison. It's not about least and more sophisticated. It's more about the math that can come out of that (unclear).
Right, or a model (unclear).
We also might... Yeah, or a model (unclear). That's exactly what I was just going to say. So, we would suggest that before you give a Rich Task that you have thought about, "What is the purpose of this Rich Task, and so therefore, what kind of work am I going to..." And I think Peter mentions this. That if there are things that you want to come out of it, that you might drop hints and extensions to help that particular thing happen. But work towards that goal that you know. In other words, it's not a random Rich Task. There's a reason you're doing it, and so you're going to work towards making that Math Congress, that discussion be productive. You're going to orchestrate that productive mathematics discussion. And there are different ways to do that. So, maybe maybe we'll have a whole episode some other time just on Smith and Stein's five practices, and we can kind of parse that out, you know, different purposes for Math Congresses. So, we would just sort of push back that it's not always consolidating from the bottom. Maybe I'll mention one other reason why not, especially if what you mean... I don't think Peter means this, but but listeners might hear. If you consolidate from quote "the bottom", they might hear you always start with the sort of worst group or the less sophisticated group.
Kids will feel that, and pretty quickly if they're the first ones sharing, then they're like, "Oh, we're the dumb ones. Okay. Great. Fabulous. Yeah, we're happy to share." And so, I would be careful about making that too much of a tradition. Don't make that too much of a tradition. Cool. Alright. Last one, Kim, that we want to talk about today is where he suggests to "asynchronously use hints and extensions to maintain flow". So, there's kind of a lot there. I don't want to spend too much time defining "flow", but I'll just sort of say, you know, if you want to keep kids in...I'll say...right at the edge of their zone of proximal development, in a place where they're not too bored and they're not too frustrated. If you want to keep kids in that place, then a thing you can do is use hints and extensions. And so, Kim, you and I talked about this a little bit before. Hints and extensions. What's on your mind, right now, when I say "hints and extensions"?
Well, this is the thing that we talk about the most, right? Where we said, "In order to give good hints and extensions, to do that well, you have to know the math," (unclear)
(unclear). Yeah. What do we mean by that? You know like, I could be a third grade teacher, and I could say, "Well, I can add and subtract and multiply. I know the math." We would suggest there's maybe more about the math. Like, I don't think I knew enough math as an early teacher, as a young teacher because I was in that rote memory place where everything to me was disconnected sets of facts to memorize and rules and procedures to mimic. I didn't own the relationships enough to be able to give hints and extensions that actually help students continue to learn. I could give hints and extensions. My hints looked like, "Well, use that formula." Or, "This is the next step." Or, "Remember this fact." Like, my hints were very fact based, memory based.
Well, you mentioned the fraction workshop earlier. I think a lot of those teachers, too, would say that if they needed to give a hint, it might have been about what to do, a procedure to do. And now that they've had an experience of really wrestling with fraction understanding... I think it just deepens your understanding, and you have a bigger view of the topic that you're talking about, and so you're able to give hints that aren't just, "Let me tell you the step to take."
You know, I'm picturing the t-shirt that we have that says... What does it say? "Be dense". It has a brain, and the idea is that you want to have lots of connections in your brain. You want to have lots and lots of relationships and connections, right? And so, we want our brain to be dense. It's like "Be dense". It's kind of just like a funny way to say that. But if you have that dense network of understanding, then hints and extensions become a whole... Well, let me just talk about extensions too. When I was a rote memorizing teacher, my extensions were always vertical. If I'm thinking about a landscape of learning, or if you think about like the time sequence, it was always, "Oh, you're done with that? You can do that rule? You can you can mimic that procedure? Well, let's see. Do I give you the next one ahead of everyone else?" That felt weird because kids are in wrong. Like. extensions to me were, "Why would I do that? It would almost wreck my classroom, because then I have kids in different places in the curriculum. I didn't have a sense and a feel for what it meant to extend horizontally on the landscape. Thank you, Cathy. Fosnot for that vision of we need to work horizontally sometimes. How are things connected with concepts, models, strategies, big ideas that are around them. Not necessarily just moving forward in the curriculum. But to do that, you have to have that landscape of learning. You have to have that web, that interconnected web. So, Kim, this really reminds me of the Success Map we created for our Journey online implementation support. We love our Journey people. And when we created Journey, we wanted to help people and help us sort of think about. We kind of kind of got our thinking out about how we kind of think about a landscape of teaching, and where are teachers in their sort of Journey about learning to mathematize. And so, the way we kind of think about that is, first of all, for the thing you teach. So, you teach third grade. If I give you an addition problem, can you solve that addition problem for any of the numbers that make sense for your grade? Can you solve that reasoning? Or the only thing you have is an algorithm? Or if you're a fifth grade teacher, can you do a division problem? Or if you're an Algebra 1 teacher, can you write the equation of a line not from rote memorized steps? Like, can you reason through it? So, that's a first kind of thing. And then, we say, "Okay, great. Now, that you can do that reasoning, cool. Do you know the major strategies to be able to do that? Not just kind of any way to do it, but do you know what the major strategies are and the major models we would represent those strategies with? Can you do both of those?" And then, once you have that, then we're like, "Hey, now can you elicit that thinking from students? Can you pull that thinking out of students?" And when they tell you how they're thinking about it, can you choose a good model? Maybe the best model to represent that thinking? And can you name it? Can you describe their strategy? And then, think about... Yeah, like how would you represent it? And where would you kind of nudge them next? Can you represent their thinking? And then, once you can do that, then we help teachers really focus on the teacher moves, the neutral response, and the way they question kids, and the way that they facilitate a Rich Task or a Problem String. We help them sort of get better at that. And then, once you've kind of gotten all that down for the stuff that you teach, then we think about, "Okay, now, what's the next move?" How do you sequence tasks to make this all happen? How do you extend the math from there? Look, I just said "extend". Because we're talking about extensions here. Like, we kind of see that. Like, you have to know the math. Everything I just talked about, you kind of have to know then, in order to really be able to extend the math. Like, what's next? How do I help a student really deepen their content? I don't know. So, that's kind of a way that I'm thinking about. As we say, "Oh, just go give a hint and an extension." Yeah, but in order to do that well, I really have to understand that math.
Yeah, yeah. And listen, you and I love Peter's "how", like love the things that he says in Building Thinking Classrooms. I'm so happy that it's been so popular because the "how" is so fantastic. It's called Teaching Practices for Enhancing Learning. But also, what I love about what you bring to the world is that you talk about the "how" but also the "what".
Yeah. Yeah, it's not just the "how", which is super important, but it has to be combined with the "what", the "what" of the math. You need to know the math to go along with those teaching practices. That's why we do the math in the podcast, right? People are always like, "Oh, I love it when you guys do the problems together in the..." Yeah, that's why we actually do. Like, so many of the math teaching podcasts that I listened to are all about the "how", but we have to actually talk about the "what" as well. That's why we make the Building Powerful Mathematics workshops. So, like Krystal's comment we talked about earlier how it completely changed her math, we need teachers to have the opportunity to deeply own the math, the mathematics, so that those teaching practices can really be effective. Ya'll, everyone's telling you and talking about the Building Thinking Classroom "how". I'm telling you that you might not be getting the results that you're hoping for. You might have tried it and you're like, "Why isn't this working for me?" It might be because you're doing it with a framework that math is rote memorizable. And that you have an opportunity now to learn more and more math, more and more math being figure-out-able because when Math is Figure-Out-Able for you and your students, it makes all the difference. Alright, ya'll, thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!