SPEAKER_00 0:00

Welcome to Room to Grow. I'm Curtis Brown. And I'm Joni Fenderberg. We work together at Texas Instruments, and we're glad you're here. We're looking to continually improve our practice, and we understand that you are too.

SPEAKER_02 0:11

We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better. In this episode, Curtis and I discuss one of NCTM's effective teaching practices: use and connect mathematical representations. We describe the five representations and share why this instructional practice is important for all learners. We talk about how teachers can anticipate, select, and sequence the different mathematical representations that students produce, and we emphasize how important it is to help students make connections between the representations so they can begin to internalize the mathematical structures and generalities that lead to deep understanding of important math concepts. And we ground our conversation in some actual mathematics involving pancakes. What could be better?

SPEAKER_01 1:00

Let's get growing. All right, Joni, I'm really excited to be sitting down here today again with you and having this conversation. We've gotten into some great conversations coming up into today's episode. And I think this is a topic that's near and dear to your heart as mine for sure as well. And so let's go ahead and dive in with this idea. We're going to be talking about mathematical representation, but really I think it's important before we dive too far into that is what do we actually mean when we say mathematical representations?

SPEAKER_02 1:36

Yeah, I this is such an important topic. And there's lots of research around mathematical representations. So I think if folks, you know, have done some reading or have had some professional learning around this topic, they may have heard some different indicators about what that means. But what I want to focus our conversation on for the podcast today, Kurt, is the five representations that are outlined in NCTM's Principles to Actions. And they have um NCTM actually has like make connections, use and make connections between mathematical representations as one of the eight effective teaching practices. So in that book, and there's a great diagram online too on the NCTM website, the five representations are physical, visual, symbolic, verbal, and contextual. So physical being like something you could actually pick up and touch and manipulate as a representation. Visual being like I might draw a picture of the thing I could pick up. So not actually tactile touching it, but representing it visually. Symbolic is kind of what we think about mathematics is, you know, numerals and operations and variables and how we manipulate things. Yeah, exactly. And then verbal obviously is describing with words, and that could be out loud or written. Um, and then contextual is putting um uh an authentic context around the mathematics. So it's grounded in some real world experience or phenomena.

SPEAKER_01 3:11

So I just want to make sure that I'm clear on this because this is a little different than the than the five um representations that I kind of went around when we were working at laying the foundation talking about. So um the contextual really is different from the verbal because you're dropping this into a context, right? And the verbal is really describing what the mathematics uh are and what the mathematics is talking about.

SPEAKER_02 3:37

Correct. Exactly. Thank you. Thank you. It's perfectly possible we could see situations where these overlap, right? Like where something could be verbal and contextual at the same time. So it's not like these are discrete. Um, and they interact with one another too. You know, how we draw um a visual representation and connect it to a symbolic, uh, there's gonna be similarity between that of the physical and the symbolic as well. So there's this the diagram that NCTM shows is like um is like a pentagon. It's so it's like five-sided with each of those representations and then arrows pointing between every single thing. They're all interrelated, interconnected. And most importantly, they're all important across all grades. I think, you know, you and I are both former high school teachers. And I know, at least in my experience in the classroom, attending to physical, physical or visual was not very common. You know, most math teachers don't bring out something for kids to actually manipulate and touch and move around on their desks when they're in high school. I remember when algebra tiles came out in the early 90s and it was like, ooh, this is so innovative. You know, whereas in the early grades, it's almost all that tactile, you know, that that's very common for kids to engage with mathematical ideas tactilely. So I think the first thing that's important to realize when we take on this idea of mathematical representations is that these are important across all grades and ensuring that students have those experiences to engage with all representations is hugely important, regardless of what grade level they're in and what level of mathematics they're engaging with. So um, and then back to the research piece, I think there is a lot of research around the effectiveness for attending to all of these mathematical representations and the impact on learning. So, you know, it's not uh it's not one of NCTM's eight recommended teaching practices because they think it's interesting. It's because the research says this actually makes a difference for learning. And I think when we think particularly about our populations that uh tend to have struggles in mathematics, our English language learners, our um, you know, students with identified learning challenges, for particularly for those groups of students, engaging intentionally around multiple mathematical representations really shows to have a significant impact on their learning.

SPEAKER_01 6:08

I was just gonna kind of address or at least ask a question related to that, uh Joni, thinking about these multiple representations, really diving into the connections. We know that that looking at these um from different ways increases um students' comprehension and understanding of the topic, right? I mean, that's that's shown, that's proven, and it just makes sense, right? I mean, if if uh we had we're all a little bit different, we address things a little bit differently. But I think in in particular, when we start thinking about uh the language learners and the and some of the the students who really struggle um maybe with that symbolic representation or an analytic representation, right? Because the language the it just feels a little bit bizarre. It's certainly abstract. Um, and and when we think about abstracting things, that makes it, you know, that takes the language up a little bit even more than maybe what we're used to. Um and so just being able to draw on a physical representation and a f and a visual representation when language is in fact a barrier to help jump that hurdle just a little bit, right? Um we we do that when we're trying to communicate other types of things. So why not in mathematics, right? Right, exactly.

SPEAKER_02 7:26

And I want to add too, you know, there's a lot of uh school of thought around the sequencing of engagement with a different representation. So you may have heard of this idea of going concrete, then representational, then abstract, this sort of CRA approach. And I just want to uh not to challenge that, you know, there's some research around that, but I I want to just say that like just because a student might be comfortable or proficient in the abstract, maybe they're, you know, this was me growing up for sure. I could catch on to the symbolic really quickly. Like, give me the pattern, give me the rule, and I can crank it out. Um, even if they're already proficient in the more abstract representations, it still benefits uh students to engage with the other representations as well. So there's power in ensuring that all of those are happening, even if, well, oh, they're already ready to engage with the symbolic. That doesn't mean you can say, oh, we're we're good. We don't have to engage with the physical. It does deepen the learning and the understanding to ensure that all five are attended to.

SPEAKER_01 8:30

For sure. I I would even tie that back to some of our modeling conversation from earlier. I know we talked a lot about in that session um about just mathematical modeling. And really, I think what a lot of us think about there is coming up with some sort of equation or something that we can plug numbers into and whatever. But I think there's also a section of modeling which is determining a way to physically represent what is happening uh or visually represent what is happening in that analytical model, right? Or in that analytical situation. And so it certainly is a benefit for a student who maybe, and that was me also, right? Um, I I was kind of a math person, um, if you will, if there is such a thing, um, that I really related to the symbolic uh representations of mathematics and and procedures kind of made sense to me. But the challenge, even, of determining a good way to represent things physically or to represent things visually, or even to take that to the verbal and contextual uh you know representations was worth it for me. It deepened what I had to know and understand because I couldn't just rely on those procedures that I had sort of uh mastered, if you will. Um so I I really I I'm with you on that. I think there's a huge uh benefit in challenging our students who maybe are um being successful with the symbolic representation. Like let's continue to push them or give them opportunities for both these or for several of these other uh representations as well. We definitely do want to make sure, and I think that actually leads us into our our next kind of conversation here, thinking about you know, those students. So we do want to make sure that students um share their different strategies, right? I mean, I think this idea of having multiple representations lends itself then to a classroom experience where, hey, Joni, what did you do on this particular one? I represented it like this. How do you um represent it? Or those kinds of conversations are the things that we want to try to maybe encourage in our classrooms. Do you want to talk about that just a little bit?

SPEAKER_02 10:56

We we definitely do want to have students share their different strategies. And that connects to another one of the eight math teaching practices from NCTM, which is elicit and make use of student thinking, right? We want to be sure we're, you know, letting the students share their language and their way of thinking. However, I think there's um there's maybe a misconception that happens among educators that this is just about multiple strategies or multiple representations for multiple strategies' sake, right? It's not like, okay, who has another way of solving this problem and who has another way of solving this problem? It's not about how can we can we list, you know, 17 different ways for solving every problem that we encounter. It's really about intentionally sharing a diversity of strategies and not just sharing uh, you know, diverse strategies, but making connections between them. And I think for me, that's really the key around this as the mathematical teaching practice is it's making connections between the representations is where the powerful learning really, really happens. So it's not just, okay, can we do a um, you know, a physical representation of this? Now let's do a verbal representation, now let's do a symbolic. And it's not checking the boxes. It's not, you know, just iterating to have a list and say, yep, yep, I did all five. It's about intentionally drawing those out and using those different representations and students' different solution strategies to deepen everyone's understanding in the class, right? So it's making those connections intentionally between um, between what students did. So one of the mathematical topics that I think lends itself to this idea just really easily are the thinking around uh ratio activities, ratio thinking, and proportional reasoning. So I um, you know, we were looking at some tasks together before we started recording. Um, so let's talk. I'm gonna turn it, I wanna flip it back to you because you're gonna give us kind of an outline and we'll link to this task in the show notes too, so folks can uh jump into it and gauge on their own. But tell me about uh the pancake task you found.

SPEAKER_01 13:09

All right. So I found this pancake task. Um it's part of uh a larger project that we're gonna uh link in the uh in the show notes here. Um and in particular, this was a lesson called Comparing Uh Strategies for Proportional Problems. Um, and in specifically, I I got down to a task called a sense of scale. And so here's kind of the situation. I'll set it up for us, and then Joni, we can kind of talk through um some multiple strategies. Cause I do think, you know, I really do want to draw out um sort of that intentional connection among multiple strategies, not just, hey, everybody's got a different solution, or there's maybe multiple ways that we can enter this conversation. But but more than that, it it deepens our understanding of the mathematics as we look at the connections between um multiple different types of solutions, representations of this. So um let's check this one out just a little bit. Um, it's called a sense of scale. And there's in particular, here's a problem about pancakes. It says this here is a represent recipe for making four pancakes. And then it gives us a breakdown of the units in here. And I'll just bring up the first one that in those four pancakes, we need six tablespoons of flour. Um, and the the question then goes on to say, you want to make 10 pancakes. How much flour do you need? And I love this this representation or this type of this problem that we've got here. Um, it it lends itself so greatly to lots of different strategies for solving this problem, lots of different ways to represent the problem. Um, I can I can in my head, I've already got, you know, pancakes being drawn out and some amount of flour being thrown at each one. And how am I gonna figure that out? Um, that is not certainly uh uh being a high school uh teacher, that isn't how I addressed the problem uh initially, but that's there's certainly things that could be done there, right? There's certainly things. So when I worked this problem, and Joni, I thank you very much for um just kind of throwing me in uh and saying, hey, go work this problem and uh and come back and let's talk about it. Yeah, let's talk about it. Um and so I'm gonna throw some things back at you on this. So what I decided to do was set up a ratio uh six to four. So I have my six tablespoons uh of flour to my four pancakes, right? Six tablespoons of flour makes four pancakes. So that's what I I set up. And I thought to myself, okay, so now I want to turn that four into ten somehow, right? I need to to come up with uh ten. So I wrote down just right immediately below it in standard ratio notation with my little colon there. I put a colon and then I put my 10, and then on the other side I kind of left a blank. And I I I was looking at that and I thought to myself, well, I know that if I double four, if I just make four twice as much, then I'll get to eight, and so I can double six. So I've immediately just gone to this multiplicative uh thinking, right? So this was the first thing that was in my head. I just jumped right straight to that. And then I thought, okay, well, I can't double again because if I double again, I'm gonna get too far. So I need to add uh two more. Well, two more is half of four. So I'm gonna add half of six also. So I I I then added sort of this this additive piece, but it's still like an additive multiplicative sort of in together, right? So I'm I'm adding half of four and adding half of six. And that got me to my 10, which gave me 15 um for my my tablespoons of flour, which was, you know, the that worked out well. And then I thought, oh, but wait. And in fact, I even realized this while I was working it out. I was like, no, I should have done it this way. I was like, no, I need to finish it the way it was first, instinctive, and then I'll come back and I'll say, okay, this is what else I that can occurred to me while I was working the problem. I also thought, oh, you know, I could just multiply four by, you know, 10 is two and a half times a four um two and a half times four. And so I can just do my two and a half times times six. Now those two things being presented by two students separately, everybody could say, oh wow, those are great. That's one, you know, this is one uh solution, and here's another solution, and wow, you know, whatever. But there's a connection between the two things of what I just did. Uh seeing that that two and a half really being two plus one half, right? Seeing that that broken up into two pieces, um, now I can make a connection between the two things and and and show that maybe the first one is sort of like that that first instinctive step, and it's something that I can do, right? Because I'm doing part of it and then I'm gonna add on another uh half as much. But now I can look at that that broken up two and a half and then and see that connected directly to what maybe what a uh maybe a another student who maybe has a little bit quicker grasp on the numbers um might have thought of right away.

SPEAKER_02 18:36

Right, right. Awesome. Well, I love that you shared your thinking. I I really appreciate how you shared the initial thinking and then the sort of your mind jumped ahead before you were completely finished with your first thought. Um and I also appreciate you reflecting for us for a minute there on how you saw connections between your own two ways of thinking about that and how you could see, you know, like, oh, I the first way I thought, I thought this sort of like a two-step thing. And then, oh, how could I condense it down to get there in a single step in a way, right? Right. I really like that. So so I want to offer a different um mathematical representation in terms of the five we outlined at the start of the podcast today. Sure. And I want to offer a different representation for how a student might go about solving this. So I've been working with my 10-year-old nephew, and I think, you know, he's this is this is seventh grade content, a little bit sixth, seventh grade, a little bit beyond where he is right now. So I imagine he's gonna want to do um either a physical or a visual representation. You know, he's gonna want to touch the table spoons and he's gonna want to imagine the four pancakes. Not that I have to make him pancakes to do it, but we could have, you know, like something that physically represents that for him. But he also might just draw it, right? Where he could draw a picture of six, six little spoons, and then underneath it draw four circles to represent six tablespoons of flour as the recipe for four pancakes. And then I could see him doing that again, right? Another six tablespoons of flour and another four pancakes, and similar to what you did, right? But actually doing it with his drawing and saying, okay, I'm at eight pancakes now. If I draw another set of six and four, I'm gonna have more pancakes than the questions asking. So I'm only gonna draw two more pancakes. And then he'll I imagine he would have to think pretty hard to get his three tablespoons of flour. But these are these are nice numbers, and you could imagine the picture in such a way that you could talk about that, like this is half of that original group. So if you imagine that physical or that uh visual drawing, the the power of making the connections comes when in a class you could ask a student who drew out, you know, two and a half drawings of six tablespoons. So a group of six, a group of six, and a group of three, followed by their four pancakes, four pancakes, and two pancakes. You know, where do we see the two and a half that Curtis's approach had? Where is that two and a half in this drawing? Two and a half what? Like what does the two and a half mean? Where does it show up? So those ideas around making the connections between the picture and the symbolic representation and even the thinking, the breaking down of the approach of, you know, first I doubled. Well, what does that look like in terms of the picture? To double the picture, what would that look like? What would that mean? So having the taking the time to have those conversations and have students look for you how does your thinking align with mine? What do what do our approaches have in common? You can see, I can see people can't see your. Face right now, Curtis, but I can and I can you can see, right? Like your face just is lighting up with the like, wow, this would be so powerful for a kid. Whether I'm the kid that drew the picture or whether I'm the kid that just multiply them by two and a half, right? The power in being able to think about how these things connect to one another. And um, I think it also brings in, you know, another thing we've talked about is kids relying on their own common sense and understanding that their natural thinking about the world is uh grounded in mathematics.

SPEAKER_01 22:32

I love it. And and I that actually really leads beautifully into what we um want to talk about next. But before I get to that, I just want to comment that my mind is absolutely racing right now with all the little images. I love the pancakes. I watched the the uh images go through my mind and I'm seeing that. And I'm also seeing, man, if I was doing a lesson where we were going to now try to establish unit rate, I'm working that image in the I'm working that that set of pictures backwards in my mind right now. All right, how do we get back to one pancake? Um, what what do we need to do to get ourselves back to one pancake? And the numbers don't work out quite as beautifully that way, but it's just, I mean, I'm immediately going, oh, how can we do that? How can we do that? Um, and and seeing that, and I I can imagine students being excited about that as well. This idea of um students and and getting their um their thinking validated, I think what you were just talking about shows um a validation of student thought, right? That that if my uh drawing, your your nephew's drawings of those pancakes, we can now see that two and a half times that I was saying in my little abstract uh writing over here, um, we can make that connection and see the two and a half and two and a half what and all that connection. Now it's like, oh, I don't have to, man, I can really see my own thinking up there in multiple different ways. And I feel and I find that connection. So that leads us into the question then, what other impacts, what impacts might we see on student learning because of using these different representations, because we've attended to that uh that sort of methodology.

SPEAKER_02 24:36

Well, I think the one you mentioned is maybe one of the most important. And again, tying back to, you know, a few episodes ago, we talked about students who struggle in mathematics and really reaching all learners. And that opportunity to not just acknowledge but really validate and value the thinking that every student brings is so important. And in doing that connection piece, we're really taking the opportunity to do that for students. So, and not only that, asking the direction, asking the question both directions, right? So, from both the student who approached the problem by, I'm gonna multiply both values by two and a half, and the student who drew out pictures to represent, you know, how does yours connect to the others? How does Andrew's thinking connect to Curtis's thinking? Each of each student verbalizing that is such a rich opportunity for them to rehearse the mathematical language, for them to speak precisely, um, and for them to acknowledge that the classroom is a community of learners, that, you know, it's not only the teacher validating each student's thinking, but it's the students validating one another's. And, you know, I know you can imagine uh, you know, sixth or seventh graders working on a task like this, they're not going to see the connections as quickly as you and I did. So Andrew might not see the two and a half from Curtis's approach in his work. So being able to ask each other and to talk and to dig into unpacking one another's thinking is such a great opportunity for mathematical discourse. And it's and it's uh, you know, it's coming organically from the students rather than, you know, a script that teachers hope to have. I know a lot of the teachers that I worked with in my district position believed in the power of mathematical discourse, but really had a hard time. How do I get kids to talk about their thinking? And a great task like this that lends itself to approaches and multiple representations is a great way to encourage that dialogue. And you can imagine too, as kids are struggling through that and you know, very new in their learning of these ideas, we have the opportunity to then introduce the formal mathematical thinking. I heard you talk about additive and multiplicative thinking when you were describing your own solution. Um, I think this would it's it's a great chance to, you know, highlight those things and name them for students or to talk about unit rate, you know, these are these are prime opportunities to drive home that um deep important mathematical language in ways that actually make sense and that and are authentic, right? And that's when we know that learning's gonna happen. So, and then again, back to what we talked about at the beginning for language learners, for kids who um, you know, may not go easily to the symbolic representation and therefore may have viewed themselves as not as good at math as the kid who, you know, does the symbol piece really easily, all of this is validating and valuing, you know, every student's approach and every student's learning opportunity. So I think that is is so important.

SPEAKER_01 27:48

And Joni, that that um idea of validating every student's learning and every student's uh solutions and and their work, um, I think creates great ownership of the topic too. You know, I I have this, I have this passion about creating ownership of the mathematics that I learn. And I know we've kind of seen that flow through all of these uh sessions and episodes, this idea of the mathematics that students know. And and I really, really, really like this idea of getting someone else in my classroom and and they happen to see, hey, I did it this way, and I see that you did it that way, and I see my solution in yours. And here's how your solution, here's how the pieces and parts of your solution are are represented in mine. And you hit on a couple of things there, just creating that classroom camaraderie, right? That that we're in this together, right? And that mathematics makes sense, but it also says, hey, like I own this, I know this now, right? Like this is mine. I've I've come upon this and and and seen it, and now I've seen it in a couple of different ways. And I love that you brought up also um the opportunity as a teacher to introduce and and continue to encourage that mathematical discourse. We've taken a problem that that really could have been found anywhere. Uh, you know, this was this is a pretty simple problem. This wasn't, you know, some big huge task that was, you know, well written, all this stuff. Like it's a it's a pretty simple problem. You got 10 pancakes, you know, you got six pancakes or whatever it was. Now I've forgotten. Four pancakes. You get four pancakes from from six tablespoons uh of flour. Um, and you want 10 pancakes, how many, how many tablespoons of flour do you need? I mean, that's that's a pretty quick problem to just write out or or whatever. Sure. Um and then this whole kind of representation, allowing students to have some time to come up with representations that are more than just, hey, we're practicing writing ratios today, or hey, we're we're gonna set up proportional relationships and solve these uh relationships, or heck, in my algebra one class, we're gonna set up a linear relationship and and solve this equation. Um, you know, whatever the whatever the system is that we're we might might want to put it in, instead just giving them the freedom to go play uh is is a is a big deal, right? I know I talked a lot about that in our last episode, like this idea of play, but I think this is what we were talking about. Like, let's propose a problem and let our students go. And we will, as teacher, then have the opportunity to sequence these solutions and be able to kind of build those up for student uh understanding and allow them the opportunity to see those connections.

SPEAKER_02 30:53

Yeah. And and I appreciate you, you know, bringing light to the fact that this is not hard. Like these this is a typical kind of problem you'd find even in a you know textbook published in the 90s, like the ones that I used in the classroom years and years ago. So it's not necessarily about, you know, a significant amount of the teacher's effort around preparing this, but they're easy to find. But executing them isn't trivial. So I think it's really important. You know, you said like you have to think as a teacher about sequencing. And before you can think about sequencing, you almost have to think about anticipating the different approaches. What's it gonna look like when a student is is taking a visual approach to this? And what if I what if a student is stuck and you know, what question might I ask that will help them create a verbal representation or get them, you know, moving that way? So I I love the um the five practices for orchestrating productive mathematical discussions as a framework for doing that. And it ties really nicely to what we're talking about in today's, uh, in today's podcast with anticipate, then monitor, then select and sequence those um, you know, share outs and then making those connections, which again, that connection piece is is um, you know, that's the exclamation point on your lesson. And I wanna, I wanna kind of dive into that a little bit deeper here for a minute before we as as we wrap up. I think this is kind of the this is the kind of POW thought I want to end the podcast with. That really it's it's not just about, oh, research says, so you know, we got to do what research says. That can be such a little trite little catchphrase in some cases, I think. Absolutely.

SPEAKER_01 32:34

Yeah, for sure.

SPEAKER_02 32:35

I think part of part of the reason that mathematical understanding is deepened by making connections between representations is because it provides us an opportunity to identify the underlying general generalities and the underlying consistencies that exist in the mathematics. And that's really that's the heart of it, right? Like understanding what is the math that is consistent, regardless of the representation. What is the thing that is the common thread as we're talking about this idea, this mathematical idea from different perspectives. So really helping students understand those mathematical structures and the essential features of what they're talking about, regardless of what form it actually took or what initial approach or initial thinking that the student had. So that insight into mathematical structure and general generalities, I'm stumbling over that word today, generalities is at the heart of deep mathematical understanding and is why attending to mathematical representations as a teacher is so very important for learning. Well, that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator. See you next time.