 Hello, I'm Anne Watson and I'm going to talk to you about variation and its use in the teaching of mathematics, its power in the teaching of mathematics. And I'm going to give you lots of examples, so I'll share a screen with you, but I may not give you time to think about all the examples, so I hope you'll think about them again afterwards. Variation is a word that's around at the moment that's been complexified with lots of meanings and lots of writing, but it's actually a very simple idea, which is at the heart of mathematics, and therefore could be at the heart of the teaching of mathematics. You need to identify what it is that you hope people will learn from the lesson, precisely what it is, and then think about how, how does what I do help them understand this thing? How am I going to communicate this thing? What am I going to say? What am I going to draw? What am I going to show them? What am I going to focus their attention on? And hopefully the lived object of learning, what they experience will be the thing that I hope that they were going to learn. How can I design my teaching for that? And the key to that is thinking about, well, what am I going to vary, and how am I going to vary it, so that their attention falls on this key idea that I hope they'll get to. Let's start by looking at this diagram. Now my guess is that immediately you're trying to make some sense of it. Does it look like something? Is it symmetrical? Are there any congruent shapes? All kinds of things you're probably doing. But when it boils down to it after a while, you'll see that length and vector are probably the things that you can say something about from this diagram. And it's on grid so that you can say some sensible things about length and about direction. So the diagram limits what you can do that is of any interest and focuses the mind, focuses the class towards thinking about what can we say about direction. This diagram, there are some things changing and there are some things staying the same. And you could have a look the way the things are labeled helps you perhaps to make connections between the B's and the A's and the D's. And you might think what's changing? What is kept the same? The areas are changing? Oh, they are all squares, I should have said that, but the positions are changing. So how is everything changing? What is changing as a result? What can I conjecture from this collection of diagrams? It's the variation that draws your attention to what's the same and what's different. I hope you're thinking about this. You may not have time to do all of these, but you might have enough time to realize that there are some patterns going on. And here the variation is used so that you can say something general about the relationships here. You can maybe write them in terms of X if you want to, or a little cloud if you want to. You can maybe generalize about these relationships. So here variation is used to offer a mathematical structure that can be generalized and then extended because the first generalization doesn't carry through to the second set and the third subset. But there is a generalization of generalizations that you can do over the whole screen. This one is rather fun because first of all, you can decide how it is that you've found the number halfway between because most of us, however sophisticated we are, will have some kind of idea of bouncing from the sides to the middle. That idea of bouncing from sides to middle isn't such an obvious thing to do when you get to the second example, but oh look, I don't have to do much to this. What is it that I have to do? Can I use the thing that I've just done in order to help me with this? And what's the generalization that I'm drawing on behind it? Similarly here, is there another general rule that I can use to get to this one from the first one? What about this one? Does it work with negatives? There's a great big question, the Key Stage 3. And then what about this? Well thank goodness the 9000s are well out of the way, but what about this one? So you can think about that after this presentation. Here's a diagram. What is it a diagram of? Is it compass points? Is it two straight lines on a graph? I know what those lines are. No, because there's no labelling, so I don't know if they're equal aspect. I don't know anything. Can't say much really. Could be just beginning of a pattern. This one. Oh, I can begin to say something here. The axes, I can call them axes now, are kept still, but the slope of the lines varies, so I can talk about slope. But with this diagram, I really don't want to talk about slope, because there's far too much going on. I'm much more attracted to the fact that there is an invariant point that they all go through. So I've got two different sorts of variation here, and they happen to be two kinds of variation that you can look at when you're looking at straight line graphs. And then you know, because you're good mathematicians, that they're going to be embodied in the equations that you end up with. What about these? What of these sequences got in common? How do they vary? How can I generalize? I only need to generalize the starting point. How can I then write general terms? Here. Oh, we start the same, but then we go different. How can I generalize? What varies? The starting point is the same. Then what happens? Oh, this is kind of similar, but kind of different. What's varying this time? So from this kind of thing, you can build up generalizations about arithmetic progressions. You can build up rules because you're controlling variation. This is an interesting collection. You might, if I give you time, observe that both of these factorize. I tell you free that these also factorize. Oh, where's she going with this? What's the same? What's different? Find as many examples as you can which factorize whatever sign the constant has. Have you ever thought about that? And what's the effect on with myself around a bit of changing the sign of the coefficient of X? So the initial examples have promoted the question, if you like, and then you're asked to make up examples of your own to use variation in order to find out what's going on. So if you see an interesting set, sine square plus cos squared, you can find out that you can try examples, you can bounce numbers in for X, you can find out that it's always equivalent to one. So instantly, you can know what the second, third and fourth are, and maybe you can even know what the last two are going to be. So you can generalize in a sequence of examples, maybe examples that have been explored on a calculator or graphically because this is quite nice to look at graphically. Variation theory focuses on what is available to be learned. And it's made available by the focusing of attention by using variation. So alternative generalizations possibly available for learners, because you never know what they're going to see or how they're going to make sense of it. But you can anticipate some of those beforehand, because if you control the variation, you control what there is to pay attention to. And it's also mathematics. But people say things about it. And I'm not going to tell you, read out for you what they say, because the 10 minutes that I've been given up. And I will leave you with two slides that give you some references about meanings. So the task for you as well as perhaps going back to some of these and working out what it is the author made happen by the task design is to perhaps open some textbooks and to ask yourself, what is this author doing? What does this author intend people to experience, or is it just mechanical practice of a method? There's loads of stuff online. A lot of it is mechanical practice. Sometimes occasionally you can see, oh, this author wants me to notice this bit of maths. I know.