Mode, median, mean, and more. In this episode, Robin and Adam are joined once again by Roger Hitchin to discuss if there is a case where in maths, as in life, 2+2 sometimes equals 5. Why do we have certain conventions in maths? How do we go about explaining these to pupils? Plus, Adam reminisces a beautiful piece of writing that sadly didn’t fit the marking criteria.
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Hi, I'm Andy Psarianos.
Hi, I'm Robin Potter.
Hi, I'm Adam Gifford.
This is the School of School Podcast. Welcome to the School of School Podcast.
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So welcome back to another episode of the School of School Podcast. I'm Robin. With me is Adam Gifford. Hi, Adam.
Robin, how are you?
I'm great. Thanks for asking. We also have a special guest with us today. He's certainly a good friend of Maths — No Problem! Roger Hitchin, welcome to the show.
And I would love for you to tell us a little bit more about yourself.
Hi, Robin. Well, thanks for that. And thanks for having me. Yeah, we are a good friend of Maths — No Problem!, and we've been wedded together since 2015. We were early adopters in our school down here in Wellington, in Somerset and yeah, it's transformed our maths and many children now over some years. So yeah, it's bit of a journey, but we're certainly knowing near the end of it. And it's nice to be here, both with yourself and Adam, just to share some ideas and some thoughts about what's happened along the way for us as well.
Oh wow. I bet you, you could share a lot and we'll probably hear more today on this episode. And this is a surprise episode, everyone, because my two friends have decided to leave me making the decision on what this chat is going to be about. Now, Roger had given me some discussion topics ahead of time. So it's not a complete surprise, but they didn't decide which one we're going to talk about before we've pushed the record button. So here it is. Is there a case where in maths, as in life, two plus two sometimes equals five? Sounds a little familiar for all of you literature fans out there. Roger, this was your topic. I think you need to start us off.
It's a funny one, isn't it? It's taken from 1984, I suppose, George Orwell, but I'm verbalising, I suppose, on behalf of some of the pupils that I've taught over the years, that at some point, and it can happen on any particular topic, get flabbergasted or frustrated at what they consider to be the ludicrousness of maths sometimes. And they look at me or around and they go, why is that? So a kind of low level example would be maybe some of the language of maths that we use, but in no way hints at what you have to do. So let's take maybe a straightforward one, maybe not the best example, but I think it works for the purposes for this.
So if you're looking at a set of data and you want to find the mode, what is that? Now we would know, yes, that's the most common, but of course you've then got the median, your middle number from the set of data. And then you've got the mean, which is something your brother or your sister might be to you, but also means that you're adding up your numbers, dividing it by how many there are to find the average. So you've got these three M words and that's before you've even mentioned the range of a data or gone in any deeper than that.
And I think sometimes children, and I think it's a valid point that maybe see it as a bit of a plot in that these words or some of the devices in maths are there deliberately to trip them up, to make them almost put two and two equals five when they know the answer is four, but it's kind of phrased in a way, a question is phrased in a way, or there it's a term that is easily muddled with another unnecessary complication that makes maths harder than it actually needs to be.
And it's different for different pupils in terms of what frustrates them. It's normally those things, we've all had them and it may not be in maths. It may be a particular word you're using in a language or a particular spelling that always defeats you, like beginning or something like that. And you have to come up with a mnemonic or something like that. But I think maths can get a bit of a bad press sometimes because you do come across these things, particularly as you go through the years higher up where you have to know what certain things mean in order to get to the answer. And if you can't remember what they mean, then it may as well be two or two equals five because it is essentially a foreign language and it's sometimes out of reach. And I think there are those frustrations with maths on behalf of some of those pupils. I'm maybe too old to think about those things that kind of affected me, but I know they are there. They're in my children. So that's where I am on two and two equals five. Sometimes that's just the way it is.
Right. Rather than having a real comprehension about a certain subject or as you were referring to in maths, understanding an equation, it's, you've been told that two plus two equals five and that's the way it is. And unfortunately, sometimes for the student and I certainly can speak for myself as a young person that sometimes it was just a matter of that's what I was told to solve the problem. And so that's the way it is rather than really gaining that understanding that this is how to do it. And I get it. It's more like, okay, you've just told me, this is what I use, and that's how I'll get the answer.
Yeah. That's a huge discussion point is there's no relational understanding there. This is how you go from A to B, there are various parts, isn't it? It may be a division, but in order to get to that answer, you need to multiply, but hang on, it's a division. Yeah, I know. But, and you need to just multiply to get to the answer. Why is that? What do they? And I suppose it then leads to, I mean, algebra is also a nice one, isn't it? When people go, well, where do I need to factorise? I mean, what use is that? When I'm actually going to do that? If you're a bit tired and you're having a bad day, I think it could defeat children sometimes before they've even maybe given it a fair chance. I think certainly in the higher up you go and the more advanced it goes, but I think it's still there even at a young age that you still come across those frustrations.
And the reality exists that there are maths conventions that are there because that's just what we do. This is the way that we do it. This is the way we may label something. Or if it's an order of operations thing, this is what we do first. And I think I've listened to Ben Has say this on a number of occasions where he points out these maths conventions and it's kind of like we do it because someone way back when decided this was a good idea and enough people agreed with that person and therefore it becomes a convention. It's just something that you do. There's no deeper understanding to it. It is just simply, this is the way it goes. And if you can convince someone to do it a different way. So if you convince people to add or subtract first, then maybe if you get enough people on board that will become a standard sort of convention.
And that's just the way it works. So I think that there are these things in mathematics that we'll often talk about. We want to make sure there's a deep understanding, but sometimes there's these situations where it's just like, well, that's just what you do, but isn't there any more to that? Isn't that terrible teaching, Mr. Gifford. Isn't it always something that you should understand on a deep level? No, in this situation, it's not actually, it's just simply a convention and this is what you have to do. And if you don't like it, that's cool. Get enough people behind you and see if you can change it and see if you can change that to happen. But I also think there are some or a colleague of mine used to do this quite a lot in training and he'd put that... Now, I don't know if this is going to work without any graphics or anything.
So I try to make it as simple as I can. He used to put one was equal to 0.999. Discuss. So you're thinking why as a charter and moving through school and you learn about a third. So that's a reasonably, you start to learn about thirds quite early in the piece in your primary school life. And by the time you get to year six, you can express a third is 0.33. The children by this stage are also able to add three digit numbers. They're able to add 3, 3, 3. And so if I had one third and it was 0.333, it would make sense that two thirds, 0.666, therefore three thirds is equal to one. Hold on, hold on. This is just twisting my head a little bit here because is that right? Is that? And then it becomes a little bit more complicated. And I think sometimes when we think actually what you really need to understand is a far broader context within this.
And you need to understand about how decimals work. You need to understand that this can go on forever and ever, and ever. And what we're talking about is a larger conceptual idea. Then I think that it stands to reason that A, children can look at it and go, I just don't get it. Teachers can look at it and go, I just don't know how to teach it because I don't know how to explain this well enough to say, no, you are right that 0.333, and then 0.666 and then one. Right. Okay. Where did that thousandth come from? Did it just, do we make it up? Do we just throw it in there? Lob it in because it fits or what's the story? So I think there are these things that when you're on the cusp of some of the ideas, they can be quite difficult to...
Certainly I'm speaking for myself as a primary school teacher when I first started, no, I wasn't a specialist in anything. I'd been to primary school. I got through that. And I'd been to secondary school and I'd been to my uni and I'd been trained the best that people possibly could to get me into the classroom. But by no means gives me the understanding in every subject that I taught as an expert, not even close. And so I think that's where it becomes difficult. I think that's where it becomes difficult. And then on the flip side of that, and this isn't a maths thing, I may have mentioned this in another podcast.
One of the single most beautiful pieces of writing that I have ever, ever read in a primary school was the most simple, it was simple. And it didn't tick all of the boxes for the year six sets, but you knew that what you were reading, for me was the best piece of writing I have ever read in a primary school. Possibly right up there with some of the best writing I've ever read as a beautiful, moving simple, wonderful piece of work. Yet, if I looked at the rubric, if I looked at the marking criteria, if I looked at the schedules, that is a level, whatever.
And to me, that's a two plus two equal five moment. It's just that two plus two, it's a crap moment. Because it's like, there was nothing that could be added to that. No big words, fancy punctuation, complex sentence structures, blah blah, blah, blah, blah. There was nothing that was going to improve that piece of work, yet when you marked it, you're being told that. So I think these things do exist. I think that we need to be mindful of that and we need to be mindful of our limitations. And so some of these ideas that crop up and some of them come out of the blue, hey, a child asks you a question, it floors you. Just understanding that this is with some things and mathematics and possibly this is true for every subject. You're only a heartbeat away from some really big, heavy, complex ideas that people have been studying and arguing about and discussing for centuries, which I think is wonderful in some parts, but also one of those that can do probably they need to be well managed as well.
But I think you've hit upon the idea actually that is you talk about a piece of writing. They're also the joys of it. It doesn't all fit nicely into that box. You still have a choice about how you react to these out of the blue moments that kind of hit you between the eyes. Maybe some teachers feel they should have those answers at their fingertips. Personally speaking, I like the children to know that I don't have the answers, not even on my arms and alone fingertips, because then I can walk around the classroom where I want to go rather than them coming to me. So it kind of frees me up a little bit.
And I always say to them, when we're kind of tackling these things, the answers are in the room, the answers in the room somewhere and they they'll get to that. And when you do come across those frustrations, you do get that dynamic where they all kind of jump on it. Yeah. Why is that? That actually is annoying. Yeah, it is. Hey, I didn't invent it. I didn't write this, actually. I didn't live 500 years ago with that long beard. It wasn't me. It wasn't designed to inflict deliberate pain on you all these years later.
But good for them to be asking the why.
And challenging that even if you don't have the answer, as you said, if it's in the classroom somewhere that all of you can be trying to figure it out together and rather than just being accepting of these things. That's the way it's always been, and so it will continue to be that way.
I said that lovely story about the piece of writing. And a few weeks ago, I used part of the maths teases towards the end of the lesson. And there was a deliberately, very hard one, a two star one. And as the children finished their workbooks and their journals and the things that they had done, they all tackled this problem and using the idea of the answer being in the room. As they finished more and more came to this problem. So by the end of the lesson, they were all working on one problem, the whole class were just working as one big group all around this table, kind of discussing, arguing, laughing, joking, going out blind alleys or all the way around about it.
There were some great moments in that. I'd love to say I had a happy ending, but it came to lunch and we still hadn't solved it. So we just let it go at that point really. But that was okay. We didn't find the answer, but we enjoyed the journey of going along to our two equals five there and it could have equaled six for all we know, because we still didn't get there. But it was nice to have things like that. Yeah. It's going to be frustrating sometimes, but it's okay. That's part of life, isn't it?
Yeah. And I think it's a really good point. And I think that maybe we should remind some children, there's these amazing proofs out there, and there's these amazing people who have studied mathematics for a long time, going back hundreds of years. And they probably spent day after day after day after day getting results that two plus two equal five. And they knew that somewhere in amongst it, they'd find two plus two equal four, but they persevered. And sometimes they passed on the batten and sometimes it's taken hundreds of years to be able to prove someone's theory or an idea that someone's had because that's exactly what they got time and time again. And it didn't quite make sense, but that curiosity, I think that has kept mathematicians going and reminding children, that's what mathematicians do. They're not human calculators. They get curious when they get that result and that's a good place to be. And you keep on looking. And like you said, with that problem that you've given them, yeah, you know the answer's in there somewhere. You know the answer's in the room somewhere.
So that answer will be there and it will keep being there and keep being there. And then it's worth the effort at times. And that is the role of mathematicians to have that tenacity and curiosity.
You can never finish. You can never finish. Yeah. It goes on. Yeah, absolutely.
So stay curious.
Yeah. Stay curious.
All right. Roger, thanks so much for joining us. I'd love to have you on again.
Yeah. Thank you.
Robin, I've loved it. Thanks for having me. Thank you very much.
Thank you for joining us on the School of School Podcast.