Rice and bean counters, Number crunches, and more. In this episode, Andy, Robin and Adam discuss all things manipulatives with special guest Emma Potter, Director of Maths at Leo Academies and Vice Principal and Teaching & Learning Lead at Cheam Park Farm in Sutton, England. Just how important are they? Is there a perception that you only use them if you don’t understand? Plus, discussion about whether manipulatives are needed at secondary level.
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Hi, I'm Andy Psarianos.
Hi, I'm Robin Potter.
Hi, I'm Adam Gifford.
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Welcome back to another episode of the School of School Podcast. What a proper treat today team. Robin's here, Andy's here, but special guest, Emma Potter, is also joining us today, which is really exciting. And I can't wait to... I was about to say, get the party started. Is that a cliche? Get the conversation started at the very least, but Emma, just before we do start, would you mind just introducing yourself and just telling us there's a whole host of things that we can pick that you do, but perhaps best coming from you?
I'm currently a primary teacher, so I teach Year 6. That's probably my first role, but then I also am a vice principal in a primary school, which is called Cheam Park Farm. And it's part of the LEO Academy Trust. And then I also am a mass director for LEO Academy Trust, which means that I get to work with other schools for their school improvement and support their teaching and learning of maths and other subjects as well. So they're probably what you might call my day job is that as well.
You've pretty much got zero spare time on your hands. That's a handful. That's a lot.
Yeah. Yeah. I'm pretty busy. I'm mainly seen walking from place to place and on Google Meets and stuff like that. But yeah, I am pretty busy, but all very fulfilling work as well. So I enjoy all parts of my job.
That's great to hear, Emma.
Yeah, brilliant. Oh, lovely to hear it. And actually, what I hope we can have a bit of a chat about is to really dial into the sort of fine detail of one part of your job and it's the use of manipulatives and maths. And I just wonder, maybe if you could talk to us a wee bit about that to start us off and what we mean, maybe, by manipulatives. So, for people listening, they may not know or...
Manipulatives are concrete materials, so things that you can hold, but we might come onto some other bits about manipulatives as well, because sometimes you don't necessarily hold manipulatives. Because we found in the global pandemic that we had that actually sometimes we had to move some of our thoughts on our concrete manipulatives, but manipulatives are things like things or place value counters or plain counters. I'm just trying to think of what we have in our boxes at school could be bead strings, pretty much anything that's going to help to support the learner to actually be able to explain them a mathematical thinking or just to explain something that they're actually doing and something that we definitely promote at LEO and something that I hold really quite close to my heart is the fact that manipulatives aren't just for your least able children and actually they are for everybody and everyone should get that experience.
Even your most able Year 6 child should be able to have a go with manipulatives to be able to show their understanding and actually find that they are your children that need the most support with the manipulatives because sometimes they haven't been as used to using them or maybe they think that they don't need them anymore. And actually, we've found some of our more able pupils have actually been able to show us their deeper thinking using the manipulatives. So that's something that we really promote, something that I'm always banging on about to people is actually it's for everyone at all ages and at all levels as well. It doesn't matter whether you are down in reception or our youngest children up to our Year 6's who think they're too cool for school. They still use manipulatives, which we think is really important.
So Emma, you said a lot of things there that are really, really, really interesting and worth exploring. So, one of the things that you said, which I'm glad you said it, you said about using manipulatives as a mechanism to explain things. And I think that's a trick that a lot of people often miss. They just automatically assume that manipulatives are merely a step in towards the understanding and that we often overemphasise, I guess, what I would call the abstract element of mathematics. And we think if a child can do something in an abstract fashion, a good example would be giving us some kind of difficult calculation, if they can just generate an answer that they are at the pinnacle of their understanding.
If I could just crunch the numbers and that's that.
Exactly. But like you said, a lot of those children who show signs of being able to do those types of things remarkably quickly really struggle to be able to show what's happening with a model. So they can't build a physical model of what they're doing and we think, "Oh wow, they're just beyond that."
But in actual fact, that ability to number one, visualise something in a physical model. And number two, being able to explain what's going on using a physical model are two very important skills, very core competencies in what we would call an advanced mathematician. Someone showing promise can do the abstract but can also proficiently do the concrete and the communication as well.
So, I'm really glad you said that because all too often, especially Year 6 and it's more difficult with Year 6. People just jump into like, "Wow, Year 6, you don't need to use manipulatives anymore." They should have a sound understanding of everything. Yeah. Just so thank you for saying that, we've taken the conversation from this base level, right up to the most complex aspect of manipulatives right away, which is great.
Adam, do people use manipulatives in secondary school at all? I mean, you've worked with secondary schools quite a lot, right?
I think that there's a growing movement of realising that they need to be able to model problems. And I think that both primary and secondary have had aspects where presenting maths in the abstract was seen as the most sophisticated. Now, that's fine if you've got that understanding and that's the notation of mathematics. But I think that what I find really interesting is asking people to show me something. Show me that formula with a piece of paper. And it's like, "You look that as a footy talking about why'd you even bother with that?" This is ridiculous. This isn't the way that we teach. We teach, but we tell them the formula. We give it to them in the abstract and they memorise it.
I had that with my son. Came home one day and said, "Oh, we've got a new maths teacher. And he's told us that we have to memorise 23 things before the end of the term, and that's going to help us with our maths." I was like, "Okay, I'll temper my response to that."
So now to your question, Andy, I think there's a realisation that children are coming up increasingly with a greater language of mathematics, an ability to explain mathematics, an ability to generalise when some of the models that are being used can be applied to, say, algebra. The models that we can use for algebra are amazing and help us understand it beyond just working in the abstract.
So, I think it's a shift. I think there's still a long way to go, but it has to be a shift as well. That I think it's only until you put in that position, "Can you show me something," that you just realise A, how difficult that is, and it's difficult because your conceptual understanding isn't as deep or as wide as it could be about that particular idea or concept.
And you don't get questioned a lot on that too often.
Yeah. We often say, "Well, prove it to me, prove that the answer is what you say it is." And I said, "Well, it just is." But "it just is" isn't good enough. So tell me why and show me how it can be like that. And I think it's that shift in teachers actually facilitating the discussion with pupils and actually saying, "You need to prove it and you need to show me how you're going to do it." Because I think as teachers, we just quite often are just pleased they've got the right answer, especially higher up the school. I mean, we have the pressures of the SATs tests and whatever your thoughts on those end-of-key stage tests are, at the moment they do exist. And actually you want to set the children up to be able to approach them as well as you can.
So, if you are not then getting them to unpick how they've got there, then actually the checking process for them is going to be really difficult because they just see it as, "Oh, well, I've got an answer. So therefore it's okay." And actually, if they haven't got different skills, different methods, different actual tools that they can use, even though they can't actually use the physical manipulatives, they can actually think about how they're going to work their way through their problem to actually check their answer as well.
So I think that's just really important that you don't take away the use of manipulatives too early, or even talking about them because, actually, children need lots of tools in their toolkit, if you like, to actually fall back on. And that unfortunately is sometimes in a test situation where they have to try and check their work themselves. I just think that's just such a crucial thing that they have to be able to do themselves.
And I keep thinking when you're talking about this, that you get them at Year 6, and I'm assuming that most of them have been doing this at the earlier levels, at least at your school. So you can probably see how that's progressed once they get to Year 6 and then beyond, I would have to think that a lot of this working with manipulatives would be kind of ingrained. Not that they're going to pull out some beads in Year 10, but more so that maybe just the whole process of how they get to the answer, how they check their work would be greatly influenced by how you've been teaching. You think so?
Well, what it does is it gives them a visual model, right? So, and the visual model can then be represented in their minds without even having to manipulate anything. But I guess it's being able to use the sort of spatial reasoning aspects of your mind and the visual aspects of your mind to do mathematics is really what we're attempting to develop and also the communication. Here's an interesting thing and an interesting way to think about the communication aspect. A little bit of research we were doing and we were talking about stuff and I was working with some people who were much more clever than me and that they're not hard to find by the way. And-
I don't believe that.
Yeah, it's true. Believe me. And somebody pointed out, look, we're doing an exercise with manipulatives and the person articulated and said, "There's a child that I worked with who basically doesn't speak any English, but he can do this. And he understands it by seeing what others are doing it, and then he can do it. And not only can he do it, not only can he succeed at doing it, he can also show me and with the little bit of language that he has, he can succeed at communicating this quite, for him, challenging mathematical concept using the manipulatives."
When you realise that, you realise how powerful that actually is for children who maybe don't speak the language. But it's not just them. That's just kind of an interesting experiment because it shows you that, because usually what happens is the verbal communication kind of gets in the way. So it dominates. And then you just assume that the verbal communication is the stuff that's going on, but actually you can do it without any language, real language, or what we call the language, the language we communicate. And you start to recognise that mathematics actually is its own language. It has its own constructs, and it has its own vocabulary and everything. And manipulatives is one of the best ways to communicate that language.
I think also because in the textbook, Maths — No Problem!, the children can see the methods that are coming up. So after the beginning part in the in focus, they can see in the next section, "Well, this is some of the methods that I could use." And then they start to try and replicate that. They might not fully understand it, but they can see the pictures. They can see all the different diagrams and actually you've then enabled children to be able to do something that at the beginning, they had no idea of how to begin. And giving them those pictures that show the manipulatives, they can then start replicating themselves.
And I just think that's so important for some of those learners that perhaps aren't very confident or perhaps are starting at a different point to their peers that actually they've got a step in as well. And they've got a method that they can start to use. So that's why I'm always banging on about manipulatives and just trying to get children to use them. Because also I think there's sometimes children think, "Oh, I can't use that now." And it's all about us as adults facilitating that and supporting them that actually, "Yes, you can use this," and don't take it away because actually they need it for as long as they possibly can basically.
Well, how about the parents, Emma, are they involved in this at all?
We always show them at our parent evenings, where we get parents in to look at them and to sort of in our lower key stage, in our key stage one, they actually come to a Maths — No Problem! lesson. The teacher then teaches the adults. So they see what their children are experiencing. But what we really found that was actually quite good for us in terms of actually parent engagement was during the pandemic. Obviously, a lot of our learning went online and we were having to rely upon parents supporting their child in whichever way that they were. But then also they were being exposed to the different manipulatives that they were using. And so actually the parents were seeing, "Oh, when you are talking about place value counters, this is actually what a place value counter looks like."
And for some families, they perhaps were lucky that they actually had some of those resources at home, but also we were then able to use different technology to show the parents what these were. So we had different things on different Google networks where we were actually able to use place value counters. We could make what we would have in school as a concrete material that then the children could manipulate at home using technology.
So then our parents were able to see, "Oh actually you are able to do lots of things with these counters, for example. And actually, this is helping you to understand." So some of our parents were then able to engage with that as well through the use of technology. And obviously, we are really lucky at LEO because technology is at the real heart of everything that we're doing at the moment because that's where we've come. But during that pandemic time, when we were doing a lot of online learning, it made such a difference to the children to be able to actually still have those manipulatives but in a different form.
So it's exposing parents to it because I know when I was at school, if you use the Dienes, it meant you didn't understand. And I'm only 33. So some of our parents are older than that. And actually, they've never experienced them and their experience might have been very limited. They may have only ever seen one type of manipulatives. And also I think something that's really important is although we want them to use our maths place value counters, it's then children being confident to use other things as manipulatives. Things that you've got at home, beans or whatever that is rice to actually count with, actually that becomes manipulative as well. And it's not putting this barrier on things if that's what they've got at home.
But I think you raise such an important point and so important when the learning was remote, because I'm just thinking about it. Say if I write something in the abstract, so it's just symbolic, right? Just up on the board in the abstract. The children are relying on me to unlock that, right? The secret to be able to read it and make sense of it. And it will rely on my description and what I say, so that may lend itself to... Or sorry, the environment that can only really work in is if you've got all of the audience together at once. Otherwise, you're saying the same thing, 30 times over in a remote situation.
But obviously, if you are set up like your schools are and you are using these and it's children's second nature, what I find remarkable is just asking children to describe what they see. So if I just put down, I said, "This is 23, right? This is number..." And I'll put down two tens counters and three ones counters to say, "Well, tell me what you see." "Well, I see two tens." "Okay. Is that true?" "Two tens. I wonder what two tens are." And you start asking those and it's literally just a description.
Now, they can have that same conversation with their parents or sisters or brothers or to themselves, or to their friends online, all of those sorts of things. But I couldn't imagine, it must be so difficult if you don't have that, because how do you develop that language? And just merely by describing it, "Oh, did you know," at the end of it and you write it down the abstract, that's a way that we can write that down. So now exactly what you said, we can read it. Yeah. True story. Do you know that? Amazing. But I imagine schools that aren't set up in the same way. I imagine that would've been a huge challenge because it relies on one person to unlock it. The secret behind these symbols, which is...
I think also it's just encouraging that everyone's a mathematician. So we have parents some that say, "Well, I wasn't very good at maths when I was at school." So that's, "I can't help my child," but actually you were a mathematician. You are a mathematician. So actually, whatever level that you are at, you can still support your child through that as well. And in the classroom as well, everyone's a mathematician, the teacher, the TA, whoever is there, the children, everyone's a mathematician.
Awesome. I'm going to go around and start telling everyone, "I'm a mathematician." I haven't said that. I don't think ever.
I'm a bit ashamed now because someone actually asked me if I was a mathematician on Saturday. And I said, "No," and I should have said, "Yeah."
You are a mathematician.
Emma Potter from Sutton has told you, "You are a mathematician." And I just think that's so important because I always say about the children in the book are mathematicians. The characters are mathematicians. We are mathematicians. So let's approach this problem as mathematicians. And I think that's just really important for everyone.
Yeah. That's great.
So Andy, you are a mathematician.
You can let people know now.
I might get a t-shirt.
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