 Hello, I'm Julian Brown and in this short video we're going to take a brief look at some aspects of visualising in mathematics. The starting point perhaps is to give a sort of framing of visualisation. The value of visualising in doing mathematics and in the teaching of mathematics feels like it's quite strongly linked to what I'm interested in. If I am concerned with an answer, then it doesn't really matter what's in my head or in the heads of the learners I'm working with. If I'm more concerned with ways of thinking and possibilities for thinking differently, then it feels worth exploring the images that are being used. We'll need to communicate something about those images then and that requires drawing or writing something or describing something. So we have these modalities of the visual and the verbal and I'm certainly not trying to set them up in competition with each other, but if we see value in an awareness of how we are thinking, then it seems to make sense to discuss the imagery that we use. In a classroom that might lead to some frustration, finding ways to talk about the images that we're carrying, but that in itself, the doing of the communicating is part of drawing attention to what we see as the key features of the mathematics, I'd like to suggest. And finally, in terms of framing, I think it's worth attending to the fact that the images that we use are going to be representations of things that we have already experienced through previous physical or mental activity. Therefore, there's a call for me there to think about the experiences that we're offering to learners that we're working with. So let's take an example. If I ask you now please to pay attention to what imagery occurs to you as I ask you to find the roots of the function x squared subtract four. So you might have thought about the roots of that function x squared subtract four and I could speculate since we're not physically present with each other. I could speculate that you might have had images associated with manipulating an equation or perhaps an image of the product of two brackets or perhaps a graph of the function. Perhaps this is a function that you just know or perhaps you get to that by translating the graph of another function. All of these are going to be products of your previous physical and mental activity in terms of what's accessible to you. So I would like to focus here on visualizing as the summoning of a mental image. And that image is one that can be unrestricted by physical constraints and it allows us then to work on patterns of ideas. So I might ask you now to try again another short visualization. So I'm going to ask you to to visualize a semicircle and to inscribe a triangle into that semicircle and then to take the vertex of the triangle that lies on the arc of the semicircle and to drag it around that arc. And as you do so just attend to what's happening to that triangle as you move that vertex around the arc of the semicircle. I suspect that you'll be seeing things that change. You might also be seeing things that don't change in there. But now suddenly with that image you have access to an infinite number of inscribed triangles. And so if I were to ask you to think about the triangle that has the greatest area of all of those inscribed triangles, there's something that we could work with. So we're representing a situation here in order to work on it. You might want to pause the video now if you'd like to work on that further right now. Another way of representing a situation in order to work on it might be to to see the unseeable. For example the inside of a 3D object. If you haven't worked previously on the painted cube investigation now that's something that you might you might look up. Use your favorite search engine to look for the painted cube investigation and try visualizing that. But we might also visualize in order to test possibilities. So I have here perhaps an image of thinking about solving a Sudoku puzzle let's say. And thinking about strategies and testing possibilities. The link at the bottom of this slide is to a manipulative provided by Enrich. This manipulative allows you to explore a situation I'll try to set up with you very briefly here. So if you imagine a square and then a circle that is outside the square. And then bring the circle so that it's just touching the edge of the square. You might then roll your circle around the perimeter of the square. And follow the center of the circle as it rolls around the square. It's not going to lose contact with the square and it's not going to slide. You might be seeing that locus of points in a particular way now that supports you describing the length of the locus with particular features of the locus. And once you might have been able to access this problem without this imagining without this visualization. The visualization itself might well be drawing your attention to certain features what happens at the corners. You might be thinking if there's a manipulative here then surely we could just look and absolutely. But I think that using the verbal description for a visualization allows us to direct attention to particular features of the mathematics. And in a similar way we might say offering an image such as that manipulative allows us to direct attention in a different way. And we might actually offer an image and then remove it and invite people to reconstruct the image. Again that might seem to you that that's a bit pointless if I've shown the image. But again it's an opportunity to direct attention to particular features. One way of exploring that is to use silent animations and there's some links here to various silent animations around mathematics that you may or may not have met before an invitation for you to make use of those afterwards. I will briefly share with you this one. And I'm going to invite you to watch it twice. Firstly without any prompts from me. There's lots of things that you might have been noticing. And now I'm going to invite you to watch it again but this time to attend to the number of lines. So you might pause again here and work on that the number of lines and here is an opportunity to work with reconstructing an image that you've just seen. So what next after this very brief overview when you may choose to go back and work on some of those aspects of mathematics yourself based on what you observed what you saw. You might use those silent animations to consider what the people who made those films were directing your attention towards and then an invitation to think about what you might do in the classroom. There is a link on the next slide or a list on the next slide to various bits of further reading with other examples of visualizations you might work with. If you follow them up again my invitation to you would be attend to where your attention is being directed as you work with this visualization.