Ep 9: Strategies and Models

Pam Harris  00:02

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

Kim Montague  00:08

And I'm Kim. 

Pam Harris  00:09

And we answer the question if not algorithms, then what in the world are we supposed to teach?

Kim Montague  00:15

Hey, Pam, before we start today, remember how we asked listeners to share a mathy thing that one of their friends does? 

Pam Harris  00:21

Uh-Hum.

Kim Montague  00:22

Alright, we got a couple of responses and I wanted to share one with you. So Robin White from Pine Tree said that she's got a friend named Clay Nyval. And he messes with numbers a lot. So she shared two things that he does that she thinks maybe not everyone thinks about. One of them is that when he adds numbers, like say 241 and 316. He thinks about 24 tens plus 31 tens to get 55 tens or 550. And then he adds the ones to that. Isn't that cool?

Pam Harris  00:54

That is kind of cool. So y'all, you might want to get a pencil and write those numbers down. Like he was thinking about 241 and 316. But instead of thinking about 200 and 300, he was thinking about that 241 is 24 tens with the 1 leftover, 316 is 31 tens with the 6 leftover and what did you do with those tens? Again,

Kim Montague  01:15

He just added the tens together.

Pam Harris  01:18

Because 24 tens and 31 tens, he can get 55 tens. So he knows that's 550 and then he just tacked on the ones. That's pretty cool. 

Kim Montague  01:25

That's great place value. Right? 

Pam Harris  01:27

Cool. So what's the other one?

Kim Montague  01:28

She also said that they had a conversation where he sometimes just comes up with an idea and then finds a lot of examples to test his theory to prove or disprove it. It's kind of like the Wonder game. So he just, something will pop in his head. And he just runs through a bunch of numbers to just see if his theory is correct or not.

Pam Harris  01:47

Pretty cool. Pretty cool. So shout out to Clay and thanks Robin for suggesting Clay. So if you have someone you'd like to shout out that does some mathy things, we'd love to hear from you. So send them on and then we'll give a shout out to your friend. Alright, so we have some other exciting news. We've just posted video of Problem Strings on the website. Whoo. So you may know that my favorite instructional routine is called Problem Strings, because it is the coolest ever. It's the least used out there. But it's so powerful as a way of helping construct relationships in the learner's head. So you've been asking for video of real teachers and real kids, and we have made it happen. So head on over to the blog at mathisFigureOutAble.com. Not only do we have the video posted, but we're also posting whole write ups with teacher instructions so you can facilitate those strings in your own classroom. Very exciting, and we're thrilled to be putting that out.

Kim Montague  02:44

Yeah, it's gonna be such a great resource. Okay, so onto today's episode. Today Pamela, we are talking about the ever confusing models and strategies. It's not a new topic, right. There's been a lot of conversation around this topic.

Pam Harris  03:00

I mean, so you say it's not a new topic. It's not for us. We've been talking about it for a while. But often when we meet teachers, it's something that they really haven't thought about. We've been posting for a while on social media, some graphics that asked people, you might have seen them on Twitter or Facebook or Instagram, that ask people whether the examples they see are two models, two strategies, one of each. And what we found is that not only are parents confused about the difference between a model and a strategy, lots of teachers are too. Even though the standards call for different models and strategies. We've got a lot of confusion out there about the difference between them and how we can use them to better teach real math.

Kim Montague  03:42

So how can we help teachers and parents?

Pam Harris  03:46

Yeah, so in today's podcast, we're gonna define models and strategies, and let's start there and then we'll talk a bit more about how we can use them better. So let's start with definitions. So first of all, Kim, what is a strategy?

Kim Montague  03:59

Okay, right. So a strategy is going to be the way that you mess with numbers. It's how you solve the problem, the relationships that you use in your head.

Pam Harris  04:09

Okay, so if a strategy is how you mess with the numbers, then what's a model? 

Kim Montague  04:14

The model is the way that you represent those relationships that you've used. It's a way to represent to others what's happening in your head, so that you can communicate the mathematics.

Pam Harris  04:24

It's kind of the picture of what's going on.

Kim Montague  04:26

Sure.

Pam Harris  04:26

Yeah? Alright. So we'll do a podcast later on the word model in math, because there are a lot of different ways that word is used. But today to parse out the difference between strategy and model, we're going to really focus that the strategy is how you mess with the numbers. And the way we're talking about model today is that it's what your strategy looks like: the way you've represented how you mess with the numbers.

Kim Montague  04:51

And I think what's so confusing for so many people is that given a problem, you can solve it with a few different strategies and each of those strategies can be represented with a couple different models. And so it's kind of like what's going on?

Pam Harris  05:04

Yeah, exactly. So let's give some examples. 

Kim Montague  05:06

Sure. 

Pam Harris  05:07

For example, if you're going to solve an addition problem, and ya'll if you want to get a paper and pencil out to sort of follow along with the relationships, this might be a time to do it. So if you're solving addition problem, like 48, plus 36. 48, plus 36. You could, in fact, you might want to pause the podcast and actually solve 48 plus 36. And then come back and hear some different strategies. We always like to have people mess with the numbers before we ever superimpose someone else's strategy. So for the problem 48 plus 36, you could think of the strategy that we describe as "Add a Friendly Number", you could start with the 48. And say to yourself, "Instead of adding the 36, I'm just going to add a friendly 30. So 48 and 30. That's 78. And now I still got that six hanging around. So 78 and that six, let's see, that's 84." So you could think about that 48 plus 36 as adding that friendly 30 first. That's the strategy: Add a Friendly Number. I can say that with words. I can write it with equations. I can represent the relationship on an open number line, which is the one we prefer, especially with beginning learners, because the open number line is so nice to show those relationships that makes that thinking really visible. What would that look like, with the strategy I just used? I'd write down the 48. And then I would draw a big jump of 30. And I would write down that we got to 78, right? And then I would say, "Okay, so that leftover six." And so then I might add the two to get to 80. And then the leftover 40 gets 84. But the big point of that strategy is that we added a friendly number. And the model that we used was an open number line.

Kim Montague  06:46

Sure, but you could also solve that problem 48 and 36 by using something that we call "Get to a Friendly Number", right? That's how I'm thinking about it. So like 48 plus, just a little 2, would get you to a friendly 50. And then you still have 34 left to add. So now you're adding 50 and 34, to get to that final 84. That's how to do what I'm thinking about is called Get to a Friendly Number. And again, I could model that with equations or on an open number line. And my open number line would actually look different than yours, because I would still start at 48. Make a little bitty jump of 2 to get to 50. And then a big jump of 34. Your way of thinking could be represented on two models. And my way of thinking could be represented on those two models. But we're thinking about the numbers differently.

Pam Harris  07:38

And when you say, "way of thinking," that's strategy. 

Kim Montague  07:40

Right.

Pam Harris  07:41

Yep. 

Kim Montague  07:42

So what I'm realizing is that there are a ton of people who mess with numbers. They really do use a variety of strategies in their real life. But they don't necessarily model that for others. Right? My husband's a builder and he has great relationship with numbers because he does a lot with number. But he doesn't represent those on the number line. It would be all equations if he were asked to record something.

Pam Harris  08:06

Yeah, and he doesn't probably do either. He probably just does the, use the relationship... And he probably doesn't use either he probably just uses the relationships in his head. Gets the answers needs and moves on. But in a math class, we want to communicate our thinking to others, and that's where the modeling comes in. Let's do a multiplication example.

Kim Montague  08:26

Okay, how about 16 times 9? Okay, I might think about that as 10 times 9, and 6 times 9. And I could represent that with equations. right, or on a ratio table or on an open array. So my strategy would be 10 times 9 plus 6 times 9. 

Pam Harris  08:47

So, you're breaking up the 16 into a 10 and 6, okay. 

Kim Montague  08:51

And I could represent those on an equation model, a ratio table model, or an open array. And on the open array, which is going to be my choice, I might look at that as a 10 by 9, and then a 6 by 9 put together to make that 16 by 9 rectangle. And then you can add the areas of each of the small rectangles together to get the area of the big rectangle. While a different kid might think about the 16 by 9 as 16 by 10, and get rid of the extra 16. That would look like a set of equations or entries on a ratio table or a big array where you hack off part of it. So again, a different strategy. But the same models, but those models might look a little different because it's a different strategy.

Pam Harris  09:39

Yeah. And so you described two strategies, sort of where you'd kind of chunk two smaller areas together was one strategy. And the other strategy is where you found a bigger area and kind of hacked off the extra that you had. Those are two different strategies and you were describing them on a rectangle or an area model, which is a really a nice way to sort of integrate multiplication and area, and dimensions and factors, and the area with the product. And so that's one of the reasons we like to use the area model or the rectangular array is that it brings those all together. But it also makes the thinking visible, which is the power of modeling. And we'll get into more detail about this later. But I wanted to mention, if you are doing those rectangular arrays, those area models, then those rectangles should try to at least be somewhat proportional. Like don't make a 16 by 9 rectangle look like a square No, don't do that. Because that's bringing sort of in Proportional Reasoning that teachers should have that rectangles should look like rectangles, squares should look like squares. Part of the power of that model is that students can develop spatial sense, but if we're not actually representing them correctly, or at least somewhat correctly, that spatial sense goes out the window and it becomes just another set of steps for kids to memorize. So we really want to represent rectangles as rectangles somewhat proportional to what they really are.

Kim Montague  11:03

Sure. That's a really, really good point. So we started with some numeracy examples, talk to me a little bit about a high school example.

Pam Harris  11:12

Alright, so a high school example of what strategies are versus what models are, could be that often we ask kids to find the equation of a line given some data. So they've got some numbers, some data, and we say if you were going to fit a line to this data, then what would the equation be of that line? So one way that students might use data to find the equation of a line that would fit that data, is they might find the rate of change. And then they might walk back that rate from a given point until they find the y intercept. They might choose a friendly point, the rate that they found, and then walk it back one at a time, because the unit rate, until they get the y intercept. And now that they have the rate and the y intercept, they can write the equation of a line. But a student might do that in a table. Or they might do that on a graph. Two different models, but they're using the same strategy of using the unit rate from a given point to walk back to the y intercept. That strategy is a Unit Rate strategy. And the model could be a table or a graph. Now a different strategy might be that a student might find that rate using that data, but they might find a non unit rate. Say they might say, the kid was traveling six feet in two seconds. And instead of simplifying that to three feet per one second, and using that and going back by one second, every time, they might go back two seconds, and six feet all in one fell swoop. They might use that non unit rate, and then that way, they're jumping to the y intercept even faster. That's a bit more sophisticated, a bit more efficient. So that strategy, we could talk about that strategy as a Non Unit Rate strategy. And that student could use either model. They could represent their strategy on a table, they could represent the strategy on a graph. A third strategy that we've seen students use when they've been given data to write the equation of a line is that they might say, they might find that rate of change again. And then once they have that rate of change, they might say, alright, so if the kid was walking at a rate of two feet per second, so let's say that that's the data that kid's walking, and we've got time and distance data. If the kid was walking at a rate of two feet per second, I can think about that line, y equals 2x, or y equals 2t. And I can think about that line. But I know that it needs to go through these points. So how high up would I have to shift that line? So I sort of start with the rate of change. And I think about the line that would have that blanket rate of change, how high would I have to shift that line to be able to go through most of those points? So that's sort of a Transformation approach. Their strategy is to think about transforming that rate of change line to go through the points. It's kind of we would call that a Transformation strategy. Well, they could think about that, most of the time, we see kids thinking about that on a graph, but they could think about that in the table. Two different models, a table or a graph for three different studies. strategies a Unit Rate strategy and Non Unit Rate strategy or a Transformation strategy, all to find the same equation of the line.

Kim Montague  14:08

So, you know, I'm listening to you share a little bit and I'm thinking about how we're really just honoring the way that kids think about things. 

Pam Harris  14:16

Yeah. 

Kim Montague  14:17

Just their own approach to solving mathematics. Okay, so we defined model and strategy. And you and I are clear that a problem can be solved with different strategies, and then also they can be recorded with different models. So I'm going to ask you a big question now. 

Pam Harris  14:32

Okay. 

Kim Montague  14:33

Which of those is more important? Which should teachers and parents be more concerned with, strategies or models?

Pam Harris  14:41

So in a nutshell, if you force me to answer this question, teachers should care about both. But if I have to make a choice, it's more about strategy. Like you just told us earlier about your husband, who has some strategy. That's so important. He's not just mimicking a bunch of steps, regurgitating a procedure that he learned. Like he's got some ways of dealing with relationships. That's so important. Now, if he was my student, ideally, I would want to build on that. I would want to represent what he was doing in such a way that he could then build other strategies. That would just help him gain more relationships and have a more dense brain structure. So models can help bring the thinking forward, make it visible, so more students can pick up on that particular strategy. More students get more dense, they own more relationships. But it's those relationships. That's the strategy. So models are important in communicating what the strategies are doing.

Kim Montague  15:37

It's kind of like that models aren't going to do us a lot of good if we don't have a variety of strategies.

Pam Harris  15:43

Yeah, totally. The thing that we do in our heads, the strategy, the way that we represent it, can totally take longer than doing it in our head. So of course, it might take longer to put it on paper. You might have seen some social media stuff that sort of poking at, "Why are we doing this new math stuff, look how long it takes." But in reality, it might not have taken very long to do in your head at all. It just takes longer to put it on paper. But part of mathematizing and mathematics is community and it's communicating our thinking, posing solution strategies and defending our choices.

Kim Montague  16:17

The social media stuff just makes me crazy, right? When they make fun of how long it takes to represent your thinking, I want to yell at the screen and go. "Of course it takes time to communicate, but what's going on in my head is so much quicker."

Pam Harris  16:28

So when you ask a student, "How did you solve that?" And you hear students say, "Hmm, I did a number line, I did an array". This shows that the student is, thinks we mean the model not the way they're thinking - the strategy. So we can help this confusion by pushing back and saying, "Oh, I see your number line. I see your array, but what did you do with the numbers? How did you use the number line? How do you use the array? How did you mess with the relationships? Let me help you represent that." if need be if they need the help. But it's focusing on not that you did a number line, but how did you use the number line? Not that you did an array, but how did you use the array? It's a little challenging to talk about strategies and models without having some visuals. But we wanted to start the conversation here.

Kim Montague  17:11

Right. If you want to learn more, you can check out the blog on Pam's website mathisFigureOutAble.com for a blog on strategies versus models. And if you're a leader helping teachers, you will especially want to check out our card sorts that help clarify the difference between a variety of strategies and models. Please join us on Wednesday on your favorite social media for MathStratChat to talk about models and strategies with other teachers and parents. And also if you'll like our podcast and give us a review, that would be fantastic.

Pam Harris  17:45

So thanks for joining us today. We've had a lot of fun talking about models and strategies. If you're interested to learn more math and you want to help students develop as mathematicians then the Math is Figure-Out-Able Podcast is for you. Because Math is Figure-Out-Able!