Episode 67: Looking at pre-number on children’s mathematical development

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. Are you a maths teacher looking for a primary school assessment tool that can give you a detailed look into learner or class achievement? With Insights, it's all in one place. Make sense of assessment data so you can strategically plan and teach lessons. Insights, it's assessment for advancement. Visit mathsnoproblem.com for more information. Welcome back everyone. We're here for another episode of the School of School podcast, and we've got Katy Reeve with us. Hi, Katy.

Katy Reeve

Hi, thank you very much for having me on.

Andy Psarianos

We're so excited to have you on, and we're here to talk about pre number on children's mathematical development. So what does pre number mean? Like is there maths before numbers?

Katy Reeve

Yes, there are... Well, it is part of number, but there's definitely maths before number, and it was something that, when I found out about it, really changed how I looked at mathematics and teaching mathematics.

Andy Psarianos

So you're going to have to give us a bit more than that. I mean, what is this? Like surely numbers... I mean, maths, as you say, or math as us north Americans say, starts with counting, doesn't it? Isn't that where it starts?

Katy Reeve

Well, traditionally, I think it starts with counting, but that just is a rote exercise where we are learning names, but they need to have some meaning. And in order for it to have some meaning, there's a bit of work that needs to be done first. There's part of the early years curriculum that we teach, and for example, sorting and classifying, that whole section, which I had taught for many, many years.

And then when I discovered what pre number is, so the learning that needs to happen before, I realised that I hadn't been teaching sorting and classifying, I'd been doing it, but I didn't know the intent behind it. I didn't know why children need to gather objects and sort them in different ways. So what's the same, what's different, what's the same, but different. And when you have that knowledge, when you come to look at numbers, you can look at what's the same about it, what's different, what's the same, but different. And having that language helps you to gain the understanding, especially when we start to look at things like what a 10 is.

Adam Gifford

The word language you've just used is so, so important, because I think that that's what separates out sort of something that's just task completion, can the child sort it in this way or match it in this way. But the language that's generated in order to describe what's going on, that goes on for keeps. That underpins so much of the mathematics, and will continue to underpin the mathematics that we do. I mean, patterns and mathematics, for example, I mean, mathematics is patterns. Simple.

Andy Psarianos

Yeah.

Adam Gifford

And these are the sorts of things that I think that when we're developing the language and the ideas, I think perhaps we underestimate just what a huge role these early ideas play, but they never stop playing that role. It's not like you kind of leave them behind. And I think that if you concentrate on the language aspect of, if you like, the tasks that you could categorise as pre number tasks, I think you'd be blown away. And then look at something further up the school, and look at that and some of the language that's used, and I think you'll find that it's still the same language, just in a slightly different context. And I guess that should point to just how crucial these exercises are and why they're being done.

Andy Psarianos

I think maybe a practical example might be useful for some of our listeners. If you imagine this concept of numbers. So for us, it's quite easy to understand because we already know it, but it's hard to empathise with someone who still struggling with some of these notions of what a number can represent, for example. Let's just stick with the topic of numbers. So you've got cardinal numbers, right? So numbers that you can... counting things. When you're thinking of cardinality, you're thinking of 1, 2, 3, 4, 5, 6, 7, right? Is one way of thinking of number. And then you can get into discussions like seven is greater than five.

But then you've got this other notion, which is, let's say more positional in nature. Right? So now you're talking about ordinal numbers, things that come in order. Well, if you came in seventh place, it ain't greater than coming in fifth place. Right? So now all of a sudden the context changes, the language changes, all kinds of things. And when you look at nominal numbers, I can tell you without a fact that my phone number is not greater than Adam's phone number, right? It doesn't even make sense. Bus number seven is not bigger than bus number five. So all the language starts falling apart when you start looking at these different contexts.

And you might say, "Okay, well, Andy, you're just going off on some intellectual rampage here", but it isn't. Because by the time you get to some of the topics like fractions, if you don't understand this concept, you're never going to understand fractions, right? Because fractions are made up of one nominal number and one cardinal number. Right? And it's the relationship between those two things that makes a fraction. So the top number, which we call the numerator, is the cardinal number. It's the number that we count. How many of these pieces, the nominal number, do we have?

Now when you start thinking of it that way, well, wow, that's kind of a crazy idea, right? And it's a pretty encoded way of writing an idea. We were talking about stories at some point in the past, maths stories, so now the stories are embedded in the way that we write a number down. So if you really, really can't get fractions, there's a good chance that you didn't do enough of that stuff, pre number stuff, and that's why you're struggling with fractions. Right?

And then when you get into things like what you guys were talking about earlier, like categorising things, sorting and classifying things, those early notions of equality are so critical to so many components in mathematics, and the requirement to explore what's the same and what's not the same. And that comes into all aspects of mathematics, whether it's patterns, generalisation, whether it's just cardinality, you need to know that so that you can count. It's as simple as that. Like when you're counting people, you don't count the chairs, right? Why? Because chairs are different than people. That all starts with that early concept of being able to classify things and say, all these things are three-sided shapes, and all these things are green. And some of the green things are three-sided shapes, and some of them are not. So they're similar in some ways, but they're different in other ways.

And all those ideas are really critical to maths, because when you get into these notions of things like equality, equality is, oh, wow, that's a whole mind... we should have a whole podcast just on equality. Sorry, I feel like I'm going on a rampage here. You guys really pushed a few buttons in me. When you look at geometry, which is really fresh in my mind right now for a variety of reasons. If you look at geometry in year five and year six, and this notion of what's equal, what's congruent, what's the difference between the two, that is so tricky and so difficult. And then when you look at this inclusion principle of like, well, okay, a square is actually also a rectangle, but it's also a rhombus and it's a polygon and it's also something else. What is it? It's a quadrilateral and it's also a parallelogram. Wow, that's kind of hard to understand. If you can't do that early classifying and sorting stuff, this idea that one thing might fall into many different categories is difficult for a lot of people to understand. Right?

Adam Gifford

But maybe it's because we don't feel we need to understand it. And what I mean by that is, is that of course we do, but what I find, and I've heard some pretty crass statements before from practitioners saying something like, "I don't need to understand fractions. I work in reception." "Oh, I don't need to understand matching shapes because I work in year six." So I'm not saying that that's indicative of all teachers by any stretch, of course it's not. But I think that if you were to sit down a group of teachers and say, "Tell me why it's important, those early ideas of writing a sentence, what are we doing that for in year six", for example, you could kind of go through the path a whole lot easier to say, "Well, initially we need to write a sentence down to get a single idea initially, and then it grows into this and we can kind of map it out."

But I'm not sure that there's that same, that just it's generally accepted that we need to know what we are doing in reception, for example, how that impacts on year six. And do we ask those questions as to, what am I doing today in reception and for year six, and would that be as easy for people to answer? Now, I'm not convinced that there are as many people who would be able to do that as they would explain why they might be, the progression in writing, for example, in English. And I think that that's a worry. I think that that's the part, that's why it's fantastic hearing Katy, and what you're talking about, Andy, about the importance and how it does link, because I don't know that those links are known by as many people.

Katy Reeve

Yeah. I would agree. I think we've had staff meetings in schools that I've been into, where people have wanted to discuss it and are very interested in it, but I don't think it's something that people feel as confident with as we need to be, definitely.

Adam Gifford

And I know I've done a huge amount of work around interventions and different diagnostic assessments and these sorts of things. And they often, it's when unfortunately the children are most obviously struggling, so years five and six. So for those people listening, that sort of nine to 11 years old, right? And they're really, really struggling with mathematics. I reckon at least 90% of the children that I've worked with, the areas which are the basis for their struggle were from reception and year one. And it's phenomenal.

But when you come back and explain that to parents and teachers, I think that there's general shock, because they see the issue is actually with the year six content. That's the problem. The problem is the year six content. No, it's not. The problem is they can't do that, but the actual where the problem stems from are these early ideas. And we should be asking those questions around early ideas, far, far earlier, before reaching a stage. But I think this all comes back to, if we understand that sort of path that mathematic takes, it's easier to assess at different stages along that path. If I'm only looking at a single year group within that path, then how can I assess these early ideas if I'm not looking for them?

Katy Reeve

I think there's also work to do within that year group as well, where, I put my hands up and said, "I've been teaching that for a number of years." And actually, when I found out more about it and developed my understanding and the penny dropped, that made such a massive difference to my teaching. So it's about how can we get that training into schools? How can we help the teachers to be having those discussions within early years as well, and within those formative years, so that you're really aware of what you're teaching. Your job actually becomes an awful lot easier, because you're questioning the resources you're putting out, how you're assessing it, but just becomes really natural because you're so aware of the end product that you want to get to. Rather than thinking about activities that children are going to do around something, it just becomes a lot sharper. So I think as the first point, that needs addressing as well.

Andy Psarianos

Do you think there's too much pressure on teachers to perform in a particular way, and that doesn't give them the opportunity to think about these higher level constructs in education? Like, for example, how critical the didactic path in mathematics is, from pre numeracy skills all the way through to let's say calculus, right? Because you can easily draw a line on all these different ideas that you learn in calculus, right down to something that you learned before you learned how to count or add. Teachers don't have enough time to ponder that, right?

Adam Gifford

I think that's true. And I think the other thing is, is that my performance as a year three teacher is based on the year three objectives. So I'm going to be looking solely at those, and how can I get the children to do the best there? And there might be some things that I can do that allow children to reach that particular objective in that year group, but that may not address the issues that are needing to be addressed. And I may not be thinking about my colleagues down the road. I might just say, "When you multiply any number just by 10, add a zero on it."

Well, that's not going to do too many favours when my mate down the road has 2.40, if that's how it's written, or if it's $1.50 and there's 10 of them, if you whack a zero on that, how's that going to help them? But it might help me initially, because I'm only dealing with whole numbers at that stage, or something like that. So this is what I'm talking about, is if we don't understand and we're not thinking about what happens further down the track, then we can really set our kids up, but also our colleagues up, for escape. And I think that accountability set against that year group, perhaps we're not quite as holistic in our thinking, because that's what determines whether or not I'm a good teacher in some people's eyes.

Katy Reeve

Absolutely. And it's that pressure to get to the end result can result in shortcuts, but those shortcuts have quite big consequences for the children. Especially you sometimes have to be really brave and say, "We're going to stand firm and do what we know is right." And you will see accelerated progress. You will get there. But I think it takes somebody with the confidence and some years' experience to have that, and a very supportive senior leadership team to be able to do that. So it's where assessment can really get in the way of doing what we need to do for that long term progress of the children.

Adam Gifford

But I think too, if I can jump in off the back of that, Katy, and just the last thing, but I think this is probably a subject in itself for another episode, is I can't imagine the allocation of money every year in the run up to sats, that have been put into resourcing in schools for these intervention groups to get the best sats marks possible. Right? Now I think we all understand that sometimes there's tricks and tips that are taught within those to get the right scores. But what's this doing when they go to high school? They're gone then, but we've got the marks in primary school, but are we working alongside our secondary colleagues and our high school colleagues to build on those things. And I think that this is a really tricky situation, because like Katy's saying, there's a lot of work that needs to go into it to create mathematicians and have children that are confident in mathematics and that enjoy the subject and feel like they can be part of it.

Andy Psarianos

So it doesn't stop there though even, Adam, it goes all the way up to university. I was having this conversation, literally having this conversation with my son, Sebastian, yesterday, where he's just started a new semester, and he's considering whether or not he should drop this course, linear algebra, because it's really hard. Right? He's like, "It's really hard, and it's going to bring down my GPA. It's going to bring down my average and close some doors for me." And so he's talking about transferring campus and all these things that's going on that makes this really complicated.

And in this whole discussion, what I was hoping he would say at some point, was he would talk about whether or not he's interested or he thinks learning the topic is valuable. And that never came up into conversation. It was all about positioning for the next step. The only reason for taking this course is because it'll get him to some other thing, other course, other level, whatever, the GPA that he's looking for. And he never talked about, "Yeah, I really like this topic. It's interesting. I want to do this later on in life", or "I'm not interested in it", or whatever. Right?

And all of education smells a little bit too much like that to me right now. Right? And we got to maybe sometimes go back and think about, why do we even go to school, right? Like what's the purpose of this? Is it really just to get good grades and be ranked highly? Or is there another reason, like maybe we should be learning stuff. And I think we forget about that all too much. Right? And I remember, I don't know who said it the first time, but Tony Gardner's the person who said it the first time to my ears, weighing the pig doesn't help the pig grow. Right? And I'm sure it's an old saying from somewhere, but it's very true. We spend a lot of time weighing the pig, but weighing the pig doesn't necessarily help the pig grow. Right?

All right. You all look stunned. I say we wrap it there. Yeah? Okay. Well, look, Katy, you got to promise us you're going to come back one day and do this all over again with us. Yeah?

Katy Reeve

I'd absolutely love to. Thank you. It's been my pleasure.

Andy Psarianos

Yeah. It's been a lot of fun.

Adam Gifford

It's been great. Thank you, Katy.

Katy Reeve

Thank you.

Adam Gifford

Appreciate it.

Andy Psarianos

And thank you everyone for joining us.

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