Episode 231: Generalisation — key for fostering adaptability, independence, and problem-solving abilities

Ep.231.txt

Robin Potter

I can do this.

Robin Potter

Ready? Okay, so welcome back to the School of School podcast. I am here with the usual suspects. Here they are. Andy, Adam. It was, but let's not get into that at the moment. That's a whole other podcast and we're not doing that. Well, I don't know about you two, but anyway, how are you both?

Andy Psarianos

Isn't that a movie? Usual Suspects?

Okay.

Andy Psarianos

Weren't they all criminals?

Adam Gifford

Wonderful!

Andy Psarianos

I'm well. I'm well.

Robin Potter

Fantastic. So we have been just starting to talk about teaching to competencies. And we've already talked about visualizing and how important that is. But we are today going to talk about generalizing what it is, why it matters and how we use it in practice. So who would love to start off the conversation? Andy, I can tell. I can tell.

Andy Psarianos

come on. It's always me. Okay. just throw it over to me. You know, I'm just going to talk nonstop, right? So

Robin Potter

We'll cut you off. It's okay.

Andy Psarianos

All right. You guys cut me off. Okay. So what is generalizing? So again, let's just go back before we even jump into this. Cause you know, uh, we said this in the previous episode, but let's say it again, you know, why, what's, what are competencies? Why are we talking about? Like, you know, surely kids need to learn stuff like how to add and you know, their multiplication tables or what, you know, that's usually what people think of when they think about mass education. So what are we talking about? We're saying why generalization? Well, the important thing or competencies, the important thing about competency is to understand

is that competencies are really what you're teaching. So you're not teaching the content, you're or sorry, you're not teaching the competencies while you teach the content. You're teaching the content while teaching or through the competencies. And that's a really important thing. The competencies is what you need to focus on.

the content almost obviously it's important because they need to learn it but it doesn't matter as much as the competency. So what does that mean? Well that's what we're going to talk about and now we're talking about generalizing and what is that what is generalizing mean? Effectively generalizing people have all kinds of different ideas and a lot of people not really sure what it means but if you're really in the context of education we're talking about generalizing we're talking about the ability to recognize the underlining mathematical structure across varied examples.

and to be able to articulate those rules, those patterns or those principles. That's what we're talking about generalizing. Often people mistake generalizing with patterns. Patterns is a part of generalizing but it's not what generalizing is. Does that make sense?

Adam Gifford

So good, I think we're winning.

Andy Psarianos

Okay, so what's it all about? Why do we need to teach this? Well,

Robin Potter

Yeah.

Andy Psarianos

If a child can't generalize an idea, they don't know what rules to apply or what things to do in a particular situation. And that's really what this is all about, right? So when you talk about something, whatever it is, you know, it could be any mathematical context. It's not even only about mathematics. It's just, again, this is about logic. It's about thinking. It's about being able to connect things together. You know, there's a lot of generalizing that needs to happen in things like language as well. This is not just a mathematics thing, but it's really crystal

clear in mathematics and really crystal clear how important it is in mathematics right so the the idea is is like if you take a bunch of isolated facts right how can you connect those things together and then how do they connect together so that's really what generalizing generalizing is so you know what I mean that I can give lots of examples

Robin Potter

Could you give an example?

Andy Psarianos

So you know what's a so an example that's a really easy one for a lot of people to understand is if you think of what's a triangle right and and how how do how does a triangle so what is a triangle. You know.

When people think of a triangle, they usually think of like an isosceles triangle or they think of a equilateral triangle or possibly a right angle triangle. That's the kind of idea they have in their mind. But there's so many, there's like a limitless amount of triangles, right? They don't just all look like that. They can be all kinds of sizes. They can be represented in different ways, right? So how do you generalize the idea of a triangle?

Okay, so how do you connect all these various things together? you know, to give you a more, let's say, idea of a triangle. Okay, what about when people talk about, you know, locating something using somebody's mobile phone? How do they know where somebody's mobile phone is?

Well, they use triangulation. Okay. So you need three different ways, three different points where you can measure the distance. And then from there, you can figure out exactly where this thing is. Right. Well, that's a bunch of triangles. It's called triangulation. Okay. How do you tie that idea of a triangle with, you know, a very simplistic picture on a piece of paper? Well, the fact that you can make the reference and see, I see, I

visualize what that means I can generalize the idea of a triangle all these things are interlinked the things that are true in the triangle that I draw on a piece of paper let's say the calculations that I can use with that triangle I can also apply to this other thing which is I have three distance measures

Andy Psarianos

you know, location measures or whatever, three distance measures, from that I can figure out the location. That's using the principles of a triangle. It's triangulation and that link between those two things is a generalized idea. And that's what you need to develop in mathematics, right?

Robin Potter

So you're seeing what's the same in that situation. Yeah. Okay.

Andy Psarianos

visualizing through visualization you're generalizing an idea okay so another example would be what's the link between mathematics and addition i see that repeated addition right is the same as multiplication i see that

You know, if I, if I scale something in size, if I take something that's this big and I make it that big, that's multiplication, right? That's visualizing and generalizing too. They're both use multiplication. It extends your idea of what multiplication is. Okay. Through these different scenarios, right? Through these different kind of,

Well, let's just call them scenarios. We don't need to get into the mathematical jargon behind it, but it's like a different story, but they're both multiplication. So when we're talking about generalizing, that's what we're talking about being able to do, right?

Robin Potter

So from...

Adam Gifford

I think one of the things that Andy points out then is we also need to know how you can make sure that that doesn't develop. there's some really simple ways that when you say, what do you notice about this? What do you notice in that example Andy gave about scaling, for example?

and we talk about, I've noticed that the length of each side doubles. Okay, what can we use to find that out? We can use this. Okay, what do you notice about this shape? Keep going. What do you notice? If you don't give children those opportunities, why would they? Because you learn them in isolation and you're not being asked to notice those things or make those links that, you know, and like Andy said, right at the very beginning, these are competencies because we want the children to spot them before we have to point them out.

You know, we want to hear those things in the classroom and they say, hey, I've noticed this with this multiplication business, you know, like when you apply anything, it's about like equal groups. So is that true that if I've got like seven times 19, I could do seven times 20, but it'd be one group of seven less. that, can I apply this? I don't know, you work it out for yourself. What do you think? Is that how it works? You decide, it is how it works. And so when they come across these things in the future,

Because that's the things with maths, right? We build, we build, we build to be able to make these ideas or come up and secure these ideas when we don't know they're coming. And that's the idea with the competencies is so when we're put in these situations, we go, what's this similar to? know, like what have I done previously that I might be able to pull into this?

Andy Psarianos

Mm-hmm.

Adam Gifford

And I know I can test it and see whether that pattern, the underlying idea is maintained in the circumstance. And if it is, brilliant, because every other one that looks like this now, I can test whether or not it's the same idea. But if you, again, coming back to rote learning or learning in isolation without, say, if we don't know what the relationship is between other mathematical ideas or the same mathematical idea presented differently,

Andy Psarianos

Yeah, that's right.

Adam Gifford

If we don't see that, then how can we be in a position to prompt the children into what do you see, what do you notice? Dig a little bit further into that. Come up with that.

Andy Psarianos

Absolutely. And then, you've got to kind of wrap your head around with all this stuff is that, you know, as a teacher, you need to prompt and ask questions and build learning experiences that...

enable the generalizing to happen. So the question is how do you do that? What's the theoretical underpinning behind all this stuff? And that's where we go to Zoltan Deans, right? So Zoltan Deans is like his big thing is, you know, he had a lot of big ideas, but one of his most important contributions to education is this idea of variation, right? So you vary things and there's two types of variations. There's perceptual variation and there's mathematical variation.

to do both and you need to do the perceptual variation before you do the mathematical variation and in doing that what you do is you extend the idea of what this thing that we're talking about is so if it's triangles what is how do you vary things so that children have multiple experiences of triangles so let me give you a perfect example you know we and this is like you know the the kind of like trap that people like text

publishers and editors can easily fall into is or creators of digital tools or whatever it's like hey let's create an equilateral triangle

The base as the name implies is at the bottom and then the triangle goes this way, right? Yeah, but if you flip that around or put it on some weird rotation, it's still an equilateral triangle. But for a lot of children, they only experienced an equilateral triangle in one orientation. So they say that's an equilateral triangle. If you flip it upside down to them, that's not an equilateral triangle.

Andy Psarianos

Right? So you need to, that's like perceptual variance. Okay? That's perceptual variance. Mathematical variance means it's the same thing. It's varying the idea of a triangle, but varying it in different ways. So another triangle, it might just be like, Hey, you know, if you're looking at a triangle, there for sure, there's this idea of like triangles where all the angles are the same or all the sides are the same or only two of the angles are the same or whatever. But there's all kinds of crazy, you know, scaling and isosceles and there's, you know, there's like triangles.

with like really small angles, there's triangles with huge angles, you know, they're all triangles, right? They all look different, they can all have different orientations, all this kind of stuff. So that's the mathematical variation of the idea of a triangle and, you know, the perceptual, and you need to do both, right? You know, but be able to, again, generalize what's the rule. The rule is all the internal angles always add up to 180. There's three, there's three vertices, there's three sides.

Adam Gifford

And.

Andy Psarianos

That's a triangle. Right? That's what generalizing is all about.

points makes a triangle. Doesn't make anything else.