 and so we're recording as of now. Thank you. Wonderful. Thanks for the reminder. Well, let's start then. This is rather weird. Much of this session, originally as Mark said, originally planned as a face-to-face session, is very much built around the assumption and the expectation that by immersing ourselves in some mathematical activity, talking with other people about it, we can catch ourselves in the moment of being mathematical and try to look at it and try to get some fresh insight into what working mathematically means. Particularly for this session, I would have been very keen to stress if I'd have been with you at the Institute and I'd have looked at you all the whites of your eyes. I would have said what is really important is you attending to what you are doing while you're working, while you're engaging in the task I offer you. So I'm going to start with this. I don't know whether you've seen this before. One of my great heroes is John Mason and he uses this image to signal the two parts of us, though the part of us that's doing something and the part of us that is watching ourselves doing something and thinking about what it means, i.e. our cognition and our metacognition. And he has a nice little quote that he's quoted in his book there. One of us is going to eat the fruit. One part of us is going to eat the fruit. I hope I'm going to offer you some recently sweet fruit to eat. And the other part of us is going to look on. That's why I've suggested that it's going to be really helpful to have a notebook with you. And what I would suggest is that you try and engineer it so you've got two bits of the notebook which are acting as those two different things. So you'll be doing some maths and so you'll be just writing the maths then on the left hand side. But I will also invite you regularly to think what's going on there. What are you noticing inside of yourself when you do that? What are you thinking is significant about what you've done? So I'm going to signal the little notebook thing every now and again as we do stuff. Okay. But in a way, the first activity that I plan to offer you as face to face is probably the most ideal for this situation. Because I don't want you to talk to anybody else. I want you to. In fact, if I'd have been standing in the same room with you, I'd have asked you to close your eyes and you might want to do that. So I want you to imagine a rectangle and I want you to imagine a triangle drawn inside that rectangle. I want you to get a sense of how the triangle can change. So shrink the triangle down, scale it up again. It's got to remain inside the rectangle. But get a sense of how the triangle can grow and shrink while still saying inside the rectangle. Move the triangle around inside the rectangle. Get a sense of the freedom and the limits that you've got as to where you can put this triangle. Always staying inside the rectangle. Now become aware of the vertices. If you will use geogibre, I guess, you can get a sense of maybe you're going to pick up a vertex and you're going to move it around. So get a sense of how each of the vertices can move individually. Again, get a sense of the freedom and the constraints that you've got. Now manipulate the triangle until you've got the largest possible triangle that you can fit inside your rectangle. When you feel you can't make the triangle any bigger than it is, start to move the vertices. What freedom do you have when you move the vertices of the triangle but it remains the largest triangle you can fit in the rectangle. I'm wanting almost to invite you to think of a little film in your head of how the triangle can change where the vertices can move to and yet still be the largest possible triangle inside your rectangle. Pause, fix the image in your mind. Maybe now draw some things on your notebook. Maybe some little stills in your film, some little snapshots of the different positions, the different forms of the triangle that can fit inside your rectangle. So draw some things and then ask yourself what proportion of the rectangle is filled by the triangle. Okay, so imagine the rectangle. Did you all manage to find a number of different triangles that would fit in that? This is how my imagining has went. I certainly wanted to get one of the sides up against one of the sides of the rectangle. Did you? I think I found out quite early that I wanted one of the sides of the triangle to be the side of the one of the sides of the rectangle. How soon was it when you really discovered you knew intuitively that the other vertex had to be on the opposite side? I wonder whether you've got a sense of this. Was this part of your film? When I was doing this in my head, I was quite, I thought this here, this, was quite an important moment for me. I knew that was half. Did I know that was half? I've sort of got a kind of rotation thing here that the vertex can move and now this vertex can move and then it's going to snap. That's rather nice. I never thought of that before. I could have said to myself, I know something about triangles and how they fit inside the rectangle because I know all about that. I know what the area of a triangle is, but there were some things when I engaged in this activity that were new to me. What did you do in your mind? What did you say to yourself about what you thought about the triangle? I wanted to draw this line. I wanted to say to myself that that is the same as that, and that is the same as that. That's how I know that the area of the triangle is half the area of the rectangle. I don't know whether you know this publication. Well worth looking up. It's a fantastic article. I went for it before you read it, but there's one lovely little bit in that article where he says about this sense of why the area of a triangle is half the area of a rectangle. The relationship between the triangle and the rectangle was a mystery. Then that one little line made it obvious. One couldn't see and then all of a sudden I could. Somehow I was able to create a profound, simple beauty out of nothing and change myself in the process. He then says, which I really like, isn't that what art is all about? And there is something creative and kind of like bit spinting really about, blow me. There is no way that I can draw a triangle inside a rectangle. That is more than half the area of that rectangle. What a fantastic thing. And then it goes on to say, and yet this rich and fascinating adventure of the imagination has been reduced to a sterile set of facts to remember I think procedures to be followed in place of a simple and natural question about shapes and a creative and rewarding process of invention discovery students are treated to this. I don't know what your memory of this or whether you can even contact your memory of when you first encountered this, but I remember wondering why you had to have half the base. I couldn't work out why you would want to half the base of the triangle. What I wasn't aware of is it's not half the base. It's half the base time time. It's half the rectangle and nobody told me that and nobody put me in a position where I could tell myself that. Isn't that what art is all about? I quite like it. So this is where I'm going with this that there are very useful summaries of what we know mathematically. That can be written down as symbols. But if you start with that, then you've completely got rid of the journey that ends with the realization that that's what it is. So should we engage kids in the journey a bit more rather than starting with the symbols? And I quite like this thing, the symbols and no more that world than musical notations and music. So let's play our students the music and start from that position. That's kind of what I want to. I was going to say if I'm looking at you face to face, I was going to say that's the sort of conversation I want to have with you tonight, whether we have a conversation over this platform, I don't know, but we'll try. And the other thing I just want to push is this one of my favorite quotes from Gatenjo. The only thing that we can value more than knowledge is experience. Let's have an experience. He has this lovely notion that knowledge is the remnant of awareness experiencing our minds. The experience comes first. We realize something and that results in us knowing things rather than knowledge being something that you've got to be told in the form of a formula to remember or a definition to parrot back or a or a times table facts a child. It's more than that experience. So I hope I'm going to offer you some experiences. So if we separate out then the symbol and the symbolized, what is the thing? I'm going to call it the essence of the maths. The essence of the area of a triangle is half base times height is for me. Oh, blimey. Look at that. You can't make it more than half the size of a rectangle. That's the essence of the mathematics. And it's symbolized with those symbols quite neatly symbolized by those symbols. I'm interested in the second question. We probably, it's important to symbolize things with mathematics. Do we sometimes lose it? Lose something by symbolizing? What do we gain by symbolizing? Let's have a think about that as we do this stuff. And then I'm also interested in what are the sorts of situations that you can put students in where they've got some control over symbolizing the thing that they have noticed. And when that happens possibly, particularly when you get into the realm of algebra, that might help significantly kids get a hand. It seems less abstract to them because they have some agency in symbolizing something that they've noticed in their own way. So that's where I'm going with the themes of this session. So here's some things to do. And so I want you to have your notebook with you and to think about the two sides of yourself, the doer and the watching yourself doing. And I want you to do these calculations. I want you to write down the calculation and I want you to write down the answer. I hope you've written down four answers. Pause. Think about what you notice. Maybe write some things down on the right hand side of your notebook. What is happening and why might it be happening. And I'm going to, let's just try this now because I'm fed up with my own voice. Would somebody like to open a microphone and tell me what the answer to the first one would be? You brave? 60. 60. How did you do that? Was that Sue? Olivia. Olivia. How did you do that, Olivia? So I looked at the brackets and I saw that the first one is seven times six and the second one is six times three. So I noticed that I could make it easier for myself by doing rather than seven lots of six and three lots of six, just doing 10 lots of six. My conjecture is that you didn't all do that initially. Thank you for the show. I actually wrote this answer straight away in a different way. Yes, please. I straight away got six times seven plus six times three and then got 60, then 90, 80, 30. So they're all the same principle. They're all the same principle. But again, I would conjecture that we didn't all do that to start with. It might have been not until we got to the third or the fourth or the fifth or maybe not until we've actually had listened to a few people talking about what they did now that they started to do that. I did it from the first one. And I suppose part of my thing here is that is going to happen in a class that some students will notice things before other students. It doesn't matter because if activity is something that we do in a classroom in order that we can have a discussion about it afterwards. It doesn't matter when we've realized this. Competitions are a very tricky thing to engineer in classrooms and it can sometimes put kids off, can't it? If some kids are very quick. But it doesn't matter if you have a discussion about it afterwards. It doesn't matter if the purpose of the activity was to say, what did you notice? What are you noticing now? Let's describe it to you. I don't know. Maybe you had to think a bit harder about that one. Oh, well, that's eight, eight and two, eight. At what stage through these calculations did you realize that it's seven and three of them in the first one? Six and four of them. Blimey, they add up to 10. It's important to dwell in these moments because that's when you're really sensing some mathematical activity here. Oh, it's three threes and seven threes. That's why it's 10 threes. Now, it seems to me that when we write seven n and three n is 10 n, the thing that we are symbolizing is that phenomenon that we have just thought about. It's curious to me. I suppose it's kind of obvious that it's very easy to look at this as a teacher and say, this is easy to teach students this. What's difficult about that? Seven n's and three n's are 10 n's. What's difficult about that? And to start with the symbols, that hiding behind that symbolization is a pretty neat idea about calculating. If I've got some of a number and something else of the same number, I don't need to calculate the two things separately and add them up. I can add up the multipliers and multiply it by the common thing that they're being multiplied by. That's the distributive law, but it's really important that kids experience that. It's summarized very neatly by seven n plus three n equals 10 n, but if we do that too soon, I think we'd get rid of the, oh, isn't that nice feeling? Okay. Thank you for jumping in. Now that a few of you have jumped in, that's easier to do. So, now we can do what you would normally do in a situation if we're in Cambridge, and you just stop me in mid-sentence and say something because something's occurring to you. So, if you're driven to do that, that would be great. Eight. Please. I'm driven to say seven n plus three n masks the seven plus three bracket's n-ness of it. Yeah, that's great, isn't it? So, now what we want to have a common, now we want to symbolize something else now, seven plus three n brackets, seven plus three n brackets multiplied by n is seven n plus three n is 10 n. That's really nice. Yeah, yeah. Okay, another thing to do and to think about. So, I think of a number, write it down, double it and add one, and then write down the result of that. Go back to the original number, but this time add one and double it. Do it with another number, do it with another number, do it with another number. Think about the numbers that you're doing it with. Ask yourself, what is happening? Is it always going to happen? Will it always happen irrespective of what numbers I choose? I'm wanting you to get a sense of pushing this to destruction, pushing this to its limits. So, think about the numbers you're choosing and do it with quite a few. Write on your left-hand side of your notebook what you're doing, write on the right-hand side of your notebook what you're noticing. Somebody talked to me about the different kinds of numbers that you chose, where you started, where you started to choose different kinds of numbers, what numbers you were trying towards the end of the process when you read it. Shall I say? Yeah, please. Well, I want easy numbers, but I don't want ones that are going to be too confused with the numbers that are involved in the question. So, I definitely don't want two or one. So, I'm going for ten, and if I'm looking for a relationship, I might try a hundred next. Why might you buy a hundred? Well, because it's still going to follow with the same pattern, because obviously you've told us there's a pattern, because I've got a notice things. So, I want a much bigger number, and I might go for zero, or zero point one, and the negative number perhaps says I can check that it works in lots of different situations. Nice. John Mason talks about particular peculiar general, the peculiar on the way towards the general. Your first peculiar, sorry I didn't know who it was, it just spoke to me then. Oh, it was me, so. And often this is students' first peculiar, it's a large number. Possibly a hundred to start with, but sometimes you get things like, well, 279, that's peculiar. I wonder if it worked for that. But then there's some interesting things like, you're talking about zero, zero point one, it's a different kind of, negative, oh yeah, it'll work for all the positive numbers, but surely it won't work for negative. That's a very strong perception in students, I think, that generalizations only work, they don't work completely generally, you know. But this idea that, wow, this is going to work for any number is a big thing. So putting students in situations where they have to think about what are those boundary numbers, what are those numbers where I'm moving from the particular to the peculiar, because they're really important in helping students to think about what the general might be. Anybody else choose any other sorts of numbers that they felt were significant to test it? I used n. Yeah, yeah. Yeah, go on, yeah, please. I assumed you were trying to trick us at the start and was going to be one of those ones that goes back to the same number, whichever one you start with, prevent and so on. Yeah, yeah. But part of this, yeah, part of this particular peculiar general, isn't it? I think we know that students can handle particulars as if they are generals, as if they are general numbers, and they need to do that enough of the time in order to make that step, which we can make very readily now, and we can say, well, we'll just operate on an m, but we need to get students to the position through an experience that we're giving them, where they can say, well, that could be anything really. Now you've got something to say. I'm often struck by the notion of our colleagues as English teachers, or if you're a primary teacher, you are an English teacher as well. But when we get children to write something down, we want to get them into a position where they've got something to say. When you write things down in mathematics, particularly when you're using algebraic symbols, what is it that you want to say? You want to say, it doesn't matter what the numbers are, this always happens. That's what they want to say. You have to get students to the position where that's what they want to say. If you give them some freedom to do that, and maybe even if you offer them some symbols that they can use to say that, they'll find that so much easier, rather than starting from, these are symbols, and this is kind of what we need. So what did I do? I started with three, and I doubled it and added one, and then I added one and doubled it. Yes, I wasn't quite as ambitious as the 100 or 273, but 27 is my favorite number, so I then chose 27. But there's something about when they get a little bit bigger, when maybe you start to attempt to structure a little bit more. Why am I getting one more? Why am I getting one more? 27, I'm doubling the 27 and I'm adding one, but then I'm adding one on to the 27 and then I'm doubling. I'm getting a smell of it, but I need some more things. Yeah, so I went to fractions, that was my first kind of thing. Okay, so if I double it, and I add one, but if I add one first, and then I double it. I definitely want, like you, I definitely wanted to try that. And this is kind of putting a bump in the road, isn't it? You've got to slow down, you've got to think about this, because I've got to add one on in a minute. I might know that double that is negative 12, but I've got to stop and think what happens when I add one on to negative 12. I don't get negative 13, I just got to think a bit carefully about that. And that's helpful, slowing down with peculiar numbers and thinking about them. And then I might think, oh no, that's, that's different, that's one less. No, no, it's not. It's still one more, just having that little extra twisted thing. What I've found quite useful is that often, oh and then no, yeah, zero, that's really important. That's, that feels like a mathematician's choice to me, that's a clever choice. Just like drawing the dotted line in the triangle. We want our students to make these choices. If we always draw the dotted line for them, if we always say to them, you know what you ought to choose now? You ought to choose zero. If we always do that for them, then maybe we're robbing them of something a bit. It's tricky because students need a nudge, but we want to give them that, those moments where they say, oh look, I double it and add one, I get one. If I add one and then double it, I'm going to get one more. Has anybody got an articulation of why it's one more? Or maybe a better question, this is the cause of the knowledgeable, maybe a better question, another question is, what would we want our students to say when they, they realise that it would, why, what would we want them to realise? Please, sorry. I have been thinking in terms of odd and even numbers. I have as well. So on those lines. You want to just amplify on that a bit? So when I put two n plus one or two bracket n plus one, it kind of looked like to me one of those algebra questions where you have to prove, like the proof question. And I kind of, yeah. So I was then playing with odd and even numbers. Does it matter if the numbers are odd or even? No, the pattern is still the same. So why when you double first before you add one, do you get less or always one less than if you add one first and then double? Why? And what would you want your students to say? Adrianne? Because the one is also being doubled? I think that's what I want them to be considering. It's the one. It's that one that you've added on first before you've done the doubling. That's going to get doubled as opposed to it's added on afterwards and it doesn't get doubled. That's a big thing. That for me is the crux of the mathematics behind something that's, I don't know what, I mean, I don't know how many of you primary or secondary, but I used to be a secondary teacher. And I've done in my time many years ago and I've seen lots of secondary teachers do, well, it's easy. You just do eyebrows, you know, it's, you just multiply out the brackets. Well, what's, what's difficult about that? It's lost. It's lost all of the sense. Try to think about a bar being a number and think about it being able to be any size you want it to be so it can get bigger and get it smaller. So then you're adding one and then you're adding the bar or you're adding one to it and then you're doubling it. Lovely. Do you just want to do that on your notebook now? Draw what we're describing? Yeah, I described it badly actually. Well, that's, that's okay. That means we've got to work hard to, to, to work out the bars. That's good. So you've got a bar that can be any length you want it to be and then you've got another one of them and then you're adding one or you've got a bar that can be any length you want it to be and you're adding one and then you're drawing all of that again because you're doubling it. Perfect. Yeah, that's quite nice. Yeah. This is a picture of an odd number and an even number. Say that again Ruth. Which gives you a picture of an odd number in the first one and an even one. And an even number in the second picture. So you also didn't, you would notice that if you use integers to start off with that you always get an odd and an even but if you, you lose that picture when you put fractions and other numbers in but for integers you always get an odd and an even. And that's lovely Ruth. And a really critical thing. I mean I used to get this one, I used to teach A level. A really important thing for sixth form is to know is that 2n is a general even number. 2n plus one is a general odd number and they don't always get that. Yeah. Yeah. I did ask yesterday with someone else so it's high in my mind at the moment. Right. And I've talked with this before, I don't know what you think about this, is that often, often with a with a pound sign actually, 1m is a million. m is a million. I'm thinking of a million. Well there's a peculiar for you. I wonder what happens when you double a million and you add one. Oh I can write that down. What happens when you add one first onto the million before you double it? And you might get two versions of that from students, might you? You might get them saying well I've got two lots of a million and one. Or you might say well I've got, that's why I've got two million and two. So I don't know. I've toyed with that. It's been useful sometimes with students because they kind of used to 1m pounds in front, million pounds and that sometimes gets, but Ruth's bar, nice thing to do for that. Pete something that I found interesting thinking about this was at least initially when I put the negative number in, I wasn't really sure completely what was going to happen. I was pretty sure there was going to be a difference of one. But which way? Yes, yes, yes. One of the things is a very seminal article, an ATM article written by a guy called Alan Wheatley. If you remember of ATM you'll find it is called the path smoothing model. I think something like that. And he talks about this desire we have as teachers to make things easy for kids, to smooth the bumps. Careful don't make that mistake. We do that a lot in the way we, in inverted commas, try to help students understand things. And you've got to go through that, oh no, wait a minute, I am learning from the stage, you know all that. You've got to keep the bumps in the road. And one bump in the road is, oh I'm not quite sure what's going to happen with that negative number. Is it going to go one way or another? That's an experience which we must have students to have. Okay, oh wonderful. This is turning into something that's bordering on the interactive. Thank you. Okay, oh I'll just show you this. Really is in many ways a bit about a homage to Don Stewart who sadly is no longer with us. He very recently died. He's got a lovely way of just finding very succinct ways of kind of recording things like that. So you could imagine that you could do this with some students and they record it this way. You stick a number in that circle and you double it and you add one and you put them in there. But then you go the other way. And you add one and then you double it and then you put them in there. And you notice the two circles together. You do this number of times. And so I thought that's kind of a quite nice way of doing it. And of course what that allows you to do is either prompt students to do it or to get them to a situation where they might prompt themselves to do it is what if it wasn't double it and add one? What if you had treble it and add one? Add one and treble it. What would happen there? What if it was treble it and add two? Add two and treble it. What would happen there? And so he's got these little grids where you could just do a few of those and see what happens. And you could almost imagine, I couldn't find these on Don's website, but you could almost have some of these things where the multiply by two and the add three could be an empty box. And the students could be writing their own versions of what they're going to multiply by and what they're going to add in those boxes and kind of thing. So here, oh, it's the three that's being doubled rather than not the three being. So I thought that was nice, nice kind of little just a way of containing the recording of it for students so they could get immersed in the activity and notice some things. So I'm wanting to offer you this idea now then that what we're saying is that any symbolism, any algebraic symbolism is symbolizing something. So it's worth certainly it's worth us as teachers. Whenever we open a scheme of work or a textbook or a worksheet or whatever, and there are expressions like this, we ask ourselves what are they expressing? And what are the ways in which we might put students in situations where they notice this thing that it's expressing? And then they know what they want to say and then we give them the symbols to say it. Or maybe even we give them some choice over what symbols they might use to say it. And then maybe have a discussion about convention and let's use symbols in the same kind of way otherwise we're not going to be able to communicate well with each other. Let's have a look at that diagram. What is that? Do you think you could symbolize that? What is that a picture of? Any authors? 12 plus 3n, 17 plus 2n. What about the 12 plus 3n and the 17 plus 2n? Equally correct. So we could write an equation on my next clip. I'll probably have written it down the way. I didn't see it like that. Oh lovely, go on. Yep. I saw it as 17 minus 12 equals a question mark. Yeah, lovely. I'm also reading. I think that's what you just said. Yeah. Yeah, so again what you're saying, you're saying, what are you ignoring then when you're saying, what are you saying? 17 equals 12 plus n, is that what you're saying? Sorry, 17 minus 12 equals... Because if i minus 12, ah, right. How are you manipulating that diagram in your mind to be able to do that? I'm kind of lining up the lines and after the first question mark and I'm just going with where I've got most information. And you're sort of ignoring, are you? You're ignoring the four question marks on the right hand side, are you? I am because I need to find one of them. Yeah. I don't know. I would have preferred if the bars were actually touching each other. I think that white point, to be quite honest, I didn't quite notice they were the same size in my drawing. The book were different sizes. Ah, so they would have been better if they were touching. Yeah, and then after I realised the quantities are obviously the same, I would have got rid of the ones that were equal. So the four on the two other... Yes, yes. You'd have done that. I would have taken the exactly, yeah. Yeah, also I can't trust that those question marks all mean the same question mark, even though they're all blue. Right. I'm not sure. I don't know what the system is, so I'm thinking I'll minimise the damage by just sticking with one. Yeah. This is interesting, isn't it, that what am I doing when I... Look at the symbols. What am I doing when I ignore the... What am I doing when I'm taking, moving those four question marks away and just concentrating on the 17 and the 12 and the question mark? You get there quicker. Yeah. Yeah. Well, what am I doing with the symbols down below? Am I doing this? Yes. So I'm going to offer you here the notion. I don't know quite how well this goes, but I'm going to offer you the notion that we often talk about manipulating symbols. But I think we have an example here of something where we've been able to symbolise a manipulation. The manipulation is remove the same thing from both sides. Now, we might learn to do that as an exercise in juggling algebraic symbols. And I found this with any students, and I think you were doing it as well. If you look at the picture, that's what you intuitively do. You just say, I don't need to know about these because they're the same. So I'll just look at the other thing. So that's an interesting thing for me, where the thing that it's symbolising and the symbols, if they're alongside, then something that's intuitive with the picture, it can then be reflected in the symbols. And so it isn't just, I've got to learn these rules. You've got to do the same thing to both sides. Well, why have you got to do the same thing to both sides? It's an intuitive thing if you stick with something that's a little bit pre-symbolic. So sometimes you can symbolise the manipulation, I think. In a way, we kind of got into that a little bit, with the double it and add one, add one and double it, that there's a sense behind multiplying our brackets. Multiplying our brackets is just this manipulation. Why does it work? There's something behind that. I need to get kids out. Sorry, I was going to say, would you say that the picture way of showing it is a more useful way of kind of seeing the process that's going on there? Yes, I think that's what I would say, yes. But my feeling is, if we can introduce the symbols alongside, well, in a way, the bars are the symbols as well. You start to think, well, what do we mean by symbols? The bars are the symbols as well. But if we have different kinds of symbols, picture symbols and algebraic symbols side by side, then what is intuitive in the pictures can then be reflected in the symbols. And so then the symbols can become intuitive as well. Because that's where we want to get to. We want to get to the place where the symbols speak to kids in the same way as pictures of bars speak to kids. So having them alongside each other for a longer period of time is important. What do you think if the bars and the question marks would be jumbled up? Yes. So to get students to line them up and then realize that I'm just thinking on top of my head, because I always kind of use this diagram, but I'm thinking now maybe perhaps sometimes I will put question mark, orange bar, sorry, two of question marks, orange bar, question mark and then all a bit jumbled up. I wonder whether what difference would that make to the students? I think that would be a lovely thing to try, wouldn't it, is to offer these sort of things that they could move around a little bit and have in different orders and then get them to realize that they can order them in that way. I think they would be more conscious of this diagram if they actually started to tie them up themselves. I'm just thinking right now because I've never done it. I actually displayed exactly what you did. I never thought about, yeah, I just was always thought it was a given that that is the right way to do it. Oh, that's good. Thank you. I wanted to offer you this. I'm grateful for Dave Hewitt to draw my attention to these kind of ideas a number of years ago. We often say kids find symbols really hard. Well, there's some evidence that suggests that if they already know the thing that they want to say something about, that reduces the difficulty. If in addition they have some degree of agency and choice as to what symbols they're going to use to say the thing that they already know, then that makes it even easier. It's hard when that's not the case. Ask yourself how often are we offering kids symbols and they haven't had the opportunity to experience the thing that it's a symbol of? And they certainly haven't been offered the opportunity to symbolise that thing themselves. And I was reminded by a primary teachers among you will kind of know this. Algebra is first mentioned in the National Curriculum documents year six. The idea that children aren't doing any algebra before year six seems bizarre to me, but there you go. They might not be using symbols, but they're certainly generalising. But this I thought was interesting. It should be introduced to the use of symbols and letters to represent variables in mathematical situations that they already understand. I think that's quite a good touchstone for us all whatever age we teach. Let's get them in situations that they understand. Very young children will say it doesn't matter which way I add up two numbers. Two and three is three and two. Seven and six is six and seven. And we think of a peculiar example for that. Yeah, 27 and 42 is the same as 42 and 27. This is something that you can get students to already know. Now they're in the position where they can write something down that says that because they've got something general to say that they've noticed. Right, here is the time where I, if I was with you in Cambridge, I would drop a little envelope on your table and I'd say tip out the envelope. So that's what I want you to do, although you probably haven't got yours in an envelope, but get them all on the table. I'm going to put you into or mark is anyway, going to put you into groups in a minute. But before we do, I'm just moving these things out the way the screen. Just before you do, here are some things I want you to think about and I want you to think about them on your own for a bit. So I'm just going to ask you to just think about this for five minutes on your own. We've got a timer on the phone and I'm going to give you five minutes quiet time. Think about how many whole circles you can make inevitably you will have to break up some circles to remake more circles when you're thinking how many whole circles can I make. So it will be quite important if the question is how many have you got, it'll be quite important to write them down. So you'll need to think about ways of writing them down. How are you going to write down that's a whole I've got, that's another whole I've got, that's another whole I've got. So I want you to think about that. While you're making different kinds of holes, you might just notice other things that aren't necessarily what makes a hole, but you'll say well this is the same as this or I can do this with things or there might be other things that you can write down. So anything else that you know to write down. As you're playing with that on your own for five minutes in a minute, there might be some questions that you want to pose yourself so so pose them to yourself and maybe start to answer them. So I'm going to give you five minutes now and then we're going to put you into into groups and you're going to talk with a couple of three other people about what you're doing. So your five minutes starts now. Welcome back everybody. I hope that worked all right and you were able to have a discussion amongst you. Let's try this. Find the chat. Find the chat. So you have to click on the three three dots over more. I think it says find the chat. Put something in that um um well something significant about the conversation. Melanie enjoyed thinking with her hands rather than her head. I like that. Hard to keep track of combinations. John that's interesting. Do you want to just open your mind and say a bit more about what you mean by that? Well I was kind of drawing them as well as manipulating them and going through starting with the halves and then noticing that some of the halves could be replaced with quarters but then you know missing some they're missing I think that two thirds could be sorry one third could be two six. The difficulty in some ways was well there was difficulty in accounting which is often quite surprising. Difficult in accounting how? Just because well to me I start to think about these um sort of like combination combinatoric problems from probability and I just know that that's not a particular strong point and so I'm trying as I'm doing it to think sort of systematically but um something about this made it a bit more difficult than just listing because there was multiple combinations for each different bit. Yes and Vincent's saying different combinations of fractions were discussed. Do you want to just say a bit about what different combinations you were discussing? Vincent? Okay not if you don't want to. I'm wondering whether sometimes okay so we've got a little bit about symbolism here. Were you writing additions? Were you writing something equaled one? Were you writing things like I mean John's just mentioned oh if I take a third away then I can put two six in its place. I can see how that fits because I put six six together to make a hole and I put three thirds together to make a hole. I'm not just when I remove the third and put two six in I'm not I'm not just oh it looks as if it fits I'm doing more than that I'm I'm reasoning why it has to fit and am I writing down there a third plus a third plus a six plus a six. When I've done this with students which I've done it from I think year three up to year 10 sometimes they'll have some funny ways of writing it down so they'll write a third comma a third comma a third comma something you know they'll well what do you mean by comma well they all together they make one okay I'll show you how we write all together they make one we write a third plus a third plus a third and there are three thirds how are we going to write three thirds and so in the kind of conversation about the way that the students are recording things you can come to some conventions about how you write things down. We probably could spend much more time on that but I'm offering you that as something where it's a this thing about if you can generate situations where oh my goodness I've lost count of how many holes I've got I've got I've got to record this in some way if I don't record this in some way I've lost count there's a need to record and then you get some discussion about how are you recording these things is that the right way to record these things you're writing it down that way but you're writing it down in a different way shall we agree how we write these things down and you come to some agreement about symbolising. Okay I'm very conscious of time and I've promised to Mark and Lynn that I'll finish your quarter past so I'm going to move on. I don't know whether you've got your triangles your four congregate triangles but I'm going to ask you to do this if you have so you've got your four triangles in front of you and unless it's an isosceles right angle triangle one of the sides is slightly smaller than the other side and I want you to try to fit these inside a square where the length of the square is so you could actually put two of the triangles together along a side because if you get a long side and a short side together they'll fit in the square so can you fit them inside the square just do that and get a sense of how you're doing that are you doing this quite nice isn't it what's that it looks like a square is it a square what would you need to do to convince yourself beyond perception it's not it looks like it what would you need to do to convince yourself that that really is a square that each angle is 90 degrees what would you need to do to convince yourself that each angle was 90 degrees subtract to be a two acute angles in the right angle triangle which make 90 so how do you know that those two I mean you know I'm playing devil's advocate with but these are the sort of conversations you want in a class and how do you know that those two angles that aren't 90 degrees in the right angle triangle how do you know that they add up to 90 degrees because angles in triangles are up to 180 so there's a lovely bit of you know reasoning here um is that all we need to know to say it's a square sides need to be the same length as well yeah and they are because it's the same triangle that's quite nice isn't it you know put your nose like that they fit inside a square and there's a square inside it how lovely is that it just occurred to me this afternoon actually when I was thinking about this now I don't know how to do this um I think what I'm going to do is I'm going to come out the I'm going to come out of the slideshow and I'm going to do oh no I don't want to do that ah when you come out the slideshow it stops sharing oh are you seeing that okay everybody yeah okay so I'm out of the presentation mode and I'm going to I'm going to copy it and paste it there it is it's another copy of that and I'm going to rotate it how far have I rotated it so that it'll fit in here how far if I copy and paste that and I rotate how far have I got to rotate it before it'll fit in the other corner 90 degrees I've got to do it 90 degrees haven't I because if I if I concentrate on the the two shorter sides that's easy to see I've got to rotate the 90 degrees but actually the hypotenuse is going to it's those two sides of the triangle are going to rotate 90 degrees then all the sides of the triangle are going to rotate now so blow me that hypotenuse is going to rotate 90 degrees so call me stupid but I've never really realized that before I've done that little bit of reasoning about oh that angle plus that angle plus something is 180 but you just rotate it I've got almost this sense that you have your triangle as a template and you and you and you push it in sand or you push it in plasticine and you make a mark and then you move it and you rotate it and you and you make another mark and then you move it and you rotate it that's really nothing and you make a square in the middle there is an additional thing because I think students uh rotation the understanding it is a rotation and a translation isn't it you're rotating and then moving it away and moving away yeah I can see how some students might be spatially a bit confused with that rotation the the one bit I see it I mean that I mean a bit like we've mentioned before it's it's where I think anyway it's well worth you know getting pupils to feel the movement of this triangle and to see what what do I have to do with the triangle to make it fit in that corner of the if you put two of them together you get a rectangle and then you can rotate the other one on the corner uh so two of them together and make a rectangle in the corner and now rotate one of those rotate the other one and it will go in the right place yeah so if you rotate the bottom one round it will go on the bottom left hand corner of the triangle so you can see that you've rotated it that's nice that's nice yeah that's so there's an interesting kind of I don't know logistical slash pedagogical question about how might you engineer a situation for students where they might find this other way of putting these triangles in the square because what we've got now is we've got two squares the gaps so we have we have here what you know there's one square and then we have here there's two squares that's rather amazing really and it doesn't matter what it is it's only the right angle triangle of it that does it it doesn't matter what size it is and that's that's quite nice isn't it I think I might start before you put the rotate things into that big square I might just rotate the triangle by 90 you end up with a windmill you can put your four triangles like rotate them each time and you look it looks like an old-fashioned windmill and that would just be a simple way to say look just to make sure they believe you you know you do you know end up back where you started and and you can see it or you don't know when you rotate it I like what I like what you're saying there about it is important that they need to believe you and I'm wondering whether and and the technology has possibly drawn us to do this more than maybe we would have done in the past there is a difference between doing this sort of thing in front of students and then watching us do it yeah and then and then doing it so we you know we want we want to be careful about how much of this is demonstration and how much of this is them being engaged in this oh look at that I can move that and it fits in that corner blow me I've got a square hole in the middle you know it's like drawing the dotted line in the triangle they need to have some access to that experience it seems to me rather than just watch us do it and so I mean yeah so so that's interesting it same triangle but you and you've got squares and those two squares have got to be the two red squares have got to be the same as the blue square because it's the same big square original so what an amazing what amazing fact that is now you know okay a squared plus b squared equals c squared but that's a bit like you know area of a triangle is half base times height it's a pale shadow of the richness and the surprise and the elegance of all of that stuff that you get when you're trying to fit some right angle triangles inside a square in different ways the thing that Pythagoras symbolizes is much richer than the symbols suggest I would suggest I'm just going to do some kind of final little things I suppose one of the things that I think we're talking about here is is intuition what can students intuitively do that we can ride on the back of and offer some mathematical symbolism to help them express but they already know it remember the year six algebra statement use symbols and whatever to represent something that they already know I think if you offer this to kids they already know that the price of a tea is the difference between those two sums of money they know that and isn't that at the heart of something that we often teach kids as something that you do which is when you're solving simultaneous equations you subtract the left hand sides and you subtract the right hand side why do you do that one coffee plus one tea is 145 one coffee plus two teas is two pound ten they already know that a tea is the difference between two pound ten and 145 if we can if we can think what does these symbolisms and what do these manipulation of symbols what do they be where do they come from can I put students in a situation where they contact that directly without the symbols and then we say are we going to write down what you already know that that seems to me quite powerful thing to do kids can solve these they know what why is or don't call them why call them something else but they know what that is because it's intuitive if you get this kind of thing where you're offering the symbols in in many ways that's what I did with the fraction circles with you is you didn't have them on yours but if you'd add my laminated coloured fraction circles that I'd have brought with me if I'd have seen you and the symbols would have been on the circle so there's a little nudge to helping students to use the symbols in the way you want them to use them so if you can do that a little bit it kind of eases them into world of symbols but at least there's something they already know that they can recruit just finish yeah I'll just finish with this and it'll miss something that I was going to do at the end but I'm just running out of time so watch this I want you to think of this as a square in its geometrical sense of course but I also want you to think of it as a square in its numerical sense it's a square number I'm going to click in a minute and I think you know what's going to happen do it in your mind before I do it what's going to happen now will it fit will it get your notebook out maybe draw one does it work why does it work you might even want to think I'm thinking of numbers as well here so I might want to write something down about numbers about a square number and some calculation about a square number what would that look like convince yourself geometrically convince yourself numerically yeah my usual trouble with all of these is I um I rush at the end so I don't know whether that's a bit too quick but you're removing one from a square and if you subtract one from five squared blow me you end up with six times four you end up with one more than five multiply by one less than five does that always work if you start with a 10 by 10 square and you remove one do you end up with nine elevens blow me yes you do will that always work happens if you didn't take one one square what happens if you took two squared away would it fit would that row of two squares along the bottom move up to the side yeah it would fit wouldn't it what would it do to the rectangle it would make it too longer and too shorter this makes sense geometrically and it also makes sense numerically I don't know how we're splitting here yeah I'm just going to say I don't know about the split um you know between primary and secondary but you know secondary focus you will know the difference of two squares that if you if you subtract one square from another so what's this is this 10 squared if you subtract two squared from 10 square you get a rectangle that's two more than 10 by two less than 10 now again when I was doing my a level I learned about the difference of two squares nobody ever told me that it was the difference of two square no one ever told me that what I learned was a bunch of symbols which was a squared minus b squared it was a plus b not by a minus b and that was the reason because it is that there was no sense that that symbolize that symbol those symbols symbolized anything not only did I not know what it symbolized I didn't know that it needed to symbolize anything I just thought it's symbols so you know difference of two squares is the difference of two squares and it doesn't matter what square it is that's always going to fit that will be three more and three less and and now I have something to say but I have something general to say so I can now offer students some general symbols how are you going to use those to say what you want to say and then maybe we'll get to then noticing the a squared minus b squared equals a plus b times a minus b and realizing the three sonners of creativity about that rather than oh yeah that's just another fact we're going to learn how are we doing I will do this very quickly then and then I will finish um you I guess you've seen this kind of notion before this is the first one this is the second one this is the third one what's the fourth one what's the fifth one make a table spot a pattern I like the notion of don't do that draw a peculiar draw the 71 do you want to just do this very quickly draw the 17th one of these attend to how you're drawing it and have a think while you're drawing it can I figure out a quick way a clever way of working out how many matchsticks there are in the 17th one without counting them one by one and this is where you really lose something in this virtual environment we get people up to the flip chart if we're all together in a room and we'd say how did you draw yours because you bet your bottom dollar that not everybody would draw it the same way so I'm I apologize I'm going to suggest that these are these might be some of the different ways in which people might draw them the 17th because it's peculiar and it's an honorary n it's an honorary variable oh this is day pure it how nice is that there's a difference between counting and watching yourself counting and something that's been very powerful for me over the years the algebra is not the statement the algebra is the work you have to do in order to get yourself in a position where you can make that statement which I find really powerful so what some people do is they start with one one and they add some threes and they ask themselves how many threes am I going to add if I'm if I'm drawing the 17th one well I'm going to do 17 threes so we might write underneath that one plus three 17th but if it was the 25th one it would be one plus three 25 and if it was the millionth one it would be one plus three million so that's what that is it's not three n plus one it's one because I drew the one plus three n but not everybody drew it that way I bet maybe you did that way well that's three 17s plus one oh no I haven't got 17 on you they didn't fit so that's three n plus one you might have started with a four and then added some threes how many threes am I going to add I've got the 17th one oh I'm going to do 16th I'm going to be one less than 17 threes I'm going to have four and one less than n you might say I'll do the top lot I'll do the bottom lot and I'll do the vertical lot but how many vertical lot I'm going to have 17 on the top 17 on the bottom but how many I'm going one more than 17 there this seems a nice way of introducing algebra which is expressing what you have done and it's different but it's the same it's got to be the same how are these things the same you can have a conversation about why they're all three n plus one the algebra isn't the statement three n plus one is not the algebra Dave's saying three n plus one out the algebra is the work you have to do to get yourself into a position where you can say three n plus one and then I was doing this the year ago with some teachers and one of them did this no I'm just going to do lots of fours I'm going to do 17 fours and then I'm going to take some away how many am I going to take away I hope some of that has been interesting so to summarize let's not confuse the symbols with the symbolise I think the mathematics is the thing that's being symbolised let's give kids contact with that through an experience which as Gatenio says knowledge is the remnant of that in in our minds and in our awareness so let's give them the experience get to the essence of what is the essence of this and what situations might allow pupils to have this kind of control over the symbolism and and and and get them to a position where they say I've got something to say I want to say something about what happens and then then experience that's difficult in lockdown isn't it difficult in in in um you know online lessons you know education is having an experience we look forward to getting that back for our kids I'll leave you with this um I don't know whether some of you remember a very seminal TV program by Bronofsky the Ascent of Man and he said this in the moment of appreciation we live again the moment when the creator saw we reenact the creative act and we ourselves make the discovery again the great poem and the deep theorem are new to every reader and yet are his own experiences because he himself recreates them I don't know whether your heart missed a beat in some of those things that you contacted but that's what we want to get our kids to do wow look at that yeah so thank you I hope some of that's been useful thank you thank you thank you paint Ray interesting thank you very much you're welcome thank you well Pete um can I say a formal thank you for um for this evening um and some some hugely exciting tasks to do at all sorts of different levels lovely to have the opportunity to chat about some math so the math um with others as well so thank you very much indeed for um I was going to say taking the time to come and join us but um for um for switching on um and um and leading us this evening I'm very very grateful you're welcome it was a pleasure thank you um and of course there would be a round of applause um for you but that's that's quite odd when we're all in our own um in our own places so I don't know then whether you have anything finally that you'd like to add no thanks Pete that was really good I think um it's a very difficult job to try and keep everybody's attention but you managed it absolutely brilliantly to everybody who was um attending thank you very much for attending and if you would like to be told about other events I need your email um I am Elin M for McClure409 at cam.ac.uk LM409 at camcam.ac.uk and then I will send you information about the next one which might also be virtual for all we know um let's hope we can meet face to face but you never know stay safe everybody and don't forget to wash your hands Lin perhaps put Lin yes perhaps put a message on the chat with your um contact good idea I'll do that I've just done that um that's in there thank you Sandy thank you stay safe everyone bye bye bye thank you thank you bye