Upside down Fractions, The learning buzz, and more. In this episode, Andy, Emily and Adam are joined again by Researcher, Teacher and Cheshire and Wirral Maths Hub Lead, Andy Ash to discuss the role of the Maths textbook in the classroom. How does a textbook support both teacher and learner? How can schools tell the difference between a good or bad book?Plus, find out who gets frustrated when they hear of teachers downloading random content from a google search.

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The school of school podcast is presented by:

Andy was one of the first to bring maths mastery to the UK as the founder and CEO of the independent publisher: Maths — No Problem! Since then, he’s continued to create innovative education products as Chairman of Fig Leaf Group. He’s won more than a few awards, helped schools all over the world raise attainment levels, and continues to build an inclusive, supportive education community.

With nearly 20 years of education experience, Emily has a knack for creating wildly successful learning content. Her past work includes publishers like Oxford University Press, Pearson and Collins Education. Currently, you’ll find her dreaming and scheming in her role as Head of Publishing at Fig Leaf Group.

In a past life, Adam was a headteacher, and the first Primary Maths Specialist Leader in Education in the UK. He led the NW1 Maths Hub’s delivery of NCETM’s Professional Development Lead Support Programme before taking on his current role of Maths Subject Specialist at Maths — No Problem!

Andy is currently leading a 'Maths Hub' that provides professional development for teachers across a region in England. Alongside this, he leads mathematics across a Multi-academy Trust and is also undertaking a part-time PhD. The main driver of Andy's work is teaching for mastery and, in particular, how teachers can use a well-designed textbook to support this. Andy's research has focussed upon teachers as they adopt a mastery approach, looking at shifting believes and mindset, dialogic teaching and textbooks, and also the use of multiple-representations.

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Hi, I'm Andy Psarianos.

Hello. I'm Emily Guille-Marrett.

Hi, I'm Adam Gifford.

This, is The School of School podcast.

Welcome to The School of School podcast.

Are you an early years teacher struggling with lack of support for lesson planning? Foundations can help. Foundations is the new reception programme from Maths — No Problem!! It's a complete reception package with workbook journals, picture books and online teacher guides, all in one place. Visit mathsnoproblem.com/foundations today to learn more.

Hello and welcome back to The School of School podcast. And today we have Andy Ash who's joined us. Andy's a very experienced mathematician, well teacher of maths, and is doing a PhD currently at Cumbria. But before we dive in today's topic around textbooks, Andy, could you just tell us a little bit more about yourself?

Yeah, I'm a maths teacher, right? I'm not, I don't know. I guess I would call myself a mathematician, but not, not in the kind of strictest sense. But yeah, I'm a teacher of maths. My background is in primary teaching, but I have worked with secondary teachers as well. Yeah. I've just got a really keen interest in research and in particular research around mathematics education and how we can use mathematics education to help pupils to become problem solvers and critical thinkers.

So we, today want to take some of that knowledge and some of your thinking, and one of the topics that we really wanted to talk about was your thinking around the role of textbooks in classroom discussions. So, you know, I'm really interested. Do you want to kick us off and get the debate going?

So the reason that I thought this would be an interesting thing to talk about is because I'm currently actually in the middle of trying to finish writing up a research paper, actually, that is about this with my colleague Pete Boyd, who is also my PhD supervisor. In the work we've been doing with schools, primary schools, who are using a maths textbook, we've been working with a large group of schools for really the past six or seven years now. One of the things we started to become interested in is to what extent the textbook itself is a key player in classroom discussions, because, of course, the textbook is not a person it's not, you know, it is just a thing. It's a bunch of paper put together or it's online as well. But we were seeing, we were seeing something in our research that kind of showed us that actually there was something more to its role than just simply being a book that was there that teachers were using.

So if I just, if I talk a little bit about how, I guess some of the theory behind it, I'll try and keep it fairly simple. Won't get too geeky. But when, when we think about maths, what happens is this kind of big subject of mathematics that exists in the world, it gets re-contextualised and transformed into what I sometimes just refer to as school maths. And that's a really important thing. We've got to recognise that school maths, i.e. the maths that happens in schools, is not the same thing necessarily as mathematics in the real world. It's a version of mathematics. And that version of mathematics has been decided by the people who write the kind of formal curriculum in England, our national curriculum, or in the U.S., the common core standards and things like that. But then what happens is teachers sort of take that and they use materials such as a textbook, and they'll turn that into the curriculum that is enacted, it's acted out on a daily basis.

And then we could also think of it as kind of the experienced curriculum. So the curriculum that actually pupils experience. So, mathematics, as a subject, gets transformed through those different processes, if you like, into this school maths subject. One of the reasons I textbook is interesting is that, on the one hand, it's kind of formal in a sense, because it's been written by maths experts. So if you give it to a primary school teacher who, who isn't usually a maths expert or, or doesn't have like an advanced maths degree, usually, they have something there written by someone who, perhaps, is more of an authority or mathematics than they are. But equally, as a teacher, they're the person in the classroom using it every day to actually create this experienced maths curriculum. So, I guess, our research was really interested in what role does the textbook play in creating or helping create this kind of experienced school maths curriculum? Hopefully that made sense. There's a lot of terminology there, but

Yeah, it does make sense. So that's really interesting, especially for me, because I just happened to be writing a chapter on fractions for a book as we speak. So okay. Is it really is kind of cool. So, so what did you find?

Yeah, so one of the things we found was that, or I guess, say we find it, our research is not research that's designed to kind of find something to say, "This is definitely the truth," I guess it's to contribute to a bit of a theory about how we think about it. So, what we're theorising is that the maths textbook and the materials that are used in the classroom are actually a part of the dialogue. So, traditionally or typically we might think of dialogue as being between just people. So, in the classroom situation between pupils and other pupils, and pupils and the teacher. And what we're theorising is that, and we're not the first people to do this, there are other people that have looked at science teaching, for example, using this idea. What we're theorising is that actually the materials used, the physical materials, which include the textbook, actually become a part of that dialogue.

And we can't separate the dialogue that's happening from those physical materials, because without them actually, it would be a very different experience. And therefore the learning that's happening in the kind of process of creating meaning it would be different. So, I'll give some real actual examples. So, one of the things we found is that the teachers, when they use the textbook, in the textbook, there are these characters, I'm sure you know. So there are characters that kind of suggest mathematical ideas and thoughts. What we found when we actually video recorded a couple of lessons and we've done analysis of the video recording. And we used that alongside interviews with teachers. We found that in the videos, the teacher refers to the characters that are textbook characters in the first person as if they're real pupils. And so if you watched a small clip of that video and the teacher saying something like, oh, let's think about Hannah's idea. If you didn't know that they were using a textbook and Hannah was a textbook character, you would think that Hannah was just another pupil in the class. And so this is the kind of thing we are finding, that the textbook was becoming a voice in that classroom dialogue. And that was kind of one way in which it was happening. The teacher was sort of referring to things in the textbook as another person's ideas, even though that person wasn't physically present.

That's fascinating. That's really fascinating. So it's like kind of all consumed into the classroom conversation for the teacher and the kids that's fa- yeah. Wow.

Yeah. It, it's interesting because this is something that, you know, I know Adam and I have talked about this recently, and Emily as well is, you know, the narrative in the book, right? So there's all kinds of layers within the, with every lesson, you know, that needs to be considered and you know, those, those prompts. So if you look at a home versus a textbook for a classroom, they're very different products. They need a different narrative, right? Because the classroom product is being used by a professional and is being in, is in a controlled environment while the whole product doesn't have that. So it needs to have a different narrative, right?

Yeah.

Mm-hmm (affirmative)

You know, and the, I guess what's interesting to me because, you know, like, look, this is, I mean, this is what we eat, leave live, breathe every day in our world.

So for us, from a very selfish point of view. From the inside, the people who make these things right? Is the wrestling match that we have between content specialists versus editors versus layout people versus, you know, designers, people, illustrators, people who make these characters that decide the positions that they stand in. Right? All those things, the little nuances that go into this, and, you know, when, when you draw something that needs to be measured, whether it has a round corner or square corner, or, you know, is it, could it be measured diagonally? And is it going to be different if you measure it straight? And all those kinds of things, you know? Or, or the perspective, right? And all these things that we wrestle with all the time. I think that people sometimes don't really recognise how much effort goes into like every little detail in these books, right?

What we're sort of saying, well, one of the things we're saying, from our research is that kind of stuff is so important because essentially what we're saying is the textbook isn't just a book that teachers use. It's not like a passive thing. Actually. It's a very active agent in the creation of the school maths curriculum. So therefore, what part of our sort of theory is that actually, when used in this way, the textbook, in a way, becomes a voice of mathematics. It is part of turning mathematics in the real world into school maths. And so, all of those little decisions about what's in the textbook, the kind of problems, how many problems, all of those details you're talking about there, Andy. That they are contributing to what pupil's experiences being mathematics when a teacher is using a textbook and therefore those details matter. They matter hugely, don't they? And they matter, in a sense, as much as the way in which the teacher is teaching. And that's one of the things we're finding. That's one of the things we're arguing. Is that we shouldn't downplay the role of materials in dialogue, even though it's a book sitting there, it playing a very active role in the dialogue.

Yeah. And people always knew that for literature, right?Sorry Emily, just to jump in. They always knew that for literature, but they forget about it for other topics, yeah? Sorry, Emily, what were you going to say?

No, no, I'm I think we both got a little animated there, Andy, because I'm just really curious now, with my educational publisher hat on. So the significance of the textbook, there is the potential for that textbook, from what you're looking at so far, to have kind of quite significant role to play. So presumably, text books that are not well created, orchestrated, considered, or content that is just sort of randomly delivered, does, have you, have you had a chance yet to investigate whether that there is a negative side to things that aren't good quality?

Yeah. So, so we haven't done that, but I think, think it's been done before because one of the reasons why textbooks traditionally have had a bad name, in terms of maths teaching, is because when you look at the traditional textbook in England and especially in the U.S., they tend to be very big and full of just lots of drill and practise type tasks. So they, they are, what, in a sense, what we're saying is that those textbooks, that perhaps were designed in a particular way to reinforce mathematics as the idea of memorization and practise, they kind of shaped the mathematical experiences of many, many pupils over many years. And no wonder they got a bad name for themselves. Go on Emily.

No, I was going to say, I was putting my finger up. Cause like, I was one of those.

Yeah. Yeah. That's it.

It didn't work. It didn't work well for me.

Yeah. So, so I think, you know, in a sense, yeah, the content of the textbook is contributing to school mathematics and the experience curriculum of school mathematics. And so, you know, that is what I remember, you know, going back to 2014, maybe it was? When I first encountered the maths, the problem textbooks. And there was a big kind of, I guess, this friction inside of me, because I had been trained to be a teacher and I'd been trained that maths textbook was, were bad. And that, you know, teaching from a textbook, that's a bad thing. And, I obviously since then have learned about all this stuff. And, and I think that actually what we realise is that it's not that textbooks are bad, it's just that certain textbooks have been bad in the past.

But what, what amazes me and I've got skin in the game, right? So, so I contribute to content that goes into the textbooks. And, but what gets me is that, is that I've worked in schools and I've worked with colleagues that, whilst textbooks were bad, and, and I've got to admit, there are still schools that to this day and teachers to this day that will ascribe to that, yet the internet and just Googling, I don't know, dividing a fraction by a fraction is somehow okay. And that quality assurance is okay if it comes from the internet, but God forbid we use a textbook, you know? And understanding, just as Andy said, I mean, we do truth with that, the amount of thought and effort and the number of people and in all of those things that go into a product because A, they are so important, right? They, it is so important that these children get a chance to learn.

Yeah.

Well, but I think the other part that I find is sometimes missing in this is what the textbook can teach the teachers. Like I don't think I, my, my sort of learning trajectory was so steep after the first training with the textbooks for me. I saw stuff that I never saw in them before. So I could be on a desert island with a textbook for 20 years and I'd see some stuff. But then you start to open my eyes to it, you start to train me how to use it, and all of a sudden I became a better teacher, not just in maths, by the way. But across the board. And I think these are things that I just, I don't know why, I don't know why we are some issues on Twitter can dominate some bits of education and everyone wants to be an expert in this. Yet, there are some of these fundamental things that we know from evidence and particularly in other countries that have used them for so long. Well, "why are we still got, why are we still got people that, oh, no, I wouldn't teach from a textbook. No, hang on. Just let me Google this. What is it?"

Yeah. I'll teach from Google.

Our prime numbers.

I'll teach from Google instead. So...

I'll just do that and print it off and away we go and that's better go. I'm not going near those textbooks that I get out of it. No, no. I've been doing this for years.

So Adam, when you were talking, I looked up dividing a fraction by a fraction in Google, by the way. So, and I took..

You get print out, is it download? You

Get KFC. Did you get KFC?

I got, I got a downloadable print out. And I, so, and I went to the page where that downloadable printout was and there's, the instructions are there. So for our readers, in case you weren't sure, I'll read out the instructions for you. Right? So, step one. So first off, it says "there are three simple steps to divide fractions". So, dividing fractions is easy, right? "Turn the second fraction," this is step one, "turn the second fraction," in brackets, "the one you want to divide by, upside down". Okay? "This is now a reciprocal."

Good.

Right? Step two.

At this point, I'm imagining like one half turned upside down. That's an upside down two and a line.

Exactly. Then step two, "multiply the first fraction by that reciprocal." Step three, "simplify the fraction, if needed."

If I, if I'm multiplying, do I turn the first one upside down? So when I'm multiplying, can flip that one round or what does it work that way? Is it like a sort of, you know, does it work like that?

I think so one, one of the ways in which I would describe, you know, one of the things which to say in this paper, is that in creating the experience curriculum of school maths, it's a bit like the teacher is collaborating with their curriculum materials and the pupils to create the mathematical meaning that happens in a lesson. And, and so a way of thinking about it might be, well, who do you want to collaborate with as a teacher? You know, do you want to collaborate with the random bunch of stuff off the internet that you've never, you don't know where it's come from or how much time has been put into it? Or do you want to collaborate with something that, you know, has had significant time and research and finance put into it in order to ensure that it's of a high quality?

And, and I don't think it's a surprise is that when you look at other high performing countries, not that I'm trying to suggest we policy borrow, but when you look at Shanghai or Singapore or Japan, they all have textbooks that have been being developed for years and years and loads of time and effort and research is put into them because they know there, there are actually typical ways that pupils respond to certain things in maths. We've learnt that. So why not? Why are we going to try and recreate the wheel when, when we know that there are others who have done that for us?

So Andy, how? So here's the problem, how do you tell the difference between something that's good and something that's not good? Because you know, the reality is like, unless you go, unless you spend hours and hours and hours and hours crawling through it, you're, you're not going to be able to tell a difference. Most, you know, I mean, there's the subtleties are... So how do you tell a difference?

So I would always say the starting point has got to be forget any of the books you're looking at or any of the resources and think about what you're trying to achieve from your maths curriculum. Because look, if I, if all I want to achieve with my maths curriculum is pupils who are really quick at their times tables and who can do loads of arithmetic, then that's going to lead me to think this resource is going to be, and that one's going to waste loads of my time, because that's the one's going to get me to do all this problem solving and I don't want to do that. So if you know what you're trying to achieve with your curriculum, that's the first step, because then you can start to look for curriculum materials that are going to help you achieve that. So I'll give you a real example in our school.

We know that we want our maths curriculum to achieve certain outcomes for pupils and they relate to pupils, being able to think meta cognitively to engage in kind of critical debate and dialogue with one another. We want them to be able to become, to problem solve and to think independently. So when we are looking for a textbook or curriculum materials to use, to inform our maths curriculum, we're looking for something that supports that. So that I think is the first really, really important step that actually in my experience most schools skip.

Yeah.

Yeah.

I think they skip that. From there, it becomes harder I think. And then it becomes having some knowledge of the research into, for example, effective representation of mathematical concepts. So we know a lot about representation. So, you'd want to either do some research yourself about that or, or find out from other people about what, what do we know about effective representation of mathematics and then other things like variation theory, which are, are useful to look for in a textbook.

So the first thing is easier. I think, well, not easier, but schools can do that. They can think about, what do we want to achieve with our maths curriculum? The second step, which is knowing a bit of the mathematics related research and know what to look for in a textbook. That's where I think, in a sense, that's why we have developing in England, lots of mathematics, professional development through maths hubs. So we can help schools make those choices in an informed way, rather than just jumping at the first thing that seems popular. It's not an easy choice.

No, it's, I'm sure it's not for a school. I mean, I guess, you know, the motivation for doing like, first of all, I guess people often get fixated on particular items or, or aspects of any educational material that a school needs to invest in. Right? And, and they don't necessarily, like you say, think about, or it may not be intuitively thinking about, what they really should be thinking about, which is what they're trying to achieve with this stuff and how they're going to use it. What are some of the motivations? Like, I guess, not being inside the school and not really knowing the pressures that schools, you know, like I know I'm obviously in a theoretical way, but you know, not, not, not having that weight of the world on my own shoulders.

Can I jump...

What are the things... I haven't even asked my question Adam!

I think I know where you're going. Keep going. What are the things going [crosstalk 00:21:01].

He's a mind reader out

He's read your... He's read your mind. Yeah.

What are the things that motivate schools when they're making those decisions? Sorry, Adam,

I didn't know it. I didn't know it.

No, you didn't know it.

Is that what you were going to say?

No, Because I know, I know from, I know from our school, because I know at the time when we made the decision to get involved with it, we were that the results just come in, we were top, top five percent for maths results. So why change it? Right? Why, why do it? I had a really good relationship with the, with the head of maths at secondary school that a lot of our children went to again, very high performing school nationally. And we would often have discussions around it. And there were two things that came up often. The first one was the participation rates in secondary schools dropped off a cliff with children who at primary school level, it should have suggested that they, they enjoyed the subject and could do well at it and find success. Right? So, so why, why was it that these children who were, who were, you know, displaying traits of strong mathematical understanding and being able to access new learning and all those things that excite people, right? Why is it that they were stopping? I think the second thing was, is that is the ability to cope at secondary school started to drop off.

Yeah.

And when, and when we started to look at those two things, we started to realise that we could almost, and this is no slight on the, on the results that came in. But what we realised is that there was a lot of memorization involved. There was, there was tutoring involved and that paid dividends that SATs, but what it didn't do was it didn't set children up for the long term. And, we couldn't, I couldn't, as a head at the time, I couldn't morally live with myself knowing that these children being shortchanged laid down the line, that we were laying the foundations for secondary school. Surely that's, that's, that's what we do. And secondary school for the next journey and the choices that they make.

And, and I think that's a thing that, really, it comes back to Andy's point is what's the point in doing it. If the point of education is getting SATs results ,that can make you sleep easy at night, because obviously aren't going to knock on your door, yeah, there's ways around that, right? We can do stuff around that. But if, if the point of education is trying to say right, all I wanted was I wanted children to be able to go to the next stage and have as many options open to them as possible, without those barriers in the way. And for that, you've got to look ahead and you've got to say, well, how do we do that?

Yeah. And I completely agree Adam. And if I take it back to the thing I was saying about the beginning, about the re-contextualization of maths, so we can even take this back to the question and we won't get into it right now, but if you ask the question, well, what is maths? What is mathematics? That is a debated thing. People don't necessarily agree on the answer to that question. So, some people would say, well, maths, it's just a, it's like a game. It's a big game. You've got to learn the rules of the game. And if you learn the rules, you can play the game. Right. Whereas other people might say, no, actually maths is, it's actually a subject that human beings created, and we created it to help us communicate more effectively about things we find in the world. But because we created it and we are humans, it's a fallible thing.

So it's constantly developing and changing. And it's a, it's a human activity. These are just two of the sort of philosophical views, there are others. But, it goes back to that. So, I would certainly say that maths, is the second one of those, maths is a human thing, that human beings created it. And we created it to use it for a purpose and therefore a way in which we teach it should be reflective of that, of that. It should teach pupils, that to do maths is to talk about maths and communicate maths and learn, solve problems together. So therefore I want a textbook that's going to help me do that and help me help make school maths more reflective of that. So, when you think, when you look, look at the different approaches to textbook analysis, for example, one of the things you can look at are if you count all of the different tasks that are in a textbook, you can, then you can then grade each type of task, depending on how the type of question.

So you can have closed questions. So it's just like simple, be requiring an answer. You can have questions that are requiring an answer, but also justification reasoning. Or, you can have open-ended more inquiry based questions. So that's another way of looking at a textbook is, well, how many opportunities within this textbook are there for pupils to do more open-ended inquiry kind of mathematics rather than just closed kind of straightforward questions? So, yeah, I think that's a really important thing to consider is that what I'm not expecting all, all head teachers or maths leads in schools to be thinking about the answer to, "What is mathematics?" But, I do think we should think about it a bit more than we do. I think we don't really spend time thinking about the subject we teach. We just sort of assume we know what it is.

Yeah. Adam, you should wrap this up. As a, put your head teacher, I know it's been a little while since you were a head teacher, but put your head teacher hat back on.

Yeah.

Right? You're running a school.

Yeah.

How, how do you build that culture in the school to get that momentum, to see it through?

I just think the teachers need to feel learning. I think that when anyone learns anything, you get a buzz from it. Right? And I think that with primary school mathematics, we've all been to primary school. Right? And we can probably do most of it or would like to think so. But actually understanding it at the level that we are asking children to, and we are granting them access to, with the support of the textbook. I think that that's something the head shift that has to happen is as a teacher. And I speak about this to this day, who's been so fortunate to be trained by so many good people. So I'm lucky that I've had a lot of time to think about maths. I still get really excited about, oh, can it be done a different way?

You know, can we do this differently? Is there something else? I, you know, I still get a buzz of walking into it. I don't care any year group. And a child says to me, oh, look at this, I've done this. And I wouldn't have come up with that. You know, I want to see that happen in my classroom. And I think that if we can get teachers like really excited about learning for themselves and realise that learning can happen from the 30 kids in front of them, you know, I can learn from them. And isn't it exciting that I'm going to be better at this tomorrow than I am today? And if I think you take that attitude, that if I can just be a little bit better tomorrow than I am today, then I think that you're onto a winner because you start to see more, you see the world differently, all of those sorts of things. And I think that's having that feeling of learning. I don't know, I know for me anyway, it's so exciting, you know, and that should be lifelong, not just for 45 minute, you know, lesson or something. And I think that's the key.

Andy, thanks so much for joining us today.

Absolutely.

As real buzz talking to you.

No, thank you for having me.

Thanks so much, Andy.

Goodbye, everyone.

Thank you for joining us on The School of School podcast.

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