 Hi, I'm Colin Foster. I'm an academic at the Mathematics Education Centre at Loughborough University. In this short video I'm going to set out an approach to professional development that is based on reflecting on small things that happen in the classroom. I call this armchair responses to classroom events. I'm going to suggest this is a useful thing to do. So sometimes people are critical of things that happen in armchairs and think that they're not relevant or useful to anything that takes place in the real world. But we all find those points in lessons, I think, where you wish you could press a pause button. Something happens, a student says something perhaps or does something and you wish that you could pause the lesson and really think about what you're going to do and maybe discuss with other people. And the question for me is what might I say or do that might be more helpful than the first thing that comes to mind? Because in the busyness of a classroom often all we can really do is just react to whatever happens in the way that seems reasonable at the time. And so what I'm suggesting is it might be too late for this occasion but to look back at some of those things and think about what other options would be is helpful for our development as teachers. And I'm really influenced by John Mason here in his book The Discipline of Noticing Researching Your Own Practice, which I think is an amazing book. And if you haven't read that book I think you get a lot out of it. So this isn't about beating ourselves up and saying that we did the wrong thing, it's not what should I have done. And if we're thinking about someone else's lesson and someone else has told us something that happened or we've seen something in that lesson, it's not about saying what would I have done if I'd been the teacher, what would I have done, or I might have done exactly the same thing or possibly something worse than what the teacher did. The point is what might I wish to have done if I'd been the teacher and if I'd had the luxury of hindsight and sitting in an armchair to really think about it. And I think the aim here is to just expand our sense of possibilities and to think more deeply about what's going on in the mathematics and in the teaching. How might I inform my perspective on my future teaching? So are you ready for a very quick armchair workout? So I'm going to give an example of something. So often it's things that students have said. So I was watching a lesson where students were calculating areas of triangles and the phrase that was going around all the time was half the base times the height. And then a student, I heard a student say, is it half of the base times by the height or is it half of the whole thing, base times height? I wonder how might you respond to that from the comfort of your armchair having time to think about it? And maybe try and come up with at least three different ways to respond. Because what I think is quite useful, if you only have one thing you can do then you'll probably do it. But if you have more than one thing you can do then maybe you can select whatever seems the most helpful from those different possibilities. And maybe even get to the point in the classroom where something happens and in your mind is not just one thing that you could do but more than one thing so that you're selecting whatever seems the most helpful. So do you want to take a moment to pause the video and think about what you might do and maybe a few possibilities. So I've given you a moment there to have a think of different possibilities and here are some things that are cared to me. So you might say well don't worry about it, it doesn't matter. You can half the base or you can half the whole thing or you can half the height. You still get the same answer, maybe just tell them. You could ask them to find out what their neighbour thinks about it. You could invite them to work it out both ways, maybe with some numbers and say see what happens if you try it both ways and compare the answers. You could ask them why we've got that half there in the first place, where does it come from, what's it doing there. That might tell us what it is that needs halving. You might want to draw some pictures. You might say this is all about priority of operations and start talking about mid-mass or whatever you call it. You might think this has got something to do with associativity of multiplication and you might want to bring in that term and talk about that. Or you might think well actually maybe this suggests that something has gone wrong beforehand and ideally students wouldn't be asking this question so maybe you want to not start from here. I wonder if any of those are any similarities with your ideas. Obviously if we were face to face in a room together this would be the point where I would be asking you to talk about that. But we can't do that. So what did I think about? Well I thought about ways in which these things are written and the difference between having a half multiplied by something or dividing by two and perhaps when you're multiplying by half it kind of looks like it's happening at the front because if you do a big long line over a two you can show sort of more of the symmetry that's going on. But you still have this issue about what's getting divided by two here. Is it just the base? Is it just the height or is it everything? And you can see why students might be puzzled by this. They might say even if you'd ask them to do the calculations and see if it makes a difference they might be quite surprised. They might say how can these be equal? On the left hand side you're only dividing, sorry on the left hand side you're dividing by everything. On the right hand side it's just the six and we didn't halve the four. So how can they possibly be equal? Or you could think of it this way where you've changed the order and you're saying if I halve the six and multiply by four that's the same as halving the four and multiplying by six. Why should that be? So these are potentially quite puzzling things for students. You could think about associativity of multiplication. What kind of a thing is associativity? Is it a rule that you just say please follow this rule? Or is it perhaps an axiom or just a fact? Is there something to understand or is it just is? It doesn't work for division so it can't be that obvious perhaps if it works for multiplication but not for division. And then there's confusions around priority of operations isn't there? Maybe the student thinks that if you're multiplying it's the same level of bid mass or whatever mnemonic you might be using and so you have to do the first multiplication first. You have to do a half times six because you're going left to right but you do get the same answer if you do the six times four and then multiply by a half. Or maybe they're attending to the word half of six times four. Maybe if you've got bod mass then maybe they've been told that the O in bod mass is of and therefore that has to happen before any multiplication. So I was thinking about where else does this happen? Sometimes students will do four times four is 16 so point four times point four is point sixteen they might say point sixteen instead of point one six which is right but with three times three equals nine you can't do the same thing point three times point three isn't point nine and they might think well I've divided all the numbers by ten I've scaled it all down why doesn't it work and that feels like that's a similar issue because point three times three would be point nine but point three times point three wouldn't be and maybe diagrams is what we need so you can take a triangle get another one identical one reflect it and shove it back and that kind of looks like a parallelogram is every triangle half a parallelogram because the error of a parallelogram is BH and so maybe we can see the thing that we need to halve is the whole thing the whole BH but we can think about that in other ways we can think about the triangle and we can think about halving just the base and keeping the height the same that's going to halve the area we could think about just halving the height and that's going to halve the area as well but if we do both half the base and half the height we're going to quarter the area so maybe there are some ways of thinking visually that are helpful here but I think the point for me isn't about finding one magic best answer that's going to be right for every student but expanding our notion of what's possible educating ourselves about possibilities trying to see things a bit more from a student's perspective getting around that curse of knowledge that we often have that things might seem obvious to us but don't seem obvious to students seeing how things are maybe more difficult than they might appear at first glance and it's easy to find examples to look at I've got loads and loads of examples of things like this that I've spent time thinking about keeping a notepad in the classroom and jot down just those small things that happen I think sometimes people try to reflect on whole lessons and for me that's really too big a unit to say much that's useful but reflecting on just a sentence the students said or one thing that they did can be really helpful and there are some great articles in mathematics teaching the ATM journal which often describe with enough detail things that happened in classrooms or with students in other situations that you can reflect on in a really fruitful way so I have fun doing that I commend that to you as a great way to develop professionally and if you want some homework I've got another one that you might like to think about here's something a student said in a lesson on directed numbers if two minuses make a plus why don't two pluses make a minus think about maybe how you would respond to that and think about multiple different ways of responding and what some of the issues might be for the student and I found that very stimulating and interesting I hope some of that's been useful and thank you