Episode 137: Teaching real world maths

Andy Psarianos

Hi, I'm Andy Psarianos.

Robin Potter

Hi, I'm Robin Potter.

Adam Gifford

Hi, I'm Adam Gifford.

Andy Psarianos

This is the School of School Podcast.

Welcome to the School of School Podcast.

Welcome back, everyone, to another exciting episode of the School of School Podcast. And today, we're here with the core crew. So we've got Robin Potter.

Robin Potter

Hello, hello.

Andy Psarianos

Say hello, Robin Potter.

Robin Potter

Hi.

Andy Psarianos

And Adam Gifford.

Adam Gifford

How you doing, Andy? How you doing, Robin? Nice to be in your company.

Andy Psarianos

Yeah, I'm doing great.

Robin Potter

Thanks, Adam.

Andy Psarianos

Hey, so today, we're here. We're going to talk about some upcoming kind of, well not upcoming, but something that could very well happen, which is a change of government in England and some new policies in maths education in England. They're making a lot of noise about it. So one of the things they're talking about is, I suppose, making maths real in the classroom, or they're calling it real-world maths.

I mean, what does that mean? What do you think that means, Adam?

Adam Gifford

It is a really interesting one because I don't know if it's just a sort of difference of perception. So people might perceive a maths lesson as I write a load of equations on a board and people come in or they memorise formulas, and that's effectively what maths is. That's the perception of maths as it stands. That's kind of, I don't know, well, just the core delivery of it.

So then real-world maths sounds like this sort of radical shift as if... I'm not actually sure what the intention when that phrase is used, but there's hints that real-world contexts, so whether it's cooking or measurements, those sorts of things. But of course, we've been doing that for ages.

I always think we can't separate the two out in terms of mathematics and the real world. That's just the way it works. But I just suspect that it's a perception shift, but also a real focus and prioritisation on understanding maths and context and seeing the world mathematically and I think that that is so crucial, and unfortunately, a lot of people have not experienced that, that have been through school.

Andy Psarianos

Yeah. I think for our generation, and I think this is maybe something that we run into a lot, is that you've got to remember politicians haven't been in the classroom for a long time, and more than likely, they were taught the same way we were, which was a heavy focus on arithmetic, calculations, procedures. That was how it was taught when we were young. Certainly, when I was young anyway.

Adam Gifford

Yeah, exactly.

Andy Psarianos

I expect both of you had similar experiences.

Robin Potter

Yeah, absolutely.

Andy Psarianos

Yeah, I don't think we've taught that way for a long time in most places. Some countries still do, but in England, there's been a move away from that for a long time. But do you think they're saying we need to go even further?

Adam Gifford

Yeah, I think there's probably models that they've seen that have been successful. So if you look at other countries, we talk about Singapore a lot, and looking at those statistics and seeing that it's happening in the UK and we worked with a lot of really successful schools and the kids win when that happens, and understanding the importance that it plays.

So I think that even if the overall understanding from a political party, I don't know the level of detail they understand it at, but I think that what is understood is that mathematics has to be seen differently. The age of doing it the way we did it, where we had to memorise X number of formulas in order to succeed in an exam, the realisations come that actually, the phone that we're carrying our pocket can do incredibly advanced calculations. We don't need that. That doesn't separate one person out from someone else, the ability to do it. What separates them out is the ability to apply it to a situation that they can see. And that's the part that I think is missing.

And global economies are looking at problem solvers and the OECD coming out and making direct correlations between mathematical attainment and not just financial outcomes but health, health of people, direct correlations. So something's got to be done and I think that that real-world context, and we've talked about this a lot, the ability to, again, I use that phrase, see the world mathematically and apply the mathematics to a context in which is appropriate or apply appropriate mathematics to a context in which we're facing. And I think that they've got to take notice of that because teaching it the way we were taught is not going to be beneficial to society. It's not.

Andy Psarianos

Yeah. I think we could probably all remember a stage where I suppose we all started getting a little bit confused in our own math experience. I think algebra is often when it happens for many people when all of a sudden, there's a real disconnect between what you're doing in the classroom and anything in the real world because we have tended in the past to just kind of focus in on, okay, this is how you juggle these variables and numbers around, an equal sign to get something that looks like an answer. And it just seems like some random process, like stand on one foot and tap on your head and rub your belly, and then the right answer will come out and it's as arbitrary as that for most people.

And so in that context, how do you make that real world? I know that I'm being a bit silly because I know we all know the answer to that because that's what we spend all our time doing already, but maybe explain, Adam, how you could take a concept like algebra and make it real.

Adam Gifford

Yeah. But I think if we are using something like that... So for example, I was doing some work with someone the other day, let's say pricing. There was a pricing model in a bakery and you had to put something in a box and the item, the unit cost or one bun cost this. Then you can multiply out that cost by the number of buns and add on the cost of a box. So we can take an idea, if you like, and we can look at the context of the problem and apply, in effect, a pattern to it. We can start to see that we can work those things out by applying the mathematics.

And if I just go back one step further, I think the most obvious thing, and it seems so ridiculous to say this aloud, but the first step is literally to associate a real-world context with what is written down mathematically using symbols and figures, and so many people's experience, that's been a complete disconnect.

So even the idea, so before you get to the stage where you say, "How can you make something like algebra real-worldy," where we've got some unknowns, we might talk about we just need to balance things. "I need the same amount of this as I do this, I use the same amount, the total weight of," I don't know, "flour and sugar to butter." So that doesn't change. It's balancing those things together.

Who would've thought of putting those together? I could have just said that A=B+C. That's what I could write something down, but who would put something together with a real-world context? And just that idea alone that that exists is something that I think we probably didn't get to experience very often at all, if at all.

Andy Psarianos

Yeah, exactly. And I think the other thing that people need to understand is that the concepts get introduced very early on. So if you want to teach something like algebra and you want to have a real-world association, what you need to do is, say, create problems early on. Maybe the age of eight or nine, you're dealing with things like, just to use your context, three apples and two pears. You know the price of an apple, you know the price of a pear, and the price of the container or the box, whatever. A bag costs 5p, an apple costs 7p and a pear costs 8p. How much do they add up? That's an algebraic problem.

So just replace the apple and the pear with an X and Y. And again, it's algebra and being able to manipulate. So you start off with real-world context and you eventually replace objects, let's say, with variables, and it's a slow progression, but it starts very early on.

So if you look at a good maths programme, and I'm going to use Maths — No Problem! as an example, because hey, we know it very well, that journey starts in year one, really, right, algebra? It just doesn't appear in year seven.

Adam Gifford

Yeah. And I think the thing, just picking up on that, and even earlier than that, we talk to children about reading maths. Now we read it in the same way we read a story.

So for those people who are listening that might not imagine what I'm saying, let's just go back to the apples and pears. We've got two apples and one pear, for example. Then if we're putting them together in a bowl or we're doing something with them, that's that physical action. We can imagine buying two pears from one, I don't know, shelf in the shop and an apple from another and we put them together into a bowl. We can tell that story.

Then the very last thing is we can record that story, so 2+1=3, and I'd expect the children to start reading it. And if I said, "Oh, I'm going to write another story, now 3+1, read the story to me," then the likely response will be, "oh, they bought three apples this time. Oh right, okay." So you can imagine, and it becomes something that we start to read in the same way we would reading the alphabet, like reading words from the alphabet.

But again, to me, it's the experience of applying maths to an experience, to something real world, and being able to make sense of it and record it with efficiency.

Andy Psarianos

That's right, yeah. So I think the key is if you were to think about what's the approach that we think people should take is that you start with the real-world context and then the challenge is to find the mathematics that applies to the context that you're talking about. So if you're adding fruit, that's one context. If you're measuring something, that's another context. You say, "Oh, Adam is twice as tall as his son," that's a context. Now you're applying measurement to a context. You're not just measuring stuff.

So you start introducing other ideas, so that's a multiplicative question, twice as tall, but you're applying it using a real-world context, which is measurement. So now you start to see that, oh, multiplication isn't just remembering a bunch of stuff. It's not about remembering all these random facts, but there's an application to these random facts. And if you start with that context and then find the maths to solve it, then that's, I think, what we're talking about when we're talking about doing real-world maths.

And the arithmetic part of it, of course, is important as well, and so is the remembering of facts. It's really a lot easier to do mathematics if you don't have to sit down and calculate, struggle your way through a calculation, but that's not what it's about. That's just knowing how to spell words or knowing what the grammar looks like. It doesn't make you a great writer just because you know how to spell lots of words and you know where the commas go and when to use a semicolon. That's the mechanics of it. We need to know the mechanics of it if we want to communicate effectively and we want to be efficient, but the key is the story is the important bit. And I like the fact that you use the word story earlier.

All those things make storytelling easier. All those things make storytelling easier.

Robin Potter

Okay, parent jumping in.

Andy Psarianos

Okay, parents. We're all parents, by the way, so you don't have dominion over this one.

Robin Potter

Yeah, no, but I'm coming from the parent perspective. So this sounds great. Yeah, I get it. I get what you're saying. But then how is this different than what we're already doing? I mean, maybe I'm Maths — No Problem! bias here, but in general, in the classroom, I thought we were already using apples and pears and then applying it.

Adam Gifford

Can I jump in?

Robin Potter

Yeah.

Adam Gifford

Can I jump in and answer that? So we will assume because this is what we do, that that approach is generally adopted. And I think certainly in primary school, the prevalence of that approach has skyrocketed. It really has. There are far more people who will do their best to provide a reasonable context for the math.

Where it tends to fall down... Now, I'm basing this, I remember the day my son came home and they got a new maths teacher, and he said... He looked really just annoyed, to be honest. And I said, "How's your new math teacher?" He said, "Well, the first thing he said to us was, 'Between now and the end of the year, the academic year, we've got to memorise 23 formulas and that's what we're doing for maths.' And that was it. There was nothing else."

Now, I appreciate that some mathematics makes it more difficult to put into context when, when the mathematics starts to become... But mathematicians will all use words like modelling. So people will model certain situations, which I would argue, replies to it. But I think the most important thing that happens is that it becomes a conversation. So the language that children are using and the understanding of what they're doing, we can start to relate mathematics in that way. And when we do, when we are able to make sense of what's going on, so if we're using unknowns or the variables, the same thing I sort of talked about, is do you understand the context in which you've used them previously? So you're not having to attend to that idea; you're already well-versed in it, and you could give multiple examples surrounding you of where that exists.

So I think for our children, knowing that mathematics has a real-world application, it's not just to pass an exam. That was the message my son was given, "Memorise this so you get a decent GCSE." It wasn't, "Memorise this..." I'm going to keep using the word memorise because that's what came out, but there was no, what was the purpose? What, just to pass an exam? Is that motivation really? Is that what we want? Like a society that just has to pass an exam, not apply it, not use it to solve problems? You know?

And the other thing that gives is a really false idea of what a mathematician is, like people who do it as a job. These people are problem solvers. They're not sitting there just to purely add something up. There's something that's been put forward to prove. It's a different proposition. They're not just sitting there memorising more and more each day. And so I think that that feeds into the narrative of the whole thing about what mathematics is about.

Andy Psarianos

Absolutely. And I think the issue that we often have, and we hear this all the time, is where people say, "Well, I left primary school loving maths and feeling like I was really, really competent and I saw myself as a mathematician. I went into secondary school and I fricking hate it now because it's all memorising this kind of stuff," whatever. And at that stage, that's when math starts getting hard. And then you start talking about nevermind getting into university, it's even more so sometimes, and we forget how important those visualisation concepts are.

Let me give you a simple practical example. From where you are, start now and walk straight to the north pole. Imagine that you can do that. And then when you get to the North Pole, you're going to turn 90 degrees, and you're going to walk straight the exact same distance, so just visualise this in your mind, the exact same distance in that direction. When you get there, you turn 90 degrees and then walk straight in the direction of where you started and keep walking. Where are you going to end up? Just think about it. So you're going to go straight up, 90 degrees, you're going to turn 90 degrees, and you're turn 90 degrees again.

Adam Gifford

Back to where you began, I'm assuming.

Robin Potter

Back to my backyard.

Andy Psarianos

Okay, okay. So you're going to end up where you started, right? Okay. Now what shape did you make?

Adam Gifford

Well, if you were to flatten it out, it would be-

Andy Psarianos

No, I'm not flattening it out. Okay, if you were to flatten it out, what shape did you make?

Adam Gifford

That's a good question.

Robin Potter

Oh man, am I going to get this wrong? Here we go. Triangle?

Adam Gifford

It's a triangle.

Andy Psarianos

You walk... It's a triangle, right?

Adam Gifford

Yeah.

Andy Psarianos

Okay, now add up the angles. 90+90+90=270. What are the internal angles of a triangle?

Adam Gifford

Not 270.

Robin Potter

Not 270.

Andy Psarianos

Right. It's 180, right?

Adam Gifford

So something's happened.

Andy Psarianos

So now, everybody can visualise this, right? Everybody can visualise this. Now you're talking about topology, you're talking about really complex mathematics here. But basically what you're saying is everything you know about triangles is only true in a Euclidean plane when things are flat. The minute you curve a surface, all the rules change.

Now, that's a really high-level concept. It's not hard to visualise that and understand that once you say, "Oh, yeah. Okay, I get that now." But we don't do that, right? We go straight into this complex fricking graphing and super complicated mathematics right away, and you're like, "Well, I don't understand any of this stuff," or why all of a sudden everything I learned no longer applies.

So that's the kind of stuff, that's why real world is so important because then you can make sense of it. And there's so many examples like that. And when you think about when do people drop out of mathematics, it's usually because all of a sudden none of it makes sense anymore.

Adam Gifford

I also think, just finishing up on that, and I know we're wrapping up, Robin, but I think that that example that you give as well, Andy, is I think part of the problem, if it's just down to sort of memorization and stuff, it's very binary. You get the answer right or you get the answer wrong, and that would suggest that whatever you've memorised is either right or wrong, and that's it.

But I think if we discuss these things, so I don't know, if you're building a sandcastle, we'll go back to that measurement one, and I was going to build a sandcastle twice as high and twice using the same skinniness, the same width, then maybe I can't do that. Now it's a different problem again. But what it gives me is it gives me now I'm thinking about mathematics differently, so when you come at me with something like that, I can just be blank sheet and go, "Right, I'm going to try to understand this," as opposed to what I know is either, "I don't know the formula for that, so therefore I can't do it," as opposed to, "how do I need to think about this?"

And actually, if I went with what I did and flattened a globe, it's not going to be a straight line. If you flatten that shape... So it doesn't work. But this is the mindset that I think that we're able to create if we can have those conversations using the real world as a context.

Andy Psarianos

Yeah, exactly. And then if you understand that, when you look at a map and you see that Greenland is the same size as Africa, you recognise that actually that's not true. That's just because what you said, you can't draw a straight line to represent something that's on a sphere on a flat surface. So somewhere you got to compromise, and then that's where all this stuff adds up. So then those are all real-world examples of incredibly complex mathematical ideas, and that's what we need to do. We need to explain it with those kinds of things, not with just like, "Here's a really complicated formula, and don't worry about where it applies. You just need to remember it to pass the test." I think that's what we're talking about.

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