Music theory, Framing the focus, and more. In this mountainously longer episode we have two Andes! - Terrible pun, apologies! Andy, Emily and Adam are joined by Researcher, Teacher and Cheshire and Wirral Maths Hub Lead, Andy Ash to discuss the false argument between cognitive load theory and teaching through problem solving. What are the big misconceptions here? Plus, Andy Ash shares two pointers that’ll make a huge difference for maths teachers.
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Hi, I'm Andy Psarianos.
Hello, I'm Emily Guillie-Marrett.
Hi, I'm Adam Gifford.
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Welcome to another episode. We very fortunate to have Andy Ash here. Andy's been associated with Maths Hubs, University at Cambria, you were in a teaching school, Alliance, I believe, and you've done plenty of research around mathematics. And one of the things that I know or I believe is an interest of yours and I'm really intrigued to about, because I'm reading about this all the time and I really want to pick your brains and listen about, is around cognitive load theory. But just to start with, can you maybe just introduce yourself as well. Because I've only just touched on it, and so just maybe introduce yourself, the bits that I've missed. And then thinking about yeah that whole... Well, there's plenty out there about cognitive load theory.
Yeah. Okay. Yeah, sure. It's great to be here. So yeah, my background is, I guess at heart I'm a primary school teacher, and in fact, I first started working in reception, so teaching very young children. But since then, I've worked with both primary and secondary school teachers to support with maths teaching and I have a keen interest in math research in particular and I'm studying, doing a part-time PhD as well where I'm looking at how teachers teach fractions. So yeah, that's kind of me in a nutshell.
Is that because you don't have enough to do Andy?
Yeah. Well, I've got a bit bored. I just decided. No, but, yeah, it keeps me interested and it keeps me busy.
And Andy, you did some work with Pete Boyd as well, I remember right? You did some research. What was that all about?
Yeah, so that's right. So actually, that's kind of how I ended up doing a PhD because Pete Boyd persuaded me in a nice way. So before I started my PhD, Pete and I were doing some research around how teachers were using the maths problem textbooks, particularly focusing in on how their kind of beliefs about maths changing had shifted, but also how their classroom practise had shifted particularly in relation to moving away from grouping by prior attainment, moving towards mixed prior attainment grouping amongst other things. Yeah, I think that was published in 2019. So it feels quite a while ago.
It's not that long ago, but it does sound like a long time ago, but that's like just before COVID really, right?
But I heard it was one of the most downloaded research papers on education, right? So, well done.
It came at a time where there wasn't very much published about teaching for mastery in England, but equally when teaching from mastery had been, I guess it had become a really popular thing and still is a really popular thing in our country. So I think it came at the right time and yeah, I think at one point it had one of the highest numbers of downloads for the journal that it was published in, yeah.
All right. So cognitive load theory, what's it all about? Why don't you give... Because some members of our audience will have no idea what that means, so maybe just give us a quick rundown what that means, yeah?
Yeah. Okay. I'll do my best. So cognitive load theory is a theory about how people learn. It is a small part of what is sort of known more widely as cognitive research. So it is only a small part of cognitive research, but it seems to have gained a lot of traction and yeah, teachers are very, very interested in it. Essentially, I think the best way to describe, for me anyway, the best way to describe cognitive load theory is that it's a theory that's related to memory, and it basically kind of suggests that a human being in our brain, we have a long term memory, which is kind of doesn't seem to have very many limits and that's the stuff we know or knowledge is stored and we retrieve that when we need to use it at particular moments in time.
But in order for things to enter our long term memory, they have to be processed through our working memory or our short term memory. And this is kind of where the problem occurs because quite a bit of research has shown that our short term memory is not very good at handling many things. So it can only handle a small number of things and the amount of things that can handle at any one time will vary depending on the complexity of those things as well.
So in mathematics, where often we're trying to teach pupils quite complex things, the amount they can handle at any one time is quite limited. So cognitive load theory is essentially the theory that says when your working memory or your short term memory is overloaded, then actually that leads to nothing being learned and this kind of idea of overload and therefore nothing entering our long term memory.
So as it stands, I think it's a really interesting theory, and I think it's a really valuable theory for teachers to be aware of. Yeah, so in a nutshell, I think for me, hopefully that has given a very brief explanation of quite a complex thing. I don't know if you've got any things that you think need clarifying there.
The question that I've got is that I see cognitive load theory being written about a lot and, and you try to read it. And do you know, you may be able to summarise this so I'm just going to ask the question, do you know some of the big misconceptions around it? Because I read stuff and there's things that are on Twitter. So just people throw things out there or everywhere.
Everything on Twitter is true Adam. Everything on Twitter is true.
Well, I know this. I know and so this is where I need Andy's help because that's my attitude that I take to Twitter. Is that if it's written down and it's on the internet, surely it's true. But do you know what I mean Andy? Are there some big misconceptions that you kind of see with your understanding of it?
I think the problems actually occur when we are trying to apply it to real life practise in the classroom. So the first thing that I think is really important to be aware of with most kind of educational writing and research is that it's a theory. So it's a theory that has got a significant amount of empirical evidence that backs it up. But not very much of that evidence is actually from classroom based research. So classroom based research is really hard to do. It's kind of hard to prove anything in the classroom because it's such a complex environment.
So what happens is, people read the research on cognitive load theory and they come to conclusions about what that means or doesn't mean for teaching and I think that's where the misconceptions happen. So for example, a common kind of rhetoric at the moment is that our working memory is fragile, that term is quite often used to describe it. That it's a fragile thing, and therefore teachers have to manage that very, very carefully for pupils.
So what actually happened in 2007, a really famous paper, it wasn't a research paper, it was kind of an opinion piece that was published in an academic journal. It came out and I can't remember the exact title, but it was something along the lines of, "Why minimally guided instruction techniques don't work." And it was based on this side of cognitive load theory and the sort of argument being made and is still made often, is that because of cognitive load theory, we should only teach pupils using this idea of direct instruction. We need to tell them exactly what they have to do and guide them through everything in small steps, rather than at times giving them a problem and allowing them to kind of try and work it out for themselves.
So I think this is probably one of the biggest misconceptions in relation to mathematics, that because of cognitive load theory, we always have to teach things using certain direct instruction techniques. Yeah, for me, I think that's probably personally one of the biggest things out there.
What is your position on direct instruction and when it comes to teaching them? Come on, Andy.
So I think that actually, as an idea of teaching, it's quite widely misunderstood, or maybe just I misunderstand, I don't know. But, so often direct instruction is kind of construed to be this idea that the teacher sort of stands there at the front and they tell the pupils, "Right, we are going to learn how to do this.? Let's say it's long division. And then the teacher models and shows, "This is how you do long division. First of all, you do this, then you do this." And I guess sometimes we might refer to that as kind of traditional teaching, although I'm not sure if that's the best way to refer to it.
So people think that that is the only way to directly and explicitly teach pupils some knowledge or some particular thing in maths. But actually I believe that you can use some of the principles and you can use this idea of teaching through problem solving or inquiry based teaching whilst being quite explicit and direct in your instruction at the same time.
So I guess my stance is that teachers do need to be quite direct and explicit in the mathematics that they're trying to teach pupils, but that doesn't actually mean you stand at the front of every lesson and tell pupils exactly what to do and guide them in tiny sort of small steps. There are other ways to do it, I guess, and maybe that's what we should start to get into.
Help me out here on one thing. One of the benefits as I understand it, and I'm talking very basic level here, but one of the benefits as I understand it, might be that if you can take some of the distraction away for the child, like there are certain patterns within the teaching that they're familiar with, so direct instruction might be seen as a benefit. The kids know what's going to happen, the teacher's going to get out. There are certain phrases, certain things that are going to happen so that learning can take place because their working memory can focus on the element that needs to be done. And in that sense, things like direct instruction and certain things can be beneficial.
It's almost around classroom behaviours, you might say. I guess there's a little bit there like take away anything around the edge. But you can still have moments of stickiness where you are trying to get into the problem solving. There needs to be that space because that's where they learn. I guess for me, I kind of hear all these different things and you can be easily persuaded when you're not an expert to kind of go, "Ooh, that's intriguing." So does any of that resonate?
Yeah, absolutely. So there are lots of very specific things that are there within the research around cognitive load theory. For example, there's an idea of dual coding, which is that we can process images and language at the same time. So by using a combination of language and images together, we can enhance how much peoples can process any one time. So in mathematics, that's really powerful because we use pictorial representations, concrete equipment, as well as language. So yeah. But equally, like you're saying, trying to reduce distractions and help pupils to focus on the key bit of mathematics or mathematical structure that we want them to think about.
So an analogy that we have used before and I find quite useful, we sort of describe what teachers can do as framing. So if you imagine a picture frame, and inside the picture frame, the teacher has thought very carefully about what they want the pupils to think about in the lesson. So they know as a teacher, as the kind of expert, they know, right, this is the bits of mathematics that I want them to think about or considering the lesson today. So that will have been decided through their careful planning perhaps using a resource like the textbook and their own sort of professional knowledge of the pupils as well. And they'll have thought very carefully about that very specific bit of mathematics they want the pupils to be thinking about it in their lesson.
And then what they do is, they use various tools in their toolbox if you like as a teacher, to frame the pupils' attention so that they've got the best chance of pupils focusing on that stuff. What they're not doing is simply telling pupils, this is what you have to do and sort of joining the dots in a very kind of one dimensional way. What they're doing is creating a space for pupils to be able to explore independently and they're framing that carefully.
So I'll give you some examples. Some of the things that teachers commonly use for framing learning is things like questioning. So the kind of questions that they plan to ask can lead pupils to thinking about specific aspects of the mathematics, or perhaps not thinking about specific aspects. So this make it real when a pupil puts their hand up in class and offers a suggestion and as a teacher, because you've got a certain level of knowledge of the mathematics, you know that the pupil's suggestion is going to take the whole lesson off on a tangent, you can kind of go, "Oh yeah, that's really interesting" and then just move on because you can control how much time is spent thinking about things and considering things.
So that's how I think cognitive theory can be used by teachers to think very carefully about how they frame the kind of precise mathematical focus of a lesson. Other things like choosing what representations are used, choosing what manipulatives are used potentially, even down to being the person who decides how much time is spent on a problem or deciding where the pupils are sitting potentially. They're making those decisions and all of those contribute to trying to help pupils focus on the main bit of mathematics rather than something else. Yeah.
So one of the things that really gets me about educators in general and I think it's true probably in every profession, is that every once in a while something comes out, cognitive load theory is one of them, and it rises to the top. This is the next big thing. And then everybody decides that they need to change everything that they've ever learned or known about education based on this new thing and it becomes a pendulum swing and it's like now... So the direct instruction camp will be like, "Well, this is just clear evidence that we have to use this" and of course the other camp, the sort of inquiry based learning camp will say, "Well, this is clear evidence" and it's because interpret the rules their own way.
But the really annoying bit of it is this polarisation. It's like the evidence is just getting used to justify people's position and people don't really actually look at it with in its own merit. They just look at it like, "How can I use this to justify what I already believe, what I'm already vested in?" And then it pushes them further and further. It's kind of like American politics and it becomes dangerous. And the reality is just like, the answer is in the middle and it changes with circumstances and it's never that straightforward.
While you guys were talking, I was trying to think, "Okay, what's a good analogy for this to try to explain this to people?" And I think for me, the best analogy I can come up with is trying to teach someone to be a musician. So musician just doesn't mean you can read sheet music and play the piano. Because to me, that's not a musician, that's a robot, right? A computer can do that. So now to be a musician, you need to have a lot of different skills, right? You need to understand lots of music theory is complex, it's very mathematical. There's all kinds of crazy rules that aren't counterintuitive.
Okay. So if you want someone to be, let's say a famous composer, you want to train somebody to be a famous composer, how do you do that? Right? And there could be an argument and this has commonly been the argument, "Okay, well, you've got to teach them all the mechanics of music." So they need to learn all the scales on the piano and then you're going to teach them all the different intervals and then you're going to teach them all the... Because only it's limited amount. Then you're going to teach them all the triads and how they're formed, and what they make and all this.
Okay. Now, here you go. Now you've got all this. "Circle the fifths, whatever it is, these are major chords, these are minor chords, these are augmented, these are diminished, blah, blah, blah, blah. Here you go. Here's all the facts. Imagine that's mathematics, right? Now go do something useful with that.
And of course it's nonsense. This is way too much disjointed stuff-
That you can't make sense of, right?
The magic happens when you listen to music and you try to play it.
Yeah. I have to say, I'm just marvelling at your knowledge of playing the piano, Andy.
Hey, I don't play the piano.
You know a lot about it. Yeah, and do you know what? It's funny that you use music as the analogy. When you start to look at the sort of philosophy of mathematics, music is actually often used as a good analogy. Because as you say, you can do all the technical stuff of music, but actually, that's not where the magic happens. The magic happens when you get someone who really understands how to play the piece and how it should sound and, yeah.
Yeah. And it's the relationships between all those things that are important, not the things in isolation. So if you teach people a bunch of random facts, they don't necessarily mean anything to them. It's how they relate to each other that's important, right? And this is going right back to what is it? 1973, 1979, I can't remember. Richards Kemp instrumental understanding versus relational understanding, right? He's kind of the one who wrote it really clearly. You can teach someone via instruction to perform mechanical tasks to get correct answers but it doesn't mean they have any idea whatsoever what they're doing or why they're doing it. Just like music, right?. "Here's some sheet music, play it."
But I think the danger, and you touched on it. You know what you're saying about the theories that rise to the top or the topics that rise to the top, what I find a lot of, is we try to become experts in these fields in five minutes. So if I can just, I'm a teacher I'm tight for time-
Yeah, exactly. Hey, listen, don't give away my source there, Andy all right? Not many people know that's the go to become an expert in five minutes. But I think it's there. I think that there's a sense of, if I don't know about cognitive load theory by tomorrow, then I'm doing my kids a disservice. And like you're saying Canadian, Andy, is the idea around understanding there is so much involved in teaching that if we know we are doing a good job with our children, it doesn't mean that because we don't know about cognitive load theory, that tomorrow I'm going to become a really terrible teacher. And to give ourselves time to understand it. Because I just think the danger is, is that of course there's that idea and I don't know where it comes...
Well, I've got a lot of opinions on where it comes from, but I think that when something like this happens because of the high profile nature, and just to, to probably phrase it poorly, the next big thing, I need to know about it tomorrow, otherwise like I said, that my teaching skills are pretty average at best and I better quickly just read some snippets real quick and then implement them with the kids tomorrow. And I just think this is crackers.
Well, okay. Let's just say Adam, that people are listening and they do want to implement something tomorrow. So I want to know from Andy Ash, if you were listening and there was something that you were like, "Actually I've done a lot of research on this and I could tell you one or two things that I do think you could do tomorrow that would make a difference," what would they be?
Okay. Yeah, I think I could pick two things and after I think I want to go just make a really important point which I think we've been touching upon there. But yeah, okay. Two things. The first thing I would say, is that when you're teaching a maths lesson, you've got to be really crystal clear in your head about what is the main thing I want pupils to think about. There's no use knowing about cognitive load theory, unless you are clear as the teacher about, well, what is it I actually want pupils to think about? And it's not enough just to say, "Well, today I want them to learn how to divide a fraction by another fraction." You've got to be really, really precise and specific in knowing how you want them to do that. What kind of representations do you want them to use to be able to do that? In what ways do you want them to be able to do that?
So I would say that if I could pick even just one thing, that is the key. Make sure you know very, very precisely what it is you want your pupils to think about and what you'll see in here when you see them doing that. Because if you know that, you can then decide on the tools you're going to use to try and frame that and encourage them to think about it and encourage them to actually focus upon it.
The second one, I think that I personally find very powerful, is actually something where there is a bit of a decent amount of research on, I think in particular, in relation to mathematics. And that's the use of what I call worked examples. So this idea of pupils having access to kind of model solutions. So the first thing about this is that I don't believe that we should present them to pupils before they've had a chance to think for themselves. I think there is a time in the lesson where they should have access to them. And that is actually after they've had a chance to think through an idea or a concept for themselves.
So once they've had a chance to explore something, try and figure it out by working collaboratively and using equipment, later in the lesson, we can show them one or two model examples that allow them to reflect their thinking off if you like. So they can go, "Oh yeah, that's kind of, of similar to what I did" or, "Oh, that's different from what I did." And then we can kind of see in the research that if teachers use worked examples regularly, it actually does two things. One of them is it can help reduce this side of cognitive load because they're able to see how someone else has solved it rather than always having to think about how they've solved it. The other thing I guess unrelated, is that they have been shown to help reduce maths anxiety as well. So that's another pro. So that'd be the two things I think. The first one being the most important, be really clear on the main kind of mathematical point.
And I think Andy, you're touching on, on some really important stuff. Like don't get obsessed with all these sort of high volitant terms like cognitive load theory. I mean, it sounds really impressive, but it's pretty much common sense. Don't give them too much stuff at once. Basically that's what you're saying, right? Small chunks, one thing at a time, right? All those things are effectively just ways of expressing what we already kind of know anyway.
You know that if you give them too much information, they're not going to remember anything. I mean, it's the same with me. If my wife says to me, "I need you to get butter, milk, cheese, eggs, Bok Choy, a couple of this, some..." Whatever it is, right? Just some random stuff to try. What's the chance that I'm going to come back with them, right? Pretty slim, right? And that's just the way it works. Everybody knows that. We already kind of intuitively know that.
But the planning, the teacher are planning is probably the most critical point for any lesson. And it's just like, "Okay, what am I going to teach the kids?" Is an interesting question. But the more important question is, what do I want them to learn, right?
Because, and I think this is a trap that teachers fall into, they make it so teacher centric.
They refer everything back to their performance in the classroom and not necessarily what the children are going to get out of that performance, right? So just framing things in that different light. And it's exactly what you said Andy. So what do I want them to learn? And how am I going to know whether or not they actually learned it at the end of the day?
What does success look like? And then I think the other thing that a lot of teachers, and probably especially true in mathematics, is really sitting down and reflecting, "Okay, what is the construct? What are the things that are important about this particular lesson?" So let's say you're teaching, I don't know, let's pick something from primary curriculum. Let's say it's area, right? What is the big idea behind area? Ask yourself that question before you get into the lesson. Because all too often, people just don't understand what's area. Well, it's breath times width, right? That's what area is to most people.
And that's like, no, no, no, no. That's not what area is. Area is measuring something, and the unit of measure is squares. Just get that in your head before you go in the lesson, because if you can get the kids to understand that going in or going out of the lesson, if you know that that's what you need to transfer as far as information goes, then you're much better suited to say, "Okay, I'm not going to choose circles," right. Or whatever, some odd shaped hexagons as manipulatives for teaching area in the first lesson. I'm going to use squares because that's what we're measuring, right? And now I can count the squares. That's a really simplistic example, but as a teacher, if you do those things, you're much more likely to get the outcomes that you hope for.
Yeah. In terms of something to take away, a really simple way of approaching this in planning we often use, is just to say to teachers, "Well, when you're planning, of course know the precise mathematical thing you want." And as you say, Andy know where it fits into the bigger picture of mathematics but then walk in the shoes of the learner. Imagine you are the learners that you are going to be teaching and try and think about how they might respond to the different questions you're going to put for them. And by doing that, we can actually learn much more about how pupils might be thinking about it rather than how we are thinking about it as an adult expert. So yeah, absolutely.
That's right. And that empathy, that is really a huge skill that teachers need to be need to have, right? And often, so when we do our professional development, the training that Maths — No Problem!! has done over the years, the first stage in professional development is always the teacher as a learner, right? Is you're going to throw them down at the table, and all of a sudden they become the pupil. Because why? Because then all of a sudden, you can relate to being the pupil. When someone challenges you, how does that feel, right? What happens when somebody says this thing to you? And that empathy, that being able to look at it through the child's lens is super important if you want to transfer that knowledge.
One of the things I think is really important to kind of say when we're talking about cognitive load theory, and I think it relates to this idea of it being a popular thing and things to learn, is that as I said right at the beginning, cognitive load theory is a very small part of even just cognitive research. And cognitive research is one area of research in a huge world of research. So one of the things I will often say is that, you could teach the perfect lesson from a cognitive load perspective and pupils can learn nothing, because if they aren't motivated, then it doesn't make a difference. So we've got to consider it alongside everything else.
So actually, could teach a lesson which uses all the little tricks that we kind of know from research into cognitive load theory work. But if I haven't taught a lesson that pupils are really intrinsically motivated to want to learn about, well, it's not going to have as much impact. So that's why I think we've got to be really careful with how we kind of just pick and choose things that we use. We've got to know how they fit into the bigger picture. And that's actually why I think this idea of teaching through problem solving is really important when you kind of do a bit more reading around it.
One of the rationales given for why teaching through problem solving is important, is that it helps they call in Japan, whole character development. So they would say that we teach... They don't teach every lesson using teaching through problem solving. They only teach some of them, but when they do, one of the main reasons is not just to help pupils learn mathematics. That is one outcome, but another main outcome is to help their pupils, develop as characters, to develop attitudes and positive kind of beliefs about what it means to do mathematics.
And I think that, again is something we can't forget. We're not just trying to train robots to come out of school who can just work their way through exams and tests. We're trying to produce people who can think critically and problem solve. And, and so we need to try and teach in a way that helps produce that kind of outcome, I guess.
Well, there you have it.
Walk in the shoes of the learner. I love that. Walk in the shoes of the learner.
That's right. Go on Adam. Last wise words from you.
I think what Andy said just sums it up. When you wrap everything up in a nutshell, be prepared for your lessons, understand, be a learner. So if I don't know how to divide a fraction by another fraction, ask a colleague. And I think that if we do that then we're prepared for it, and those things that come up that we do see, and of course people are passionate about these subjects, people are passionate when they post things on Twitter or online because it means a lot to them.
And fair enough, I think we just have to remember at the heart of it, that if we know our children well, if we know the subjects well, and we know those series of skills that are needed, that every step a child could possibly trip up on them, and we know how to support them and guide them in that, then chances that we're doing a good job. And I think more than anything, just be curious. Be curious about these things that are going on. But just, I think that where mistakes are made, is if we just try to take them on wholesale without understanding them and think that if I do this, all of a sudden everything that I've learned becomes obsolete, and this is going to be the shining light that we follow.
And I think that experience plays a huge part in what we do and becoming expert in something takes a long time. So take these things on board as we go, but don't neglect what we already know, which is the fundamental stuff that I think we've talked about previously.
Well, thanks for joining everyone.
Thank you for joining us on the School of School podcast.