Pam Harris  00:02

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris.

Kim Montague  00:09

And I'm Kim Montague.

Pam Harris  00:11

And we make the case that mathematizing is not about mimicking steps, or rote memorizing facts. But it's about thinking and reasoning; about creating and using mental relationships. That math class can be less like it has been for so many of us, and more like mathematicians working together. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:53

So we've just finished a series of episodes for parents. And we're super grateful for everyone sharing them on social media so that we can get the word out.

Pam Harris  01:01

Yeah, we really want to help more teachers and parents have access to that resource. So thanks. And please keep sharing that podcast.

Kim Montague  01:10

Parents, if you've decided to hang on with us, great! We think you'll like the rest of these too. And you are certainly welcome here.

Pam Harris  01:17

Yeah, because Math is Figure-Out-Able for all of us. So welcome, parents, teachers, administrators, kids, anybody who wants to listen, we're gonna figure out some more math. And here we go.

Kim Montague  01:31

Alright. So in the past, we've talked a lot about strategies, or the way that people mess with numbers, you know, the relationships that you use to solve problems. We've even done some problem strings to pull out strategies that we think are particularly helpful. But today, and over the next few weeks, we want to talk about models and modeling, because we think that's pretty important, too. And the reason why this is so important is that people often confuse the two: models and strategies. 

Pam Harris  01:58

Yep, yep. 

Kim Montague  01:59

We've even done a whole episode on that. And that's episode number nine. What we sometimes hear is that a teacher might say, "How did you do it?" And the student might say, "I did a number line." And the teacher goes, "Oh, okay, you did a number line." Or the student might say, "I did it on my fingers." And really, that's the model they used, but not the strategy for how the student messed with the numbers.

Pam Harris  02:22

Yeah, so models and modeling are interesting. One reason, like you said, In episode nine - if you didn't listen to that, you might want to go check that out - one reason is because models and modeling can get confused with strategies, like Kim just said, we ask kids, and they do something and tell us the model, but we're really interested in the strategy they were using. So y'all models, the word model in mathematics, I think has at least eight different meanings in mathematics education. And it doesn't even include the one walking down the runway, not even that meaning of model. So just in mathematics education, I think the word model can have several different meanings. And a few of those meanings are largely missing from most conversations, a few very important meanings. When I talk to teachers and educators of all ilk, and math teacher educators are missing a couple or a few of those meetings, they're missing from the conversation. And so I worry that we're talking past each other that when we're talking about models and strategies, we're not communicating well. So I think, and we think, it's important to parse out those meanings so that we can communicate well, we know what we're talking about. 

Kim Montague  03:43

So Pam, let's start off with what you're not emphasizing today. Let's get out of the way a couple of the models that we don't mean.

Pam Harris  03:50

Yeah, okay, so what we don't mean. These can be meanings that we could talk about. But what we are going to talk about in this episode, and for the next few episodes aren't these meanings. So in mathematics education, we can talk about the modeling process. Nope, not what we're going to talk about, we're not talking about - and what do I mean by the modeling process? It's the bit where you have a situation or scenario or data, and you decide what's going on with that data. And so you create a mathematical model for the data. And then you use that model to interpolate or extrapolate or look in the future or in between, and then you look back and you refine the model. That's the modeling process, the sort of a scientific modeling process. It's wonderful. I believe in it. It's not what we are going to be focusing on these next few weeks. I might even use the words modeling process, but to mean something different, not the sort of scientific modeling process of creating a mathematical model to find answers or predictions in the future. And then we refine that model. That modeling process is not what we were talking about. So another one, you might think about a model as like a miniature, Like I might have kids build a model house to get at scaling and proportions and measurement, you might have built a model plane as a kid. And so you might want to talk about how the dimensions are in proportion and all that. Not that, it's not that tiny model sort of as a representation of the bigger one. We don't mean that one. You also might find it interesting that we don't mean model manipulative, the manipulatives as a whole, it's not what we're talking about. So when we say today, we're going to talk about models, we don't mean, today, we're gonna talk about manipulatives. Although manipulatives are models, they are, and we promote some of them as the models that we're going to talk about, but not carte blanche all. To be clear. We don't mean the modeling process, the sort of scientific, find a model, modeling process. We don't mean miniatures. And we don't mean manipulatives.

Kim Montague  06:01

Yeah. And here's another one that we don't mean, right? We don't mean to model-demonstrate. 

Pam Harris  06:06

Oh, yeah. That's an important one.

Kim Montague  06:08

Like here's exactly what you do. These are the steps or the procedures. Here's how you copy me. It's kind of in line with the I do-we do-you do. In other words, let me model-demonstrate how to do something. Copy me. That's not what we mean.

Pam Harris  06:24

Yeah. And I'll just tell you, when we write that we actually write the word model dash demonstrate. 

Kim Montague  06:29

Yeah. 

Pam Harris  06:29

Because we're so clear that that meaning of the word model is like really rampant in math education. And that's probably the one, like those other ones we use. Like we believe in the modeling process. We believe in miniatures. We think there's a place to use those. There's a place to use manipulatives. We don't use model-demonstrate very much. However, can I give just a really quick example, we do model-demonstrate teacher moves. 

Kim Montague  06:56

Yeah

Pam Harris  06:57

We do model-demonstrate mathematical behavior. We model-demonstrate mathematical mindsets, of growth mindsets. So we do model-demonstrates some things, but not how to do like Kim said, steps. Alright, so another one that we don't mean. And this one we're going to spend a little bit more time on. We don't mean the CRA: concrete representational abstract, or concrete pictorial abstract, that modeling process or modeling setup. We don't mean that. So it is a thing in math education, to talk about how first we need kids to like, feel what's going on. So we give them the concrete things. So that's the concrete, that's the C of CRA. So we give them manipulatives. We give them the tools, things to hold, to move, to mess around with. We act out the problems concretely. And then we represent that. So instead of we give them counters and that's the concrete, and then we represent that that's the R in CRA or pictorial, we represent those with pictures. So we sort of draw them the manipulative that we just had. And now we can sort of see what we were just doing with those concrete things now in a picture. And then we sort of move that to abstract. That's the A in CRA, abstract. Now we represent that with numbers. So now we write down the numerals or the symbols that represent what we were just doing in the pictures and the pictures are representing what we were just doing concretely. That's CRA. We don't mean that today, either. In our series that we're going to start today in our next few episodes, we are not talking about CRA. We're not talking about doing something concretely, then representational, then abstract. All too often CRA leads to the algorithm or leads to procedures, it leads to step by step things. And we have said to ourselves, "I don't understand why these kids don't understand or why they can't do these procedures. Alright, maybe if we do it concretely. And then we do it pictorially and represent it and then we do it abstract. Okay, well, now we get some better results." Let's be clear, we get some better results. We don't get really great results. We do get some better results - if that's your goal. If your goal is to get kids to do procedures, okay, maybe then you should do CRA because you'll get somewhat better results. Not our goal.

Kim Montague  09:24

Yeah. So we just said a lot of, or you just said a lot of things that we don't mean, right? Like getting some of those - you said there's eight, at least eight different kinds. So let's dive into what we do mean.

Pam Harris  09:36

Yeah, so in this episode, and the next few episodes, we want to really dial into an understanding of models and modeling that has everything to do with a modeling process that we really like. And it comes from - golly a few places. I first learned about it and young mathematicians at work by Cathy Fosnot and Maarten Dolk.. They built it on the Realistic Mathematics Education, RME, work from the Freudenthal Institute, where they talk about emerging models and a few other sort of bigger ways of saying that. But in short, we like to think about this three part framework, three part framework to think about modeling. So that three part framework is that first, we make a model of the situation, a model of what's happening, a model of the context. That's part one. And part two is then as we ask kids to think we do a model of their thinking, a representation of their thinking. So however, they're actually messing with the numbers, the problem, then we make a model of their thinking. And then third, that model becomes a model for thinking or as a tool for thinking. And this is a process, let me just repeat it really quickly. It's a model of the situation, then a model of thinking, a representation of thinking, and then that model becomes a model for thinking as a tool for thinking in order to think we use that model for thinking. This is the modeling framework that we want to focus on for the next few episodes.

Kim Montague  11:17

Yeah. And you know, even in that progression that you just mentioned, I think it's important to note that when we talk about a model of thinking, that's going to mean teachers first modeling students' thinking, before students begin to model their own thinking. And I think I mentioned this before, right? The idea that what kids can do in their heads is so much more than what they're able to verbalize. And that's so much more than what they can represent. So we have to model for them first, before they can model themselves.

Pam Harris  11:47

Yeah. And I'm actually going to give you credit for that addition, Kim. In fact, if you read about this, this process we just talked about, this framework, you don't really catch that. In fact we kind of actually maybe think of it more as a four part framework. 

Kim Montague  11:59

Yeah.

Pam Harris  12:00

So it begins with a model of the situation. And then we do the model of thinking, the representation of thinking, but in that there's sort of two things that happen. It's the model of thinking where the teacher represents the students' thinking, and then as we do that we say to kids, "Okay, see, when your brain does that, it could look like this." Then we say to them, "Hey, when your brain just did that, can you represent that? You've seen me model your thinking, now it's your turn, you represent your thinking." But they're not actually using it as a tool yet. They're like, "Okay, my brain just did this. You want me to make that visible? Okay, I've seen you do that to my thinking. Alright, now I'll, okay let's see if I..." And they have to take some time to do that. It's not just, "Oh, I've seen you do that and I could do it." That takes a minute for that, a hot minute, for them to be able to do that. Then there's that sort of last bit where after time, and with lots of experience, that model can become a tool for thinking, it helps them in order to think. So Kim, you're the first one that sort of brought up both these ideas that we can all do more than we can say clearly. And we can all say more than we can represent clearly. And because of that, we sort of have to help kids in that middle step. We model their thinking first, and then they can sort of use that to model their own thinking, or represent their own thinking.

Kim Montague  13:19

Yeah. So let's give an example with a progression that we use a lot in workshops, actually.

Pam Harris  13:25

Yeah, okay, perfect. So we do a thing with packs and sticks of gum. So we call it the sticks of gum sort of scenario. And we do a Problem String with it. And so very first we list the numbers of packs and the numbers of sticks. So we might say, in a pack of this particular kind of gum, we've got 17 sticks of gum, then we give people, students, participants a lot of problems to solve using that scenario. So we got a pack of gum with 17 sticks, and we represent what they do. And so if they add numbers of packs together and numbers of sticks together, then we represent that if they scale, they double the number of packs, then we double the number of sticks, we represent that on the ratio table. So as they add or subtract packs and sticks. And as they scale the number of packs of sticks, we represent what they're doing. That's sort of step two. So the first step, this idea of the model of the situation, we write down a table, and we keep track of, hey, in one pack, there's 17 sticks. And in that table, we represent the context: what's happening in the scenario. As they solve the problems that we give them we represent their thinking, using scaling marks, using brackets, we draw all over that ratio table and we say, "Oh, look, let me let me make your thinking visible." Then we ask participants, "Hey, could you show me students, could you show me what you're doing in a ratio table?" And they might go, "Ah, I mean, I've seen you do that. I guess what I just did, let's see what would that look like?" And then they represent their thinking. Then want them, the goal is to get that ratio table to become a tool for them to think with. Then when we give them problems, they actually begin to sort of use the table as a tool to actually compute to do the reasoning, instead of just doing the reasoning in their head. And then looking at the table, they're actually using the table as a tool to think with.

Kim Montague  15:22

Yeah, and that's different from CRA.

Pam Harris  15:24

That's totally different from CRA. Let me give you an example of what CRA might look like. So if I was going to do a problem with addition, some addition problem. A CRA might look like I say to kids, "Okay, let's take the concrete base 10 blocks, create each of the numbers. Now we're going to pull them together, and we're going to trade all those." We have a bunch of extra 10s and so we'll trade them and make 100s with a bunch of extra ones or units, and we'll trade those to make some 10s. And then at the end, we sort of read off the answer, like what did we end up with? And then that's kind of the answer. So once we've done all that with the concrete, then we draw that. So the first step in CRA is the concrete, C, concrete, and then the R, the representational, then we draw pictures of what we just did. We draw all of the flats and the rods and the units. And then we do the trading. And we draw all that trading. And then again, we sort of read off the answer at the end of the picture. And then we say to them, "So now it's time for us to do that all symbolically." So we've done it concretely with all the cubes and stuff that we drew them all. And we did that pictorially. Now it's time to draw those with numbers. And so now we just write down the numbers and we say, "Look, see, look here are like the cubes over here in the tiny and then when we traded up with that's the little one that you put up here." And we then make the symbols for what we just did with the concrete and in the pictures. And now we do it with symbols. And that's the abstract. Well, but notice, notice what that end goal was. That end goal was that step by step procedure, it was that algorithm. We're not about that end goal - step by step procedures. We are about using relationships, just like mathematicians do.

Kim Montague  17:08

Yeah, so a couple of important points in kind of closing. Kids don't know how to model/represent their thinking at first. So teachers need to do that for them until they've had enough experience to do that for themselves. And we want to do it on a model that we eventually want them to use, that model, as a tool. And we'll continue to parse out how this is different from CRA in the next few episodes. 

Pam Harris  17:34

Yeah, so this is very important to sort of understand that we are kind of differentiating the goals of CRA versus our goals. We're not dogging what good people tried to do with CRA. We're not saying oh that's bad. We're not dogging that, we're saying what is your goal? Is your goal step by step procedures? Well maybe then we would support using CRA, not our goal. Our goal is thinking and reasoning and mathematizing. And so we're going to advocate this different setup for models and modeling. So if you want to learn more math and refine your math teaching, so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!