The First 5 Hours of Fractions in 20 Minutes

Hi, my name is Clare Seeley, and I'm the head teacher of St. Matthias School, which is in Tower Hamlets. And two things I'm particularly interested in are maths education and cognitive science. So today, in this webinar, we're going to be learning about the first five hours of teaching fractions and bringing cognitive science and maths education together. Okay, so to start with teaching fractions, as with teaching any maths, the really important thing is to resist the temptation to rush through things. You'll see on the slide, I've got the heading the hair and the tortoise. We need to be like the tortoise, to go slow and steady. Most things that go wrong in maths go wrong because we've rushed through content and we think the children have got it, and. And most of them haven't, or certainly some of them haven't because we've gone too fast when we learn. Cognitive science tells us that when we learn something, we build on what we already know. Now, usually that's great, but if we're trying to teach them something new that's a little bit different, or even quite a lot different to what they already know, then the old learning, what we already know, can get in the way of the new learning. And instead of our brains thinking, oh, this is new, this is different, and accommodate that, instead, they try and make the information we're getting fit with what they already know. And this leads to misconceptions. And the particular kind of misconception it leads to is really resistant to instruction. So we need to struggle really hard to overcome that. We should expect children will find it really hard because we're teaching them something that clashes with what they already know. The other thing that we need to take into consideration is cognitive load. Now, that's a term again from cognitive science, to do with working memory. And our working memory is the place where we think, and it's actually quite a small space. It can only really fit about four things at a time. So if we keep on adding another thing and then another thing and then another thing, it's like juggling too many balls in the air and we'll drop one of the balls or two of the balls. We need to keep things simple and in small steps, otherwise the working memory will be overloaded. That's what cognitive load is about. Not overloading the working memory, keeping things in small steps. So when we do that, when we keep things in small steps, that's what mastery is about. When everybody in the class really gets it. Mastery, sometimes people think it's about some super ninja level of really able children who really understand stuff, going off into some brave new world of mega math. So it's really not about that. It's about everybody in the whole class really getting it because we've built slowly from firm foundations. So that's what I'm going to be talking to you about today. Right, Fractions are hard. They're hard for adults. On the slide, you'll see here this comes from a training session that was run for adults for maths coordinators about fractions. And they were asked obvious question. If you have two pizzas and three adults, how much does each person get? And it actually got into a really heated argument because some people said, well, obviously you're dividing by three, so obviously everyone gets a third. And then other people saying, no, no, no, of course everyone gets 2/3. So which is right, well, people are getting a third of the pizza, but 2/3 of a pizza. Because in fractions, what the whole is, is really important. It's complicated. Adults find that complicated, let alone children. And quite often staff don't realize that quite how complicated they are. So why are fractions so complicated? Why are they hard? That's because fractions are not natural numbers. Now, just to remind you, natural numbers are the counting numbers. 1, 2, 3, 4 whole numbers, the numbers that we first start teaching children. So that's their prior knowledge, that's what they'll have in their brain. But fractions aren't natural numbers. They're rational numbers. Now, the word rational, as you can see, it comes from the word ratio. And rational numbers do not work like natural numbers. But when we learn, because we build on what we know, children are obviously going to apply what they already know about how natural numbers work to these rational numbers. And it won't work. It will lead to misconceptions. And these misconceptions might be quite resistant to being challenged because that's how our brains work. So we'll have to work hard to do that. So this whole unit is planned with that in mind. Okay, so what is the difference between natural numbers and rational numbers? Well, there's lots of differences. I'm going to go over some of them today. Probably the most important is to do with value. Here we have on a natural number, the value of a number is linked to one number. It's a quite straightforward relationship. Two is equal to two, three is equal to three. But with a rational number, it's much more complicated. You've got the relation between two numbers in the fraction and between the Whole that gives the value. That's quite complicated to say. Let no even understand what is more, it's not always obvious what the hole is. And the whole isn't always one. Although when we first start teaching children, usually we say the whole is one, but then when we stop using the whole is one, we don't point out to them that we've changed that. And also usually, going back to our adults in the pizza problem, when we give an answer in fractions, it's usually expressed with relationship to one, even if the whole isn't one. This is very complicated. We're not doing this in the first five lessons. But just to point out, the concept of value with rational numbers is quite different from natural numbers, other things order. With natural numbers, orders are simply linked by magnitude. One more, they have two and then you have three, and then you have four and each time it's one more. As we know, that doesn't work with fractions. It doesn't go up in order. So if you have two fifths, that's not necessarily. That's not smaller than four. This is smaller than three quarters. What am I talking about? It is smaller than three quarters, even though the five is bigger than the four and three quarters. So we have two fifths and three quarters. And children will assume, oh, well, the two fifths is bigger because the five is bigger. But that's because order doesn't work like that. Then the number names, we have half, third quarter, they're not obviously linked to 2, 3 and 4. The names don't help us. In English, then we have equivalents. So with a natural number, with ordinary counting numbers, the label, the number label indicates a fixed quantity. So 2 means 2, 3 means 3. It's always the same. But with rational numbers, that's not the same either, because if I have half of £50 and you have half of a thousand pounds, we've both got a half, but you've got a lot more money than I have. And conversely, with natural numbers, if the number label is different, the quantity will be different. So if I have. If I have two and you have three, we're going to have different amounts of stuff. But with rational numbers, you can have a different number label. Like you might have 2/3, I might have 4, 6 or something, but actually we have the same amount. All of these things confuse children because they conflict with their understanding of natural numbers. So in these first five hours, we're just going to think about value. Not equivalence, not order, just value. Now remember that value for fractions is complicated it's dependent on the two numbers in the fraction plus the whole. And it's the relationship between those three numbers that gives the value three different things. The denominator, the numerator and the whole. But in this session, we're just going to really think about the denominator. We'll move on to the numerator a little bit towards the end. But the whole intention of this block is to make sure that children really understand what the denominator is, what it actually means, because that's the thing that's very different. We're always. Our whole is always going to be one, and it's going to be one tub of play dough. Now, you could do this with making your own play dough, but the good thing about a play dough pot is it's very obvious what the one thing is. If you're making your own play dough, I'd suggest getting one of those paper cake things, putting a dough in that so it's bounded so you can see what one means. And we're going to be dividing this into equal parts. And we're going to really stress the equal because that's something that children find quite difficult. And we're going to link how we divide this into equal parts into how we write fractions. And we're going to be doing this. We're going to be dividing our dough up using a special knife. And we're going to tell the children that this knife is called a vinculum. Just in case you don't know. The vinculum is when you write a fraction, the line that goes across. The official name for that is vinculum. So we're going to tell them that our special knife is called a vinculum. And this knife means divide into equal parts. And why am I going on about equal parts? Well, I'm sure we're all used to children dividing shapes up like this and thinking that that's fine, that they're thirds, and thinking that they've divided them equally. They find it. Some children find that quite hard. So we need to really get them to understand what an equal part means. Okay, so what we're going to do, got our ball and we're going to divide it into two equal parts using our vinculum knife. Here's my vinculum knife. Here's that. I'm going to do it by eye again. Some children will find that tricky. I'm dividing it up. Just check. Have a quick look. Do I think it's fine? Make it into a ball. Then I'm Going to get my special scales. This bit isn't strictly. You don't have to do this but bit. But if they've used this before and they know that if it's like that, then it's balanced, that it's the same, then that's good. So I'm going to put them in here and hopefully I've done it enough that. Oh, no, that one's too big, isn't it? I'm going to have to change. Put more on there. Yeah, that's balanced. So they're equal. I know, but they're the same. So here we are. When we divide something into two equal parts, each part is called a half. So that's what we're going to start off with. But we're not going to talk about the numerator. We're not going to say, that's one half, that's two halves. We're not going to go there yet. We're just going to say, this is a half and this is a half. And then if it's not equal, it's not a half. And then we're going to say to the children, right, you've made some halves, now make some not halves, because again, getting that feel for equal, not equal. So then we want them to make some things that are obviously not halves. So these are not halves, because this part is not equal to this part. They're the same. They're not the same amount, so they're not halves. Right, then we're going to roll this back again, back to our hole. Going to get our vinculum knife. Again we're going to divide it into two equal parts. But this time we're going to make them into sausages. Apart from. I've dropped that on the floor, I'll open a new one. Okay. If you're doing this with a class, get the dough out earlier. It takes ages to get this stuff out. Okay, here we are. That will do. Right, we're now doing it with white dough. Okay. We're going to make it two into parts. Make it into sausages. Okay, you get the idea. So the same thing. Check it with the balance if you want to. Mine don't look very similar. Here. This is a half and this is a half. We divided our dough into two equal parts. Each part is called a half. It's important they see it with different shapes. Throw it back together again. Now we're going to do the same thing, vinculum, halving it, but this time. And some children will find this bit tricky. This one's going to be a ball and this one's a sausage. Are they still halves? Because they don't look the same because this one's a different shape to that and that will be a bit of a sticking point for some children. And then you might need to go back to the, the balance check. They really are the same and just because they're different shapes, that doesn't matter. They've both got the same amount. We share the dough equally into two parts and each part is called a half. And then the last thing we're going to do in this lesson is you can use the lid for this. But obviously if you're, if you're doing it. In fact, I'm going to have to take some of that out there because too much. You can use a cutter if you're not using play Doh. So I've got a circle here. I'm going to divide that into two equal parts and again the same thing. And then you can go on and do. Well, let's see, if I do a half, that's a circle with a half, that's a sausage, a half, that's a sausage with a half, that's a ball and so on and all those combinations and just carry that on until we're really sure they're secure with that. And that's basically the first lesson. Second lesson, we start thinking about writing denominators and linking it to our dough. It's really important that we write fractions in a way that underpins what they mean. And that's what we're going to be doing. So we're going to be drawing the vinculum, the denominator. We're not going to be doing the numerator, not yet, not this lesson, not for a few lessons. And this way of writing fractions should really become like in your maths policy, a way school wide that we do it and you always do it. You do the vinculum first, then the denominator or the denominator. And then finally when you get to numerators, they are going to go on top. First show the knife. And we say vinculum means divide things into equal parts. That equal parts is really important. Then we draw, draw the vinculum. And we have to explain we're not actually drawing a knife. We're going to draw a line to represent a knife. And then we say now we need to show how many equal parts we've divided our dough into. So we're going to have that there and our two balls there and then this gets written, you can see, like that. It's a written abstract representation of the. Of the concrete way of doing it. So I've divided into two equal parts. So I write the two under the vinculum. Each part is a half, this is a half, this is. And this is a half. There we are, quite straightforwardly like that. Hmm. Now the moment of truth. Have they got it? So you're going to show them that and see if they can show you with do what that means. Just a word of warning here. Like, you and I, we go, oh, I'm going to make that into four now. And we just halve that. They probably don't know that you can do that. They'll probably end up with four things that aren't all equal and need to jig that around a bit. Okay. So again, you're going to reinforce that this means I've divided the dough into four equal parts. So all the time, vinculum and whatever's underneath it is telling me how many parts, how many equal parts, parts I've divided something into. And then we need to teach them the word quarter. When I divide something into four equal parts, each part is called a quarter. Next lesson, dominators or denominators. So going to go back, obviously, go over what we've already taught them and reinforce it's about equal parts. When you divide into two equal parts, they're called halves. When you divide into four equal parts, they're called quarters. And then we're going to say, hey, let's do thirds, fifth, sixth and eighths. Obviously not all at once. You'll do the thirds and you'll do the fifths. And so practice making them and practice writing them. Obviously not doing the numerator, but we're going to keep an example of each fraction, maybe just one for each fraction. I'll show you why. Because what they'll discover is as the number of parts we divide it into gets bigger, each little ball gets smaller. And that, as we all know, is something children find really tricky, that as the denominator gets bigger, the actual part gets smaller. So we're going to keep a copy of this, put it up in our classroom, and we're probably going to put it in other classrooms too, because that's such a fundamental thing that kids find tricky. Then we'll do a bit of matching to have some pictures of different amounts and with their written form and they can match, check they've got that. And then we're going to tell them that the number underneath the vinculum is called the denominator and the denominator tells us that how many equal parts we've divided something into and do lots and lots of practice of the written form and the concrete form and vice versa. And then maybe in a bit we might actually do make one part very, very unequal. Oh, no, that can't be a fraction because the parts aren't equal. So that's not right. Just to reinforce about equality the whole time. Lesson four is something that's probably much more familiar to you because we're going to go away from the dough for a bit and just go on to paper folding. What we're going to use is squares, circles and rectangles. But long strips. We're not going to use triangles because we're not going to use shapes that it's easy to fold in, fold into two and not make a half with just things. I think, you know, if you can fold a circle in half into two and get them lined up, it will be halved. So we don't want to add to that misconception. So we'll go over the same language again about dividing things into two equal parts, dividing things into four equal parts, and then move to the writing. Do lots of matching and labeling activities with shapes and the written form. As I said, that bit's probably quite familiar to you. Last lesson. Finally we get to do numerators. Assuming that everything else has gone well. Right, we're going to return to the dough. And we're going to. Actually, now we're doing it balls first, aren't we? Make it into our balls. I'm just going to do it roughly and we're going to count the quarters. We're going to say, oh, look, I've got quarters. 1/4, 2/4, 3/4, 4/4. Then we're going to make with a lid, make a circle. Finally, we get to talk about pizzas. When you talk about fractions, you always have to talk about pizzas at some point. So here's our pizza, okay? And we're going to share it between four people. So we're going to share it equally. Obviously, that wouldn't be fair. We're going to share it into four equal parts. So they're going to be into quarters. I'm going to share it. Here we are. There we are. Lovely pizza. Okay, so now we're going to say, how many quarters of pizzas can we see? Hopefully they'll tell you four. One quarter, two quarters, three quarters, four quarters of pizza. The first time we've really counted each part so how many quarters can we see? Hopefully they'll say, we can see one quarter, we can see two quarters, so on, so on. Then mix it up a bit. How many quarters? How many quarters? Of course, the thing about a pizza being in a circle is you can see what the hole is even without the hole being there, because you can see it's part of a circle. Now, at this point, some children in your class might go, that's a half, that's a half, which obviously it is. And then you say, yes, two quarters is the same as a half, and then just leave it there. Those that get it, get it. Those that won't, that's fine. That's for another lesson anyway. How many quarters? Four quarters. Again, some might say, that's the same as a whole pizza. Yes, it is. Well done. How many quarters? Then we're going to ponder. I'm going to ask the children. I wonder how we could write it so that people would know how many quarters we had. If only there was a way we could write it down so that they knew. Because remember, at the moment all they know is that you write the vinculum and the denominator. Hmm. I wonder how we could do it. Now, hopefully somebody's going to come up with this by themselves, but if not, anyway, we're going to model it. So we're going to model how to write the numerator. So, but remember, starting with a vinculum. So I'm going to divide something into equal parts, then the denominator. So here's my four parts. I want to make four equal parts to make quarters. Okay, so they know this bit already, but I want two of my quarters. So I want this quarter and I want this quarter. And then I've drawn a line around it here. Here I've just quartered. I could draw an eyeliner like that because I want two of the quarters. And then how I'm going to write that to let people know is I'm going to write a 2 above the vinculum. So the number above the vinculum tells me how many pieces I've chosen, and the number below the vinculum tells me how many pieces how many equal parts I've divided it into. Whatever you do, don't do this because that looks like six quarters. And that brings it back to thinking it's a natural number and it isn't a natural number. They don't work like that. It's all about the relationship between the numbers. It's two of the four. There are four equal Parts, we are choosing two of them. We draw around it to show they're the parts we've chosen, then obviously repeat it for 1, 3 and 4. But remember, you always do vinculum, denominator, numerator, and always narrating the whole time what you're actually doing. Then we can move on to shapes, but this time we haven't drawn round them, we've coloured in the bits we've chosen. So a quarter, two quarters, three quarters, four quarters. Again, we're going to say two quarters, not a half. If they point out it's a half, that's fine, but that's not our teaching point. So we'll say, yes, well done, well spotted, but not really dwell on that. This is an important step. We need to check for understanding. So we're going to give them some examples to spot. We're going to say some of these are not written properly, where the numerator, numerator and denominator are flipped, where it's incremental, incorrect mix from correct and incorrect examples. And hopefully they'll be able to tell us why that's wrong, what's wrong and explain why. And that's really key. But they can do that. Now, some children will find this whole lesson, they'll be fine up until so far, but the whole adding enumerator thing will just be a step too far. So they'll need lots and lots of practice and will need to be prepared to go go slow. Others will be fine, will be fine to go straight to shapes, but we need to keep on doing that until we really think everyone's secure and then but resist the temptation to jump ahead to halves and thirds. Leave that until lesson six or lesson seven, but wait until everyone can write 1/4, 2/4, 3/4, 4/4 with the dough, whether it's in balls or in pizzas or in sausage shapes. So they can do all of that easily, sport when they're written wrongly and tell you why now. So that's the end of the five lessons. You may think that's so slow. Why is it so painstaking? Well, remember going back to the beginning, it's because rational numbers are not like natural numbers, but that prior knowledge of natural numbers will compete and try and take over and not really listen to the rational numbers, because that's how our brain works works and we'll have to work hard for our schema to change their schema. We should expect that to be tricky and to have to work for that. We need to reduce the cognitive load and take small steps just to go back to this just to end up with. So that's what we would do. And I'm thinking that lesson will probably be for year one, but there might be children even in year six who need to go back to basics because the underpinnings are insecure. But hopefully they'll extend this throughout the school and have this in every year group. Not do this whole lesson in every year group but have the understanding of what a denominator is using our denominators and have this picture up to remind them that as the more parts we cut into, the smaller each part because that's a fundamental misunderstanding. And also it shows them what, why you can't, you know, when they're in the juniors and they start in, you know, year five or something and they start adding denominators of different values, if you use this model, that shows why you can't do them. Anyway, I hope that's been clear and I hope you've understood how it works and have fun with it.