Curriculum for Excellence - HMIE/LTS Good Practice Conferences
I am interested in how we help kids engage with maths and make it more meaningful and part of their everyday lives. Now I have to say part of that for me is not making it all about shopping or all about cooking or all about weighing. I think there’s a lot of joy to be gained from working on mathematics for mathematics’ own sake. It can become just as meaningful when you’re working on a piece of abstract mathematics as when you’re working on a piece of practical mathematics that might have some impact on your lives. So I’m not associating maths teaching and numeracy within that, just with the everyday, and I want to explore with you this evening some issues around that.
I’ve been enjoying looking at this document the principles and practice booklet, which I have to say is delightfully succinct can I praise Scotland for the brevity of its curriculum documents, and that’s some of the stuff I’ve pulled out from that, that you’re being encouraged to develop, and I think if we treat these sorts of lists as a kind of - I don’t know how many bullets points there are there - as twelve things that we have to kind of hit each one and tick them off, then it is exhausting and it’s very difficult to actually cover all of that.
I want to suggest that’s there’s one thing that can bring all of that lot together and that one thing is Maths Talk, that the thing that actually makes the biggest difference in my experience over my years of teaching and the work I’ve been doing recently, I was privileged enough to spend a year working in New York on a project there and the work that we did looking at effective teaching, the thing that comes through time and time again is the quality of the talk that goes on in classrooms about the mathematics, and you know we focus a lot, we talk a lot of about questioning and the quality of questioning, which I think is important, but actually you can do a lot of talk without asking many questions at all and that’s something well explore as well a bit this evening.
When I trained as a teacher of course what would have gone on that slide was action, was kids doing things. I was of the generation where we used base ten blocks and Cuisenaire rods and Unifix and it was all about kind of the child as an individual explorer, exploring the world of mathematics through manipulating things. I’m not suggesting we don’t need to do that. I really think its important for kids to actually have those hands on experiences, but what’s very clear is that it’s not the hands on experience itself that’s makes a difference, it’s the talking about it. It’s how the kids make sense of what they’ve just done by sharing and conversations with their peers and with the teacher about what was behind that. If you leave them just to the action then they divert it.
So there are two things. The first is the connection between thinking and communicating and thinking, and I’m making three very simple claims, but it’s through communicating with other people that our thinking develops. The way you see the world is shaped by how other people talk about the world.
Most thinking most deliberate conscience thinking is having an internal conversation with yourself, that’s how your thinking develops, those times when you’re mulling things over and that talk is our primary form of communicating.
So that’s for me why I’m placing talk at the centre of teaching mathematics. I’m also drawing on some work of a colleague of mine at Kings called Peter Cooknik and he argues that there are three, three important ways of grouping kids. Working in pairs, his research and other people’s research shows, is a very powerful way of helping kids develop understanding. So if we want to actually get children to get to grips with the maths and not just memorise but understand it, paired work seems to be the key to that at almost at any age really.
For practise and consolidation we’re still best off doing it individually. If you try to get the kids to work together to do practise stuff they actually, it gets in the way, that paired work doesn’t work. So I’m not advocating that kids should be working in pairs all the time, far from it. They do need time to do that practise and consolidation work and they are better off doing that on there own, and small groups, groups of 4 or 6 if kids are skilled enough to do that are good for extension work, but they’re not so good, trying to get 4 kids to work on some understanding work is difficult. They often split into two groups of two anyway. So his work and the other work I’ve been looking at and my own research suggests to me that a lot of the time the best way, particularly in primary schools, to have our children workings is in pairs.
There is some evidence that matching kids too closely in attainment, trying to say I’m going to put you together because you two are about the same, actually isn’t the best pairing because their thinking is too similar and so they don’t spark off against each other. So you need a bit of difference in the pairs but not a huge difference, but in my experience it’s a real, it’s the real craft of being a teacher. You need a bit of time on your own and then you come back together, then time on your own and back together, so it’s not as simple as kind of forever working together. The whole dynamic of this is I think a very complex thing to manage in classrooms. You start in pairs and the bigger group kind of evolves out of the pair work, which is often much more productive than saying you’re going to work as a 6 and it falling apart right from the start.
All of those things, Peter’s work shows and other people’s work, you need to learn how to do it, that paired work doesn’t come automatically, you have to work with children on developing the skills of being able to do that.
The work I was doing in New York, one of the things that they were working on there was a kind of reversal of what I see as a very popular model of mathematics teaching, which is the teacher, me in this case, would do what I’m doing now in a sense, which is to stand at the front and kind of give the introduction. show how to do something, model it in the language of some people and then say to the kids okay I’ve shown you what to do, you go off and actually now do it. The trouble with that is that what often happens is the kids when they go off, spend more time trying to remember what the teacher did and duplicate what the teacher did, than thinking about the mathematics, and so if they don’t understand it they’ve got no kind of resources to fall back on, and what this work in America was doing was taking that lesson model and kind of flipping it on it’s head, so the kids would start off doing something, solving a problem, playing a game whatever, without the teacher really giving a great deal of input at the beginning, but there was a very very sustained public conversation at the end of the activity when the teachers would draw out the mathematics and refine the mathematics and because all the children had a common grounding to draw on through the activity, they could by and large take part in that common conversation.
I like the idea of developing mathematical habits of mind, that mathematics isn’t innate. There is a certain innateness. We do know now that we are born with a sense of numerosity that very small children, very young infants can tell the difference between 2 and 3 objects, and it’s kind of hard wired in that we can recognise small quantities.
We talk about teacher centred classrooms, where you know, I love it, I love being the centre of attention and we talk about pupil centred classrooms, I want to argue for mathematics centred classrooms, that what I’m trying to do is create this public conversation where what we are talking about is the mathematics, its not about me. I’m kind of stage-managing it because what I want to do is put the thinking into the heads of the learners. That’s my role as a teacher.
Notice I didn’t say “who’d like to come and show us their solution.” I’ve chosen the solution I want up there because that’s the one I want to explore. I’m very much in control of what’s going on here. I really, I don’t play the game where kids are saying, you know who’s got a solution they want to show us because you inevitably you get the same kids coming up, who are the ones that are bold enough to do it, but also it might not be the bit of maths that I actually want to talk about, so as I’m going round I’m doing an assessment of what’s going on and who’s got some interesting maths going on that I feel we can build on, so it’s not a random choice of who’s coming up there, and they explain and then the question I asked was who can explain in their own words what they’ve just explained.
Nine times out of ten I do this either in a school or with teachers, someone puts there hand up and says well the way I did it was… I don’t know about you but what I think going into classrooms we’ve got better, I think we’ve got better at kids talking and explaining. Notice when they come to the front, where do I position myself? At the back of the class, so that the conversation has to go across, if they’re going to talk to me as kids inevitably do, they’ve got go talk across the class do to it, I’m not going to stand here, again I don’t know about you but lots of kids coming up and there’s a very nice private conversation going on between and the rest of the kids could be playing with yo-yos for all, the maths that they’re engaged with.
As a teacher this is hard work because when you’re listening to those kids explaining you’ve kind of got to do two things. You’re listening to see if they have got the maths right. You’ve got to work on that, and you’re also kind of listening to see if the way they’re explaining it is going to make sense to the other kids, and I don’t know if you notice but what I’m also trying to do, as the teacher in that situation is to not put it in my own words. It’s very easy to say “oh what I think you’re trying to say there is this” and re-explain it for them, so there’s a lot of me playing the devil’s advocate.
I’m using, I’m very deliberately using the word conversation here rather than discussion, I was interested to read that the roots of the word discussion where the same as the roots of the words percussion and concussion, and discussion is often a clash and what I’m doing here is before anybody has any chance to agree or disagree with what somebody’s put up there, I’m working on we all understand where they‘re coming from or as far as we can get with that, where they are coming from so we can have a conversation that builds on that rather than kind of saying well that’s not what – I did it differently. You spin off into 6 different explanations.
Working in this way you have to, you have to have faith that the kids will bring something to the lesson that you can build on. Children in my experience, when you put them in problem solving situations and ask them to have a go figuring something out, and then you help them refine what they did, they gain far more than simply me assuming I know what I need to teach them today, without that initial exploratory part of it.
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