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Opening music

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Joanie – tease clip

Context can add or relieve cognitive challenge. Cognitive load. So if I'm just learning what slope is all about, adding context to the points we're talking about or the function we're talking about

provides an ease in the cognitive load, because now I can reason easily about that context.

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Joanie intro

In today's episode, Curtis and I continue our exploration of the mathematics teaching practices from principles to actions. We discussed the importance of using and connecting mathematical representations. We consider what this means and why it matters for student learning. And we both had some moments we're excited to share with you.

So let's get growing. 

00;00;51;00 - 00;01;29;02

Curtis

Well, Joanie, I am so excited to be, again, here together with you recording in person. We are at,

the Data Science Conference, in San Antonio, Texas. Really excited to be doing that. Today. We're going to be continuing our, kind of strand of looking at the Principles to Actions book, a publication from NCT celebrating its 10th anniversary. I still that's so great. That is really, really cool. And as a part of that, we've been kind of looking at the mathematical teaching practices that that are in there.

And today our topic is the mathematical representations.

00;01;29;02 - 00;02;23;17

Joanie

Yeah, I'm really excited about this one, Curtis, because there's so much depth in so, so many ways we can take this conversation today and see, you know what what rabbit holes we go down and

I appreciate too, that you called out that we're at the data science conference.

First of all, love being able to just sit across the couch from you instead of looking at you through a screen. But also, I think, you know, this morning sessions, data science sessions have given us so much good fodder for our conversation. So the official title of this mathematic teaching practice is Use and Connect Mathematical Representations. So why don't we start by just kind of backing up with what we're even talking about. Like what is it? What are we talking about in terms of mathematical representations? And what are we talking about in terms of using and connecting them?

00;02;23;17 - 00;02;38;28

Curtis

Well, I mean, if we look in the book there's a great visualization, of this and they've got five representations that are in, in the book listed off. And I'll, I'll list them here. But I think it's important that we kind of keep them in a context a little bit.

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Joanie

Yeah.

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Curtis

So I'll list the ones that are, that are listed there. So visual, symbolic, verbal, contextual and physical representations of mathematics. And really each of those may fall in different categories for different people as they kind of think about what does it mean to have a visual representation or a graphical representation of some mathematical context or some scenario or situation, maybe a symbolic one might have a different name for you as your, as you're thinking about particular context.

But really the important part is, is that the these are all different lenses with which we can look at the mathematics that we're, that we're talking about, right. That each of the situations and scenarios and ideas in mathematics, the topics in mathematics have aspects have faces, if you will, windows into the mansion, if you want to think about it that way, choose your analogy. But each one of the context or topics rather in mathematics have aspects of each of these different representations in them or in inherent in them. Right. So, we're here at a, at a data conference at a data science conference and, and thinking about representations of data. So one of the big things that happens in data classes is former AP statistics teachers, both of us, we like the visual space. We like to make pictures of data. Right. Because data pictures of data tend to tell a story. And it'd be easy for us to get lost in just let's make a whole bunch of pictures of these things. But it's important that we also draw upon the context and the the connections to the verbal and the connections to you telling the actual story that I see in the data, right?

00;04;21;20 - 00;05;22;20

Joanie

Right, right. I and I just kind of back up a little bit and say, although, you know, the five that you listed are what NCTM draws on the principles to actions and for our listeners who want to dig a little bit deeper into what those five mean, I would point them actually to, adding it up because that's where and CTM pulled that graphical image from. But I just kind of want to say, like as we work through our conversation, we're not going to get hung up on using those five words. Right. We're here to focus on conversation around the idea of multiple representations. Why is multiple representations of the same concept, something that teachers should be attending to in the classroom? So maybe let's think about a slightly simpler example. And again, I don't want to get into like, okay, as a teacher, I have to go through and make sure that every topic I'm teaching that…

00;05;22;16 - 00;05;26;02

Curtis

You mean I don't need to get up my students out of my seats for every single one?

00;05;26;02 - 00;05;51;16

Joanie

That again, is not the point, but I think it would be interesting to maybe call out a couple of examples like you started talk a little bit about AP statistics and you know, representations of data.

I want to I want to see if we can maybe pick another mathematical idea, you know, maybe more of the middle school level and, and talk through like what some of these different representations might be. So I'm going to throw out an idea.

00;05;51;03 - 00;05;55;09

Curtis / Joanie

okay, okay, Phil. And then I throw out an idea

00;05;55;09 - 00;06;10;23

Joanie

Not  trying to get all five. But let's talk

sure

few different ways that we represent, say linear functions and why is it important or how might we engage with different representations of linear functions. So what do you what what's like the first way you think about linear functions

00;06;10;23 - 00;09;56;07

Curtis

So okay linear functions. That's a that's a good one to jump off with because I think of linear functions. The very first thing that comes to mind for me is motion. Okay. So thinking about and and I even took it a little bit a step further back, one topic to, to proportional reasoning. So it was still linear, but with a very specific quality and thinking about positional, along some distance traveled, as I'm traveling at a specific rate. So ideas of motion and representing graphs of motion and on my horizontal axis is, is is, predictable. Right. It's dependable. I was going to say dependable. Because I can count on the horizontal axis being time. It's very natural for me to think about that, as in a representation. Right. And then as we put the vertical axis, this position or distance from a fixed point along a, an axis or along a line of some sort, students and my students, anyway, their first picture, when they I asked them, describe what's happening on this picture of distance and time. They want to talk about going up a hill. They want to talk about that, that they see this picture right of. And I say this describes Sally's path as she walked from home to school this day. Right. And they think Sally lives at the bottom of the hill and school is up at the top of the hill. So, so how do we have conversations that take that from a, a, a visual representation?

Right. So I have this visual representation that's happening there. I can write some sort of equation maybe right to, to describe how Sally's moving. Right. I can do that. But how do I actually get my students to, to grasp hold of? Well, that that vertical distance thing, what is that that, that value on the y axis, on the, on the position axis. How do I get them to connect those two things? Right. And then the equation going and the best way I know how is to get them up and actually say, here's a stopwatch, here's a measuring tape, we're going to walk at this particular rate, I'm going to have you try to be here when you're when your partner calls out, hey, I'm at, you know, three seconds, five seconds, 10s are you matching up and try to walk that? We, we love the calculator based ranger. This is a this is like tee-me-up on, on the golf tee. The CB2 is our favorite thing in the world because it gives me the opportunity to have students stand up, turn on a data measuring device, and start making the connection between my motion along some line on the floor and walking towards and away from a fixed distance.

Right. And I get a picture to draw the connection. Multiple, multiple representations. I don't think walking back and forth by itself does mean anything. If I just said, hey guys, we're just going to get up and I want you to try to walk at a constant rate. They might figure it out, but seeing it happen real time and that feedback of being able to see this thing grow all the time when I'm backing away from the wall, that distance is increasing, right? I'm walking on flat ground in our classroom or in a hallway, and I'm backing away from this wall, and I'm seeing that value on position increase over time. And then I get a feel for what is a constant rate versus what is an increasing or a decrease rate. And seeing all of that happen at the same time is really important.

00;09;56;07 - 00;12;28;20

Joanie

Yeah, I love that. And I love that you picked this example. You know, there's, a lesson and I'll link this in the show notes for this episode, but there's a lesson that I love to use when I do conference workshops.

And, and I use this lesson in all sorts of different workshops, whether I'm talking about student productive struggle or whether I'm talking about teacher beliefs or whatever. But this lesson is a distance time graph and, you know, you started by saying, like, students look at a distance time graph, say that has, you know, a positive slope, a positive rate of change and think Sally lives at the bottom of a hill and she's walking uphill. And I would just say, like, that's even teachers think that like that. It’s sort of natural thing to equate the graph with that actual physical representation. So one of the things I love about using that activity, or this lesson that you're describing of having students walk in front of calculator base ranger, which is tracking their distance from the device over time. Is that you can get to you could pick a specific point and talk about what does that point mean in terms of what's happening in the context. So like if we're talking about students actually walking across the classroom in front of the CBR, okay. What does it mean? What does this point three comma 2.8m. What does that mean in terms of Curtis walking Curtis's path so they can bring the contextual understanding of how the data that the CBR is collect ING is representative of what's actually happening in the world. Or in the case of, you know, Sally taking a walk. I think in the sample lesson I use, it's, it's a bike ride.

So what does this point mean? There's a section of the graph that's has zero slope. It's flat. And to be able to talk about like what does what does this mean in context, the start of the flat area and the end of the flat area. So again, this is what we're talking about when we're talking about representations. Right. Like we're talking about being able to look at a linear equation that has, you know, slope of two thirds and then be able to say what to where does that show up in the graph? Where do we see the two thirds show up? If we plotted a table of values around this linear function. And then what does it mean in terms of Curtis walking in front of the CBR?

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Music break

End of Segment 1

Beginning of Segment 2

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Curtis

I think one of the things that also comes out of this and I, I hope this isn't a curve ball, but one of the other things that comes out of this discussion and seeing the multiple contexts happen and being able to have students really draw meaning right from a particular point or from the, the graph overall. I can look at a representation and now be able to tell you a story specifically, right? Right. If I have two graphs on there, one's, one's red, one's blue. Right. And, and we can ask the question okay. So what's you know whose distance is changing fast or which distance is changing faster. Right. At this particular time or over this course of time, who travels the farthest. And we get to cover all kinds of ideas, right. Just from that one picture. And you can incorporate some high level mathematics in relatively low level courses, right? So kids can find areas. Right. So we can we can accumulate areas under a curve and do some calculus without telling them that that's what they're doing. Right. So I love that that all of that is there. So what I'm kind of going, oh, this is an moment even for me right now is thinking about that. The math we're doing actually means something like it. We're not just doing it for fun. We're not doing it just because we need to learn linear equations or learn about context for, for slope, because we have a test coming next week or it's on the. Yeah, it's on the state test. No, this is actual real, real life applications right. For things. And it gives rise to oh my algebra class is meaningful. The things we're learning in algebra are actually going to be applied. Linear functions are things that I see on a daily basis. And I can apply these things. I can think about them in a real context. I'm not just learning how to manipulate equations for equations sake.

00;14;43;12 - 00;14;50;16

Joanie

Yeah. And, and it ties back to what I know is maybe your number one passion project out of curiosity. Right.

00;14;50;16 - 00;14;53;06

Unknown

Amen.

00;14;53;06 - 00;15;45;11

Joanie

we didn't show up at the same time to the conference this morning, so I don’t know if you were in the opening session, but there was a college student. He's a sophomore. Okay. Great. 

And I loved I thought of you when he said curiosity equals motivation.

And so when I think about, like, how many teachers we hear like, oh, our students just don't care, and why do we have to learn this and kind of that apathy that we know is very, very real in classrooms and has, I would say has gotten worse since the pandemic. But this idea of genuine curiosity such a great antidote to that. And, and so thinking about bringing that context in any opportunity we have, you know, it provides that opportunity for curiosity and motivation.

00;15;45;11 - 00;16;18;05

Curtis

for sure, I mean, I just think about, you know, contextualizing things. Right? So we, we, we have this challenge sometimes, but I don't think we have to I think that's why we're here at this conference is because I think we get an opportunity to tackle this idea of real contexts, helping us teach the mathematics that we that we actually teach. 

00;16;18;11 - 00;16;34;29

Curtis / Joanie / Curtis

So I was in a session earlier this morning and the presenter was talking about we were looking at an example of, smokers and lung capacity. And does smoking actually affect your lung capacity or something like this? No, no, no, we are in Texas. We are in Texas. That's okay. That's okay. Now this idea of smoking and does it affect your lung capacity. And so we had this group, we had this large set of data and we started looking at the data. And at first we just looked at distributions of data. And in seeing smokers versus non smokers and kind of seeing you know kind of the distribution of lung capacities.

And it was kind of interesting, intriguing even that the nonsmokers had from this data set tended to have less lung capacity. And we were look at I was kind of sitting there and I was thinking, now, purposefully or not, I'll leave that up to our listeners to decide. Our presenter had made mention just off the cuff, made mention as we were looking through the data. I was a little bit surprised to see an 11 year old be a smoker in this in this data set, because they had they had age was one of the things that they had collected from all the people. Right? Was another one that they had collected from all other people and all these other variables. We had like five variables, I think. And so just way before we actually looked at a graph, she had made that comment. We were just kind of scrolling through the data and observing. And so my immediate thought was, oh my, wait a minute. If 11 year olds are in this data set, I'm going to hypothesize that that 111 year old that reported themselves as a smoker probably isn't the norm for 11 year olds. So does age have anything to do with this? The natural question that a student might ask, right. And so with but not if we hadn't. Right? Right. And so once we asked that question, we plotted age against this, this year, now we've got this sort of linear progression of things. Right. So there's, there's a linear model. And we, we made a big concentration of introducing the linear model as y equals some intercept or an initial point starting point, plus some function. Right. Plus error, plus you know, all the other stuff. Right. Because people talk about being tall, being short, having something to do with it. Or am I an athlete? Am I not an athlete? All of the things living in Colorado, being born in Oregon, all of those kinds of things maybe make a difference, right? But the idea of, you know, we have a function, an underlying function, and then we have all the other stuff. Right? And we can start to use that underlying function to do prediction, but then we still have all the other stuff. But I loved the idea of answering a question or having a curiosity or introducing the need for a linear function like I. I never would have thought of age. I never would have thought of age as being, you know, a confounding variable here in the data that they collected. But sure enough, because they collected it across, you know, young people. Five I think the youngest data point I saw in the list was a five year old, obviously not a smoker, but I was a little sad to see there was an 11 year old that reported as a smoker. But, just that idea that that age was a part of the the conversation gave me enough to go, oh, well, maybe we can make a function here. Maybe we have a little bit to explore and to think about and then it necessitated the function that we were learning.

00;20;06;13 - 00;20;07;03

Joanie

Love that.

00;20;07;03 - 00;20;21;03

Musice break

End of Segment 2

Start of Segment 3

00;20;21;03 - 00;20;28;25

Joanie

I'm coming back to the representations. Right. And thinking about there's a reason this is one of the eight math teaching practices.

00;20;28;25 - 00;20;29;23

Curtis

for sure.

00;20;29;23 - 00;20;43;29

Joanie

Why is it listed that using and connecting mathematical representations is one of these highly effective practices? And I'm just thinking that that context piece and the way that can serve as a scaffold for the verbal part, right,

00;20;43;29 - 00;20;44;26

Curtis 

for sure.

00;20;44;26 - 00;20;52;08

Joanie

To be able to talk about for every year increase in age what happens to lung capacity. To be able to verbalize that is really complex. And how often would we just say what's the slope.

00;21;03;19 - 00;21;04;28

Curtis

Right. Right.

00;21;04;28 - 00;21;32;19

Joanie

And so we're talking about the same level of, you know, seventh grade, eighth grade, ninth grade mathematical complexity. But really by attending to multiple representations, we're providing the opportunity for a lot deeper thinking. And I know, I know, I taught kids who could spit out slope all day long, but not be able to tell me in a sentence what it meant.

00;21;32;19 - 00;21;33;25

Curtis

Right, right.

00;21;33;25 - 00;21;38;24

Joanie

So, so I just think there's so much power when we add context In because it provides the scaffold for kids and I don't want to go too far off on a tangent here either. But one of the things I noticed in one of my sessions this morning was context can add or relieve cognitive challenge. Cognitive load.

00;21;57;25 - 00;21;59;17

Curtis

That's fair. Yeah.

00;21;59;17 - 00;22;11;11

Joanie

So if I know I'm going to do a, challenging context, then I want to maybe ease off on the cognitive load of the mathematics. Right?

00;22;11;11 - 00;22;12;25

Curtis

Okay. Sure.

00;22;12;25 - 00;22;54;07

Joanie

So I could add cognitive load in terms of making sense of the context when I'm solid and understanding the math and vice versa. So if I'm just learning what slope is all about, adding context to the points we're talking about or the function we're talking about provides an ease in the cognitive load, because now I can reason easily about that context. So I'm just thinking about this idea of the representations also give us a way to kind of offload some of the cognitive that allows students to maybe think harder about a different representation of that same mathematical problem.

00;22;54;07 - 00;25;33;17

Curtis

I think that's I, I love that you brought that up, that the cognitive load piece because we have innate reasoning abilities, right? That that things just make, make sense. And we sometimes don't know why. Right. Like we, we understand certain things before we're able to verbalize them. Right. Thinking about

moving quickly versus moving slowly and being able to get somewhere faster than I would, what does that actually mean? Being able to verbalize it and describe it is a very different cognitive load than understanding. If I run from here to there, I'm going to get lunch faster than if I walk right, like like that's unless you're in a house where running was out a lot and then you got had to like sit in timeout because you ran or whatever. But now the just that idea of, you know, we understand the physical connections to things. We've experienced them experientially because, you know, mathematics, the language of describing the world and the way world works like it, it's happening around us and we're learning it all the time. But then trying to put it into some sort of formal context and being able to do that, becomes a little bit more challenging, right? Unless we start to be able to see those connections side by side, we use those multiple representations. I just I love this idea of utilizing physical contexts and, and,

the, the, the real world. And I know that's been thrown around for so many years, this idea of real world mathematics in an, in and of itself is a beautiful and wonderful thing. And I even once titled, or put it in the description of a, of a conference session saying, you know what? If we take that idea of explore our world, through math and, and flip that on its head and, say, exploring, exploring math through our world, this idea of, hey, we can utilize the elevators in the hotel lobby and think about these, these functions. Right. Right. We can think about linear functions if we want to describe it as linear rates of change. We could even talk about steps. Right. We have step functions. Right. These, these particular times or intervals that this thing is in, different locations. Just the idea that that, the world itself presents much of the opportunity to learn the mathematics that we need to, through these visualizations and, and, different representations of mathematics.

00;25;33;17 - 00;28;22;01

Joanie

Yes. And I the, the question that's popping into my head as I'm hearing you talk about this is how come societally, we still have so many people that say, oh, I'm not a math person, or I never liked math, or I was never good at math, like you're doing math just by existing in this world. So it makes me think about I think, what is a key important part here, as we, as we start to wrap the conversation and make the connections back to the classroom and that's this explicit connection, It's not just about like, oh, hey, use context to drive student motivation or, you know, consider the context as a release or increasing cognitive load, like all of that is important. But if it's not made explicit to students then that's the missed learning opportunity. Again, I know, when I again, we, I feel like almost every episode we come back to when I was in the classroom, I really screwed this up. But this was like, I would spend hours planning these kind of exploratory lessons for my students, where I would guide them through experiences to engage with the mathematics that I wanted them to learn that day. And I was so proud of those activities. And they would do it. And I'd be, you know, walking around and listening to group conversations and I'd be like, oh, they're totally getting it. They're saying the things I wanted them to say about the mathematics and then it would be like, oh, the bell is going to ring in two minutes.

Here's your homework assignment. Papers forward. And, and I never did the like summary part of the lesson. And then they'd come back the next day and I would be discouraged and disappointed that they didn't get what I wanted. It didn't stick. Yeah.

all the right things. And and it wasn't until, you know, the end of my 20 years in the classroom that I realized, like, oh, wait, like we have to make explicit what the learning. It was obvious to me, I already have all of that learning. So I could see how this series of questions or being able to answer this, that. And the third thing leads me to this dramatic mathematical conclusion. But my students didn't see that and I didn't help them connect the dots. I let I put the breadcrumbs out and led the path. But I never helped say, like, this is it like, this is the mathematics we just learned? And do you see that? You know, the how many floors the elevator passed in 12 seconds? Is this on your graph? And in your equation? Right. Like making that explicit and not necessary. I'm giving a poor example here because I'm telling, and I don't think that telling is always the best way. But that idea of making explicit and not assume, saying students are going to follow the train of thought that we set up for them.

00;28;26;14 - 00;31;13;11

Curtis

so many things just went through my head in this last, like, 30s here. Well, I just, I mean, I thought of, wow, just trying not to, like, go. Well, I'm trying not to go and like, start a whole new podcast here, with your idea, with, with that idea of, you know, this the importance of being able to make explicit those experiences that we talk about in the math classroom. We know the answer, you and I, where we, we know where the lesson is headed, and, and there's, there's all these things that have come along in education about. Well, we should do we should be putting the this is the learning outcome at the very beginning of the, the, everybody's looking around for those things. And then now that's not the right thing. We shouldn't be doing that anymore. But there's a timing. There's a timing of when we make the learning explicit. And is it me that makes it explicit? Is it you? Is it the students that make this explicit? Like there's, there's so much we don't have time in this podcast. It's at least one podcast, right. Or maybe two, thinking about when, when all of those things happen. But I think that's an important piece of the puzzle. We think about these representations, and there's a quote in the, in the principals to actions book talking about the importance of mathematical representations as, giving students multiple lenses. And I oh, I don't want to there's a great movie that's, I don't know. Is it a classic? Is it a cult classic? I don't never I can never tell the difference between these things. There's a great movie that came out in the, I guess, early 2000’s. There's a scene in the movie where this actor is looking at something and it just doesn't make any sense. And he's just put on these funky looking, goggle looking glasses things and then he accidentally bumps part of the glasses on his face. And it changes his perspective because one of the lenses moves, and then he figures out that that was the intent of the build of the lenses and it makes the message or the thing that he was trying to look at completely clear. And I think that's what we're doing with these multiple representations as we kind of like put these representations in front of students with all these different kinds of lenses that they can look at a particular topic, this idea of having multiple representations. And then we as the teacher, we as the class, we as a student, calling out, making that kind of summary connection. I think that's I think that's what we're kind of talking about here.

00;31;13;11 - 00;31;17;17

Joanie

exactly what we're talking about. So the teacher's role is to bump the funny glasses,

00;31;17;17 - 00;31;22;23

Curtis

That's it. Bump the funny glasses.

00;31;22;26 - 00;31;42;21

Joanie

There’s our call to action. Go bump the funny glasses. I really do like that analogy though. Like thinking about throughout my math lesson, students are seeing these different components, But they’re not seeing the crispness of the message I'm trying to send. I really think that was a great I don't know what movie you're talking about. I'm going to have to find out.

00;31;42;21 - 00;31;46;16

Curtis

You'll have to ask me afterwards.

00;31;46;16 - 00;31;57;20

Joanie

I could like the visualization that that created in my mind and the connection to a math classroom. And that's like, how many times did we say, like, when you see the light bulb go on for the student, that's the bumping of the glasses, right?

00;31;57;20 - 00;31;59;12

Curtis

Yes, yes,

00;31;59;12 - 00;32;00;06

Joanie

They get it. 

00;32;02;20 - 00;32;20;23

Joanie

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!