I recently did some webinars where the topic of math strategies and math models, or notations, came up. I gave a quick answer, but I said that I would do a full video on it. 

I’m Christina Tondevold, The Recovering Traditionalist, and today we are going to investigate the difference between math strategies and math models in our quest to build our math minds so we can build the math minds of our students.

Watch the video or read the transcript below:

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5 Types of Addition Strategies

6 Types of Subtraction Strategies

Cathy Fosnot’s books

Now, when I first learned about how kids invent their own strategies for solving math problems without the teacher first teaching it to them, I was very fortunate to have training that made it very clear that how a child thinks about a problem is different than how they notate their thinking or how they model it on paper. The work of Cathy Fosnot and her Landscapes of Learning helped me to know when something is a strategy or a model. Then, as I got into this world of social media, I started seeing images like this one pop up: 

and I realized, I need to talk more about the difference between a strategy and a model, or what some people would call a notation.

Strategies

So let’s start off with strategies, what exactly is a strategy? 

Well, math strategies are basically how the students solve the problem. It’s what’s happening in their brain as they think through, and kind of the steps and procedures they are doing as they solve a problem. I like to say that it’s what happens in their head, that’s what a strategy is. 

Now, as they take what is in their head and try to put it on paper, that’s when the models come in, or what some people call notations. So their notation or their model is basically how they’re taking what they have up in their brain and putting it on paper. 

So let’s take a look at this one problem. We have the problem 27 + 39. And one of the students says something like, “I took one from the 27 and gave it to 39 to make 40. “Then I had 26 + 40, “which is 66.” Now, that thinking that the child is doing right there is their strategy, that’s the strategy. But how they choose to put that on paper or how we help them put that on paper could look very, very different. 

Models/Notation

Here’s an image that shows three different ways to notate or model that same thinking strategy.

So the model doesn’t make it a different strategy, the mathematics of what the child is doing is the strategy. But how we put that on paper, how we model or notate it, is that model or notation. So the strategy is really what matters. 

It doesn’t matter how they put it on paper, as long as they have a way to put it on paper, right? That doesn’t take up too much time, doesn’t take up too much space, they can efficiently model their thinking and take what’s in their brain and put it onto paper, okay?

It does not have to be the arrow model or the number line, or the break apart, or some people would call that first one the tree model. It doesn’t matter, what matters is they were breaking apart the number and getting to a benchmark number, that’s the strategy. They are using decomposition, they’re decomposing a number, they’re getting to a benchmark number that makes it easier for them to add.

Why we need to keep them separated

That’s the strategy a child is using, it does not matter how they put that on paper, right? We need to keep these separated so that it doesn’t confuse kids. Because in previous videos, I’ve talked about the 5 different strategies that kids can use to solve addition and 6 different strategies kids can use for subtraction. But if we then also layer on top all these different models for kids to learn, then it becomes even 15 different ways to solve one problem because not only do they have the strategies, but then now we need to use that strategy, but let’s model it on a number line, let’s model it using arrow math, let’s model it with a number bond. 

The model does not matter, it’s about helping kids have strategies and build up those strategies for students. How they put it on paper really doesn’t matter, but we do need to help kids be able to see the difference. We don’t have 15 different ways to solve it, there’s 5 ways to solve an addition problem, but you might model it different than this person over here, and that’s one of the things that we need to help kids understand. So what can this look like in your classroom? 

Now, for those of you who are just now joining me, you don’t know a lot about me, I’m a big follower of Cognitively Guided Instruction, or CGI. The basic idea of this is that you do not teach kids strategies. You give kids a problem in context, so a contextual situation, and then you let them solve it any way that they can. And as they are working on their own individual solution path, their own strategy, you’re circulating the room, talking with kids, discussing how they’re solving it. And if a kid is having trouble taking what’s here and putting it onto paper, then you can help model a way to notate it or model it on paper. 

You don’t need to stand up there and teach the kids, “Okay, everybody, today we’re using number bonds. “Everybody, today we are using the number line.” Those same models can be used for the same strategy. So one of the ways that you do CGI is you do these contextual problems, and then, as kids are working on it individually, then as you come together, you share and you have two or three kids get up and share things. And you pick those kids on purpose as you’ve been circulating the room. 

So there’s two ways that you can kind of go about this, and there’s a lot of books out there that talk about how to organize this time. But here’s two ideas that go along with what we’re talking about today. So typically what tends to happen is if you have this problem, when you have kids come up and share, we will have maybe two or three different strategies come up. 

So let’s say it was the 27 + 39. I might pick two or three kids who solved it with different strategies. Now, when you do this, you will get kids who will say things like, “Hey, I did it like so and so, “but I used a number line “instead of the arrows like they did.” So you’ll get kids who will say, “I did it like that person, “but mine was a little bit different.” So you don’t have to show a kid who did the arrow math, who did the break apart with number bonds, another kid who did the number line, and then another kid who did partial add. It can get very, very overwhelming and confusing not only for us teachers, but for the kids. 

You pick two or three different strategies that you saw kids do. Not different models, different strategies. And then, yes, you can have kids say, “Hey, I did it like so and so “but I used a number line model to show that same strategy.” We want them to be using those terms. “I used the same strategy as so and so, “but I showed it using a different model “or a different notation,” whichever of those you like better. 

Now, another way you can do it is similar to that image that I showed, where if you’re solving 27 + 39, there might be all of these kids who have the same strategy but use different models. 

So you could show the same strategy, they’re doing the exact same thing. Every kid added 1 to that 39, but they showed it in different notation systems. So during your share time, you could spend time showing the different ways that kids notated the same exact strategy, but you need to be very specific when you’re talking about it and say, “We all showed this using the same strategy, “but we’re looking at the different ways “that kids modeled it on paper. “This is the exact same mathematics, same strategy, “but some of you used different ways to model it on paper.” 

Those are big things that we want to build up for kids so they know the difference between a strategy and a model, so it doesn’t become overwhelming and super confusing, okay? So, again, to summarize everything that we have just shared here, strategies are how kids solve the problem, models and notations are how they show that thinking on paper. We need to help kids understand the difference so that it is not overwhelming and confusing when they see all these different ways to be able to solve a problem. All right, I hope that this video helped you build your math mind so you can go build the math minds of your students. Have a great day.

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