Episode 233: Understanding Number Sense

Ep.233.txt

Adam Gifford

Welcome back another episode of the School School podcast speaking very quickly. I feel like we're under some sort of time pressure. These are wee short sort of snippets. are are Speedy, Pacey. Anyway, along along the way, we've got the Speedy and Pacey, Robin and Andy. How are you both?

Robin Potter

You

Andy Psarianos

We are.

Robin Potter

Do we have time to say that we're good? Yeah, fantastic. Wonderful. Okay.

Andy Psarianos

Excited excited

Adam Gifford

No, but you have to answer really quickly. Yeah, that'll do, one word answer. Yeah, good, all right. So what we've been looking at is a series of competencies. We'll explain that in a little bit more detail. But we often talk about five competencies that we really want our learners to develop and for us to be able to support their development.

and one of those is Number Sense. So I'm gonna hand over to Andy and let's talk a little bit about Number Sense. How do we get it right? What do we do? you can talk slower than I am, by the way.

Robin Potter

It's BeadyAndy.

Andy Psarianos

boy. I, yeah. Speedy Andy. Okay. This Andy and fast forward mode. So number sense is, like one of the competencies, right? I'm not going to talk. I'm not going to jump in right away and talk about number sense, but what I'm going to do is I'm going to say that it, you know, I've said it several times now in various different podcasts, but I'll say it again. The whole idea is to teach to the competencies, right?

And as a result, you're going to use content because you have, need content to teach the competencies and that's how you teach. You don't teach the content and then as an afterthought say, yeah, what about the competencies? Right? So the competencies are what make you a mathematician.

The contents just a bunch of knowledge, right? Okay. So these are the, these are the qualities that someone needs to have to be a mathematician. And what does a mathematician mean to me? A mathematician to me is all about someone who can think logically. Okay. It's someone who is intellectually competent.

All right. Then know the difference between logic and made up stuff that has no sense. Okay. Made up stuff that has no sense is can be fun, but don't mistake it for logic. Right. It's not the same thing. Okay. So that's what I'm going to say. like, but before we jump into number sense, what, what do you guys think about this idea of teaching the competencies in general?

Adam Gifford

It's utterly crucial because that's what creates the learner, And the other thing, and it's one of the competencies we can start to generalize. It's not unique to mathematics, it's just life, right? So I always think, I don't know, there's so many, I don't want to branch off and take the time from number sense, but when you put the five competencies together and you start to say, if you're competent in each of them,

Robin Potter

obviously critical.

Andy Psarianos

Hmm

Robin Potter

Mm-hmm.

Adam Gifford

then you're going to deal with adversity, which is what life throws at you. You're going to deal with society and you're going to make valuable contributions. And I think if you neglect those, well, you're not going to be as effective at doing those things, but also it's just going to be simply harder. Like it's going to be more difficult for those children to go through as learners if they don't have those competencies sort of under the belt.

Andy Psarianos

Yeah. By the way, if you're really good at these five competencies, you might still make really stupid decisions in your life, but it won't, but it won't be out of ignorance, right?

Adam Gifford

That's humans.

Robin Potter

Right.

Andy Psarianos

important thing to understand. Okay so like number sense what are we talking about? So number sense is like that flexibility with numbers that idea that like numbers are just things that we can do stuff with and we can treat them differently in different situations right? So it's like understanding the understanding of number the understanding of the relationships between numbers things ideas like magnitude like is this bigger is that you know sometimes a number is great

greater. Sometimes it's more. Sometimes it's, you know, a greater number. Like is minus 7 million a greater number than one? You know, that's an interesting question. It's a pretty damn big number minus 7 million, right? Or it was definitely not more or is it, or could it be? Well, numbers are not more than other numbers, right? But six apples is more than three apples. All that stuff falls into the category of number sets, right? Okay. And you know what? The English language fails miserably at explaining all these things. But the idea is, that

despite the fact that you can't even often explain things in language, you understand you the sense you have the sense right of how to compose, decompose, what's equal, what's not equal, all those things. Those are all around number sense. Okay. And it's important. Yeah.

Adam Gifford

crucial.

Crucial. even that thing, like I was just thinking about that one. So if that negative 7 million was like my bank account, that would be a pretty big debt. that's a huge amount. But then if you start thinking about proximity to zero, right, and you start looking at it something different again, that different numbers are closer to others. And I think about those sorts of things. We say if you were adding 1,999,

Andy Psarianos

Yeah.

Adam Gifford

and 7 knowing that 1999 is pretty close to 2000 is probably quite useful knowing that 7 can be made up of 1 and 6 is probably quite useful these sorts of things where you get that sense and you see that instantaneously and think here's what I could do with that

Andy Psarianos

Yeah.

Andy Psarianos

Yeah.

Adam Gifford

Imagine being that child as opposed to I need a pencil and paper and I need to now write it down and it's going to be a column addition and I'm going to do loads and loads of calculation in order to reach the same point. You don't want that. It's not fair basically if you're that child that has to reach for the pencil and paper.

Andy Psarianos

Yeah. Yeah.

Andy Psarianos

Yeah, exactly. And also this idea of like the magnitude of a number, right? So if you say to someone, what's something that's a hundred miles away? Is that where your school is? Is that, you know, a faraway town? Is it the end of the continent? You know, that's number sense, right? How much is 20,000?

Describe something that's 20,000 right 20,000 Wow, where would where would you ever write? Was that how many people get on an airplane? Right is that a stadium is that how many words how many words you're in a hundred page document Okay, so if you write a hundred page document how many words is that that's about 20,000 words, right? how many characters is that?

Adam Gifford

Mm-hmm.

Andy Psarianos

I wonder how many characters, like actual characters that is, right? That's about a million characters. Okay, a million characters, a million. So somebody says, I'm gonna give you a million dollars. How much money is that? Well, think of it in the context of like, if you had to write a hundred page document, you would hit the keys a million times. That's how much a million is. Okay?

You know, it's like how much does a million ants weigh? Right? I think that's a question in one of our books somewhere. Yeah. A million ants weighs as much as what? Does it weigh as much as a car or does it weigh as much as a cat or does it weigh as much as a small bird? Right? Those, those are magnets. Those things are all number sense, but also number sense is things like, okay, well I've got the number 18. In one context, 18 might be 10 and eight.

Robin Potter

Yeah.

Andy Psarianos

because it's really like if I'm multiplying 10 and 8 makes a lot of sense right but sometimes 18 is better represented as 20 subtract 2 you know and in another context maybe 18 is 15 and 3 for a different reason maybe because you're dividing by 5 right so

Do understand? That's number sense, right? So not just getting into, oh 18, I need to divide that by 5. How am I going to do that? 15 divides nicely by 5. I'm left with 3. Okay, I can do this, right? And that's number sense. And it's critically important, it's critically important to be able to do mathematics at a high level.

Adam Gifford

Yeah.

Robin Potter

Yeah.

Robin Potter

And what you've done there is, you know, you've just given all kinds of different ways to problem solve and to, or just to really think about, because most likely the students haven't thought about it in so many different ways. And so for you to force them to do that in a, in a real critical thinking type of way, like, wow, how many, what?

What does, what do a million ants weigh? I mean, I never thought of that before. I mean, yes, Adam, can, I feel like you're going to announce the answer.

Adam Gifford

But...

No, I was just going to say, no, I'm not going to answer that one, but something happened when I was working with some teachers the other day, and this is again where it comes back to it. No, but I don't think I did, but there's another one, there's a lovely one about a division problem, and there's four different answers, and all of them can be true.

Andy Psarianos

you

Didn't you write that problem Adam?

Andy Psarianos

Mm.

Robin Potter

Okay, yeah, yeah.

Adam Gifford

You know, like, you know, I've got 27 children and in a car fits five children. So the answer could be six. How many cars do you need? You know, that sort of thing. But what I was going to say was, is that I was working with some teachers a few weeks ago and there was a question and I said to them, right, what do you think the answer is? And it was something like, I don't know, 4.3 or 2.6 or something like that. And I said, just think about the context of this question. OK, just think about that. And it was children at a sports event. And I said, you have two

2.3 children at a sports event, you've got the police coming, you've got a mess, you've got no more job, you know, all of those sorts of things. So it's also that part as well, either the reasonableness of the answer, but also in the context of what it's about. Do these numbers make sense in the context of this problem? It makes sense if I'm buying a bottle of pop or something like that, but it doesn't make sense if we're talking about numbers of children.

Andy Psarianos

Yeah, you got at least one dead child, right? Yeah.

Andy Psarianos

Yeah, absolutely.

Adam Gifford

So it's those sorts of things as well that I think we have to pick up on because those numbers aren't used. Although, you know, in some context, the average family has what is was always said 2.5 children, whatever it was. I'm sure those numbers have changed now. But it's that sort of thing that also feeds into it and why that contextual understanding of number is so crucial. And like Andy was saying, the different types of numbers, you know, where they exist and how they work. So, you know, there's a lot to it.

Andy Psarianos

Yeah.

Robin Potter

Mm-hmm.

Andy Psarianos

Yeah, that's right. And that context is so important. I'd love to elaborate on that idea of a context because division is a great example. I love the example you came up with. It's like, okay, if you got 10 children and you got three cars, you're not putting a third of a child in any of the cars, right? Like that's just not gonna happen, right?

You know, it's not, cause that means killing children and you know, so, so that's not an, that's not a census, not the right answer, right? But if you're, if you're dividing, 10 liters equally into three vessels, right? Then the answer, there's only one right way to write that. And that is three and one third.

Okay, that's the right answer. You can't even write it as a decimal. If you choose to write it as a decimal, you've done something wrong because one third doesn't break down into decimals. Okay, so that's a lack of understanding. It's only can answer that question with a fraction, but sometimes you can only answer it with a decimal. Sometimes you can only have a remainder, right? It's the only way to correctly answer the question. That's number sense, right?