Euclidean geometry, the CPA approach, and more. In this episode, Andy and Adam discuss representations and symbols. How important is it to use the right resources? Who is Jerome Bruner? Plus, hear why when learning how to count, pupils should use objects, not something abstract.

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Andy was one of the first to bring maths mastery to the UK as the founder and CEO of the independent publisher: Maths — No Problem! Since then, he’s continued to create innovative education products as Chairman of Fig Leaf Group. He’s won more than a few awards, helped schools all over the world raise attainment levels, and continues to build an inclusive, supportive education community.

With nearly 20 years of education experience, Emily has a knack for creating wildly successful learning content. Her past work includes publishers like Oxford University Press, Pearson and Collins Education. Currently, you’ll find her dreaming and scheming in her role as Head of Publishing at Fig Leaf Group.

In a past life, Adam was a headteacher, and the first Primary Maths Specialist Leader in Education in the UK. He led the NW1 Maths Hub’s delivery of NCETM’s Professional Development Lead Support Programme before taking on his current role of Maths Subject Specialist at Maths — No Problem!

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Hi, I'm Andy Psarianos.

Hello, I'm Emily Guille-Marrett.

Hi, I'm Adam Gifford.

This is the School of School podcast. Welcome to the School of School podcast.

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Okay, everyone. Thanks for coming back. Today we're talking about representations in math. So I guess, what does that mean, Adam? I'm not sure everyone's going to understand what we're talking about, what the topic means.

Let's make it all pretty simple, let's bring it back to what maths could be about and how we could use it. So what we want to try do is, we want to see that mathematics has real world application, that it solves problems in the real world. You know, we're just... The world's a funny place at the moment and it's required a lot of mathematics to get through it. And when we think about how we present mathematics, a lot of people might think that the only thing involved in maths are numbers and symbols. Now that's maths notations, so that is true. But, what we also need to think about is how can we represent situations that give those numbers and symbols meaning, and we can see a whole host of ways of doing this.

And I think that something that's relatively new to the UK at the very least, and I think a number of countries throughout the world, is something like bar modeling, where a bar represents something, a given amount. And we can talk about the area that's covered in that bar or whether there's parts to it, or those sorts of things. So, if there were two different amounts, we might look at one part of the bar being longer than another. And we could assume that depending on the context of the problem, that might indicate that one cost is more than another. And because it's together we're looking at a whole amount, the total cost is made up of two parts.

So we can use representations, we could... People might be familiar with circles that are cut into fractions, just showing amounts that it covers. So a quarter will cover one part of a circle and if we had four of those parts, it would cover the whole circle. We know that to be the case. So I think that when we talk about representations, what we're trying to do is give an idea of what something might look like so we can apply a mathematical idea to it. And we hope that from that, it supports the understanding of what's happening when we take it through to the abstract where we write it, if you like. What people will immediately, I think, equate mathematics to, which might be one plus one equals two, and we eventually write that.

So we use representations in a whole host of ways. When they're used poorly, we're in trouble. When they're used well, then they can be incredibly supportive. But the key is we've got to understand what the different representations might do and how they're used. And I know that's something that we've talked about at length, and I think we've talked to others about at length, the things like, say, number lines. We need to be really, really careful about how we represent certain things. And I would also say whether that representation is using concrete materials or pictorial materials, it depends on what you want to do, and how you get those things to work for you. I think that's the crux of it, that we use something to represent something in the real world, for example, to support learning.

Yeah. So I think for a lot of people it's not immediately obvious how important representations are. And like you said, people have an idea about what mathematics is, and usually it's got to do with some form of calculation or arithmetic, and it doesn't often have to with representations. Say, here's an interesting thing, Euclidean geometry. It's amazing how much complex mathematics you can do with just geometry. Okay, that's a whole topic on its own and probably too much for this podcast. Let's not go down that rabbit hole.

So Bruner is the guy that we attribute with this concept of what he called inactive, iconic, symbolic, what we often call concrete, pictorial, abstract, or even CPA approach in teacher jargon. And the concept is that when you present a new mathematical idea, you need to represent it in a concrete fashion initially.

Think of young children. When you're learning to count, it makes sense to count objects like apples. And if you're counting apples, you should count apples because it's too much of a stretch for the children to count just abstract without having objects to count. So when you're learning that fundamental idea of cardinality of counting, you need to count objects, because otherwise you can't really form a proper idea. So it starts concrete. But at some point you need to move away from that, and you need to move into what we often call pictorial representations. Now, pictorial representations doesn't necessarily mean pictures, although most times it does. But it's kind of the first step of moving away from using concrete objects to moving to something else that isn't the concrete object, but represents the concrete objects.

That could be a picture of an apple. So in the first steps you could go from real apples, and then you could go to pictures of apples. But then there's another step. You could represent the apples, the real apples, with just counters or beans. Now just think about that, conceptually. You're saying, "In order to count the apples, we're going to count these beans." That's actually kind of a crazy idea for someone who's got no experience whatsoever in mathematics, right? That's a big cognitive leap. And then you need to get away from that and then you need to move into representations where you can represent known quantities, but not in a one-to-one correspondence, which you're talking about, the bar model. So you say, "I've got 25. I'm not going to draw to 25 boxes. I'm going to draw rectangle and say, this rectangle represents 25." Now again, that's a huge leap.

Now what you're saying is, you're going from a one-to-one representation, to representing known quantities with one entity. That's actually kind of a big step. And then from there, you get into this crazy idea of representing unknown quantities with things, and manipulating those. So that's what algebra's all about. You have no idea what X is worth, but you can move it all over the place in these equations. That's kind of a crazy concept.

But the idea is, you don't just jump straight into algebra. There's a real journey, a long journey that takes many years that you go from counting apples to being able to do algebra. All those things are different representations. By the time you get to algebra, you're basically dealing with abstract symbols. So you've gone from concrete objects, to pictorial representations, to abstract symbols. So that's kind of what we're talking about when we talk about representations, is that middle part. That middle part. You're not dealing with the abstract, you're not necessarily dealing with the true concrete, but that middle part between the two where you're representing things with other things. Now the interesting place that I'd like to go to with this is, are all representations created equally?

And I would say no. Here's just a couple of practical examples. So anyone who's, let's go for year two, year three. What we've seen plenty of times is if people are doing a written method for subtraction and it's 33 subtract 17, right? So the child looks at the seven and the three and thinks, "Well, I can't take seven from the three, I'll take three from the seven," Easy, job done. So I'll just go one way or the other. So, using a representation for that, if you just gave them 33 counters, I'm not convinced that that would help understand the whole idea about decomposing a 10 into 10 ones. They'd be able to do it, and they'd be able to realize, "Okay, I can get to the answer." But the one thing that might not happen is understanding about the whole renaming process and the fact that that numbers compose and decompose at a rate of 10 in the system that we work in. So better still to give them a tens block, or represent 33, 3 tens blocks and three ones, and a wee pot of ones.

Now I've done this with loads and loads of children where I say, "Okay, well, away you go, is there enough there to take 17 away from then?" "Yeah, no problem at all. We can do that." What becomes difficult then is they can't take what they need from the tens block because it's physically put together. They know that they can't get out of pair of scissors and start chopping them up because they'll get in big trouble for that. So what tends to happen is that they'll cover them up. And it's like, "Move your thumb away. No, they're still there." So what they physically have to do, when they struggle with this for long enough... Now don't panic, I don't leave children struggling forever and ever until the bottom lip starts to go. But what you want to happen is for them to realize, "Oh, am I allowed to rename this one 10? Can I get 10 ones instead?"

At that point you are showing that whole idea, that concept, and the children are then able to make sense of it because then all you are doing is recording what's happened. You're saying, "Oh, okay, so you took that 10 and you renamed it as ten ones, let's write that down.", "Oh, you did this. Oh, right. Let's write that down." And so in effect, the abstract representation is recording it in the same way we might write a story about going to the museum. It's in effect recording the actions that we've taken in order to get the answer and to develop that concept.

Another example that we can use in terms of representations, where choosing the right one is essential, is if I was to say, "I've got a cake cut into five equal parts, can you name each of these parts?" And that's no problem, each of these parts is a fifth. So if I took one piece, it would be a fifth of the cake. But, I've put on a bit of weight recently so I'm thinking, I don't want a fifth of a cake. That's a bit much. So I'm going to go for half a piece, right? I want a 10th. So can you show me that using Multilink cubes. Now Multilink cubes, once again, would be brilliant to show the fifths. I can show that cake and I can represent it, rather than having a new cake, I can use the Multilink cubes. The problem comes is, if I want half of a piece, what are you going to do? You know, same problem, you can't go around cutting up the maths resources in your classroom. You can't pull out a pair of scissors and launch into it.

So what you're going to have to do is, you're going to have to either imagine what happens, and it becomes so much more difficult. So perhaps in that instance, it would've made far more sense to use a strip of paper. So I can take one of these five pieces and I can rip it in half, and then I can represent a 10th, and I can start to see what's left. So in answer to your question, "Are all representations equal?" the answer's no. The context in which the question arises or the problem arises is what we need to take into account when we decide on the representation that we use, to make sure that it supports the understanding of what's going on.

And I think the other really important part to this is that, when we are representing something, either using concrete materials, or we're doing it pictorially, we need a really clear link between what we are writing down and the abstract. We need to see that we can read mathematics in the same way. If I wrote about going to the museum and talking about the exhibits that I saw, we're just recounting something that's happened, something that we've had experience of. And we are recording that using the alphabet and the words that we know.

I think in the same way, we must see mathematics in that way. And I'm not convinced it always is, that if children learn that they're just... This is a way of recording what you are actually doing. Look, read, tell me what that is. "What did you do there?" "Oh, that was the part where I changed that tens block." "Ah, is that right? Is that what we do there? Oh, is that what happened? So if I wrote something similar, you'd be able to tell me what happens there, would I need to do that?" "Ah, I see, I see, I see," Ah right, you are reading mathematics. So I think it's those things as well, that we need to get it right when we choose those representations. Because if not, how do they possibly match up to something that is symbolic? As a bunch of symbols that on their own don't necessarily mean anything.

Yeah. And I think as a teaching professional, there's a responsibility that comes in setting up these people that are under your controlled domain, whatever, for the rest of their lives. And all too often, and I'm sure it's true for all of us. We've heard so many adults say, "Oh yeah, I'm not a maths person." What the hell does that mean? But when does that happen? At what point do people get switch off from mathematics? Because mathematics is one of the greatest discoveries of all time, it's the most fascinating thing. There's these really complex, amazing things to think about that if you just like to think... It very quickly gets into philosophy and all kinds of things.

And that's the beauty of mathematics. That's so much fun. For me it is anyway. But other people just get switched off. They just say, "I hate mathematics, it doesn't resonate with me." Now why does that happen? Well, at some stage in their learning... I can guarantee you because almost everyone has this experience, but sometimes it's more profound to some people than it is to others. You're asked to make this leap from one thing to another and it actually, it doesn't make sense. And then someone says, "Look, just do this and you'll get the right answer." And none of it makes any sense. And then all of a sudden math gets all too confusing. It becomes, "I have to remember a whole bunch of rules. Oh, if the X is on this side of the equation, then you need to do this," and blah, blah, blah, and whatever. And you need to cancel this out.

There's a bunch of rules. "To find the area of da, da, da, da, da, multiply this times that plus this." And you got to remember it, because there's a test on Wednesday. And at that stage, none of it makes any sense anymore. It's just remembering a whole bunch of random nonsense, which is how I was taught lots of subjects, by the way. History was like that for me. "In 1742 so and so did this," Like, yeah who cares? What does that have to do with anything? That's kind of the thing. If it's presented to you in that fashion, you're very likely to be turned away, right? There you have it.

Thank you for joining us on the School of School podcast.

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