Top tips, golden rules and more. In this episode Emily asks Adam to diagnose some crippling questions. How many slides does a circle have? How many vertices does a cone have? You may need a pen and paper here to help visualise the concepts discussed!

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The school of school podcast is presented by:

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.

Hello, and welcome to School of School podcast. I'm Emily and I'm chatting to Adam today. I've got three questions for Adam that I'm really curious, both as a parent, as a publisher of Maths materials, and generally, curiosity. And it's actually come from questions that we sometimes get from customer services, from teachers and parents as well. The first one is, I was surprised that this was controversial but it came up and seemed to cause lots of responses and it is how many vertices does a cone have?

Yeah, this isn't the first time I've heard this come up, to be honest, because it all depends on how we are taught and how we understand that we can reach a conclusion. And so, first of all, let's get a few things straight. So when we talk about vertices or vertex, often you'll hear them referred to as corners on shapes where straight lines meet at a point. Usually two straight lines on a two-dimensional or flat shape and where three lines meet on a 3D shape, again, if we can imagine a cube, something like that with straight lines. So often in the definition, we are told exactly as I've just said, that they're where two straight lines meet.

So then we're presented with a cone and it all gets twisted and all upside down because we're saying, "Where are the straight lines?" So we look and we say, if you imagine, now we need to really visualize here. So the easiest way of doing it is an ice cream cone. And just imagine that the factory forgot to have the hole at the top, where you sit your ice cream and it's just been sealed off with the rest of the cone stuff. So we sit that so the point's pointing towards the roof of your house and we sit that on a table and we look at it and we say, right, we can describe open in a number of ways. Right?

So it's sitting on, if you like, a circle or a disc at the bottom, and we can imagine that and it's flat, it's sitting there, it's not wobbly. And then we can see the point at the top. Yeah? So we can imagine that and we can see that. The-easiest way of probably understanding where the straight lines are, is if we were to get down so we're level with the table and look so we can see the cone in silhouette. And what we see is a triangular shape. If we don't see the curve, we start to see that triangular shape. And if we imagine we leaned a ruler against the side of the cone, and you could imagine it leaning against the side and going up past the point. Now that top point is a word, we refer to that as the apex. Yeah?

And that comes from Latin. We use apex... The Latin is meaning summit, or the top, or the extreme top. So we can imagine when that cone sitting there, the ruler would go past the extreme top and offered an angle. Can you imagine here, you're getting that so far?

There's a point in which your line that you've drawn has gone beyond the shape?

Yeah, yeah, yeah. So we are just leaning one ruler on one side. Yeah? So we got that. Now, if we were to have a second ruler and we were to lean it on just, let's say, the opposite side, we'd notice that those two lines cross at the apex. The two edges of the rulers cross at the apex of the cone. And what we say is that the cone, if you like, is made up of line segments. So if we could imagine we could make a cone... If I gave you, and you had a lot of time on your hands, if I gave you, I don't know, a hundred toothpicks, and I gave you a circle of dough and you were to stick those toothpicks in the dough, all of them meeting at a single point. So all of their points meet at a single point.

Now obviously there'd be gaps. So it's not a cone. Yeah? Because cone's that continuous space round it. But you'd start to get an idea that within that cone are all of these lines. Like we can imagine these lines that we call line segments. Like those toothpicks all leaning against each other. So where we pick those two line segments, where they intersect, and in this case they intersect that the apex of the cone, we say that that's the vertex. So does a cone have a vertex? Yeah, it's got one and we can construct it in that way. Does that make sense? Can you imagine that?

That makes sense. Why did nobody do that with me at school?

I don't know. Maybe you were just told it was one and you just had to accept it. And then the definition come, it's hard to reach it because if we apply... I think that's one of the things that we need to be mindful of. Whenever you have definitions of things, it's like when I used to teach children when they first learning subtraction and they say, "Oh, what's two take away four?" Now the answer that sometimes gets bandied about is, "Oh, we can't do that." Well, we know that's not true. We know that's not true. So I think we just need to be very careful that when these situations come up, which are they stray from, if you like, a rule that exists within the context of that year group or within what you are doing that's often a little bit more difficult because you're starting to entertain the idea of something that perhaps is harder to understand, it doesn't follow the same rules as what you've experienced.

I think the answer should be, "Well, we've not learned that yet. We can do it, but we've not learned that yet. So you will learn that, but for today we are learning four take away two." Those sorts of things. So I think that's the thing, I think that when you've got these situations, if we don't know ourselves, a really clear definition, then it becomes very difficult to manage that in a class of 30. And sometimes the easy option... And I've done the same, I'm not going to pretend that I haven't is, "Oh, Emily there's one. All right? That pointy bit at the top, there you go, that's it. All right? It's a bit complex so just let's go with that and time to put your books away team, right? Stick them in your desk. Let's go, Emily. You work well today. Thank you very much." So yeah, I think that's probably sometimes what happens.

So my next question, Adam, are you feeling nervous?

Yeah, of course.

This one I've seen come through, again with customer services or from parents or teachers or whatever. This also came back for homework with my youngest son. And it was very simply, how many size these two dimensional shapes have? And we went through and there's the square and the rectangle and the triangle, and then came the circle. Help us here, Adam, how many sides does a circle have?

I could give the Christmas cracker answer. Right? I've had this before in a Christmas cracker where it says an inside and an outside. Right? So we could look at it that way. And I guess perhaps that's true, perhaps there's an aspect of it. So I think what we have to keep in mind before I start answering this is that maybe it depends a little bit on the perspective we take. When we look at the shape of a circle and thinking about we're going to be having to visualize something, that's almost impossible to visualize. When I give the answer to this, we're going to have to visualize something that is really difficult. So I'm going to try to make it as easy as possible. Okay?

So if I just take the Christmas cracker answer away and say the inside and the outside, I suppose the next question is, is what are sides in polygons, multiple sided shapes? Yeah? And often we'll think about those as having straight lines. So when we know, when we look at a square or a triangle, again, that definition that we're very used to is we are looking to identify something familiar. And in this case, it's straight lines. So when we look at a circle, to the naked eye, we can't see straight lines. So we then say, "Is it possible that straight lines can exist within a circle? Is it possible that a circle has multiple sides?" Now just saying that out loud, I don't know what image you have in your head, but I know that if it were me and I was told that in a classroom, "Oh, just imagine this." I wouldn't have a clue where to start.

So what we have to do is do a little bit of visualization again, and for this, you need to imagine a circle. Okay? So have you got that in your head?

Yes.

Yeah? Give your circle a nice bright color.

Yeah.

Yeah? Have you got that?

It's like a turquoise blue, Adam, just so that you know.

Awesome. Yeah. Perfect. All right. So what we're going to do is we're going to start to place some known shapes inside that circle. Can we do that? Yeah. So what's going to happen is this, the vertices of those shapes, so where those two straight lines meet are always going to touch the edge of the circle. So we're going to place them inside the circle, but they're going to touch the circumference of the circle. Are you okay with that?

I've got it.

So I want you to imagine a triangle and you place it there. And so the triangle inside the circle is in contact with the inside of the circle at three points. Can you see that?

I've got it.

Okay. Now what's happened to the amount of that gorgeous color that you described? What's happened to the amount that you can see, comparing it when the triangle, before it was there and now you put it on there, what do you notice? Has it decreased or increased the amount that you can see?

So now the triangle is on, it's decreased the amount that I can see.

Okay. Yeah. So far so good. Right? So that's cool. So when we put something like that on there, the color decreases, are we good with it?

Yeah.

All right. Now I want you to take the triangle out and I want you to imagine a square. So this is now meeting at four points. Can you imagine that? All right. So would you say that the amount of that gorgeous colored circle that you've got in your head, is the amount that you see of it increasing or decreasing?

I hope I'm right about this but it's decreasing in my mind.

Okay. Yeah. Now I want you to imagine a stop sign, okay? An octagon. Yeah? That shape, can you imagine that?

Yeah.

Now that's got eight vertices, so there's eight points of contact inside the circle. So I want you to put that octagon inside that circle.

Yeah.

Yeah? Can you imagine that? Is the amount that you can see of that gorgeous colored circle, is the amount that you can see of that, is it decreased or is it increased when you've put an octagon on there?

I think it's decreased again.

Maybe I should phrase the question differently. Does it cover more of the circle compared to a triangle?

Certainly does compare to the triangle and I believe it does to the square, but I might need a little convincing of that.

Yeah, maybe. Okay. Well let's up the ante a little bit now, what do you notice about the similarity between the shape of a circle compared to the shape of an octagon and compared to a shape of a triangle? Which one is closer to being circular?

I guess the octagon is going to be the closest to being circular.

Yeah. So it's sitting in there a little bit more nicely. It covers a wee bit more of that color that you've described. So let's up the ante a wee bit and let's imagine a shape that's got 50 equal sides. So there's now 50 points inside, 50 points touching that circle.

There's hardly any of my lovely turquoise blue left.

Hardly any, yeah. Very difficult to see it, right? So this is where we start to stray into this idea that can we imagine, we talked about line segments before with the cone, is that if each of those 50 point were connected by a single line segment to our eyes, would that almost look circular? And if we were to continue that on and say, what about a thousand point shape? What about a million point shape? What about, and we can keep going through that. And what we realize, we can make these two, I guess, conclusions that first of all, when we have a huge number of sides equally spaced points on the vertices touching that line that makes up the circle, we can imagine that the amount of circle we could see or to put it another way, that shape becomes more and more similar to a circle.

To the point where it will get to a number where, to our eyes, we may not even be able to tell that this straight sided shape is not a circle. Perhaps we would need to look at it through a magnifying glass or a microscope in order to see that it's not. So the answer that's often given is that a circle could have an infinite number of sides, because if we had a billion points all equally spaced apart and all at the same distance from the center, then what we are getting is a shape that's very circular in nature and very circular certainly to our naked eye. So the answer that's often given is infinite. There's an infinite number of sides because we could create a circle in that way. But I think we just have to be very clear as well.

We get used to how we use the word sides. We get used to how we see shapes where the boundary of those shapes are very visible, and we can describe them as curved or straight. So I think we just need to be really mindful that when we start to enter the realms of infinity, then we are getting into thinking that that's pretty tricky. So the mathematical answer we could say is, there's an infinite number of sides, and we can demonstrate how we can get closer to understanding that. But I think that we need to be very mindful if we ask those questions, particularly in primary school, whether or not we've got a valid definition for it.

And what do we expect from the children to say? If we say, how many straight sides can you see? The answer is none. So I just think that we need to be mindful and we need to know otherwise, perhaps don't ask the question. It doesn't mean it's a valid question, but I think that children need something that's understandable at their level to get a handle around those sorts of things.

Final question, because actually, I'll be honest with you. I mean, I was going to ask you a whole load of top questions and I've only done two and it's taken up most of the podcast. I'm going to allow you to have the last one for yourself is, any kind of top tips, because those were two really good tips based on questions, things that people get in a pickle over. Have you got anything else, any tips that you can offer? Any thoughts around, I don't know, ways to remember facts or to help remember facts when you're doing your maths.

I think in the first instance... So here's my golden rule, right? My golden rule, it's based on the work of Jerome Bruner, and Bruner talked about concrete pictorial abstract. We need to be really careful that when we talk in the abstract, which is like the symbolic writing of mathematics, the digit five, the addition sign, those things that on their own, they don't mean anything, just like the 26th letters of the alphabet. On their own they don't mean anything. We put them together and decipher what those symbols say, in the same way like Chinese writing or writing that's pictorial. But in essence, it doesn't tell us what it is. I think that the experiences, the concrete experiences we have at the beginning of anything are the ones that I think that's where the idea stems from, the exploration in those areas.

And I get as a parent, we are not expected to be teachers, but if we want to gauge whether we understand something, I always think, can we make it? Can we model it? Is there something that we can do to make or model something? Because really, the last part with that symbolic representation is all we are doing with the abstract is just recording what we've done. And we are writing the language of mathematics. Up to that point it's just a ice cream cone. That's all it is. Or it's a piece of paper in the shape of a circle. That's all those things are. It's not until we write it down in the language of mathematics that it becomes, if you like, maths, but those experience is the basis for it.

So yeah, that would be my go-to is that thinking about modeling stuff is a really nice way of ensuring that that is something that sticks in your head. So if anyone that's listening to this has got an idea of two rulers propped up against a cone, you are far less likely I think to forget that than if I had, I don't know, just given you some numbers or something and said, "There you go. That's what it is." And I think that that's probably the way forward. Yep.

Thanks ever so much, Adam. That was fantastic. And I really appreciate your time.

No worries.

See you. Bye.

See you.

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

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