 Hello and welcome to this very short brief introduction to the notion of noticing and attention and how they play out in mathematics classrooms. Start by sharing the screen. Begin with some assumptions. To teach is to interrupt someone else's flow of attention. That's what the act of teaching always does. To teach effectively requires knowing what is worth attending to and how to attend to it and having a good idea of what learners may be attending to themselves and how they're attending to it. And in that way you can intervene effectively and efficiently. So let me ask you this. Have you ever had a teacher come around say or do something and then go away again while you simply carried on with what you had already been doing? Have you ever asked a teacher a question but not got the answer you were hoping for? Have you ever tried to explain something to someone but felt that you were unsuccessful? My conjecture is that these are all instances in which the teacher's attention and your attention were misaligned. Or in the case of teaching your student, you and your attention in your students were misaligned. So let's have an example of how attention works. You probably want to pause the tape and contemplate this diagram for a while, this picture, because I'm going to ask you whether or not you can see certain bits of it. So you need to look first for yourself. Can you see that parallel of piped? Can you see that object? Or that one? At first you probably saw oranges. At some point you recognize that the oranges have been cut in different ways. So you might attend to a particular orange and then another orange and then compare how the two oranges have been cut and how they're cut differently. At some point your attention shifted to seeing parts of even whole cuboids and other similar structures. But you and I know that they're not actually there, it's your brain which is interpreting them as being there. You may have looked at the details of how the cutting of the oranges actually created the illusion of faces of a cuboid. So if you pause and think back and see whether you might recognize periods when you were holding a hole, gazing, not just at the whole picture but possibly at a smaller part of it, but without really doing any analysis, just gazing. Or maybe you were discerning some details or recognizing relationships between details that you'd discerned. Or maybe you were perceiving properties as being instantiated, something which happens everywhere. And even you might have noticed yourself reasoning on the basis of agreed properties. But that's quite difficult in this situation. These are different ways of attending and so not only is the focus of what you're attending to matter, but also which of these ways you are experiencing. And you can switch back and forth between these very rapidly. Draw a circle now and divide it into four congruent pieces. Maybe pause while you do that. Now do it another way that's different. And another way that's again different. My guess is that most people I know draw something like that the first time. Then for the second one many people draw that. But then there's the question is that really different or is it not? Some people then do something like this. Is that really different? And so there's an opportunity in this task to negotiate what's the same and what's different. Then you can ask yourself what can be varied. And not only what can be varied, but what seems to be invariant. Often people end up with something a bit like that. And they are tempted to say oh so you start at the center and draw any path to the edge. Repeat that around four times and you'll have your four congruent pieces. Well not quite because there's some constraints on the variation. Namely they mustn't overlap. So as well as offering opportunity to pay attention to how your attention shifts and what you're attending to and how you're attending to it. There's also an opportunity to realize the power of the personal narrative, your self explanation and how you might modify that at various times. So the rest of the PowerPoints have two other examples where you can exercise and your powers of attention and try to capture your movements. And then the invitation was to think back and to recognize periods when you were attending in these different ways. Maybe there are other ways to attend but certainly these ones are ones that I've discovered are important. Perhaps most importantly is noticing shifts between these states of attention which can happen very very quickly. They're not a hierarchy. You can go back and forth, back and forth, round and around. But the important thing I think is to try to sensitize yourself to these ways of attending so that you can be sensitized to your learners. So you can notice ways that they're attending and then choose to act pedagogically in order to may perhaps shift their attention. And the way to do that is to notice opportunities to act freshly rather than out of habit. These fresh ways are informed by preparing yourself intentionally and then hoping that that will actually happen in the moment. So what you can do in the future is to notice particularly your own shifts of attention especially from specific relationships to perceiving properties because that's the essence of mathematics as far as I can see. And the purpose is to sensitize yourself to notice shifts in learners. By working on mathematics yourself, by working with colleagues on mathematics, for example working on tasks that you're going to give to learners, or by setting yourself to look out for something that you are doing unwittingly when you're in the classroom with learners.