Episode 215: Conceptual understanding is crucial for long-term learning

Hi, welcome back to another episode of the School of School podcast. I'm here with the usual suspects, Andy and Adam. Hello, you two.

I'm gonna say good morning for Andy. no, he's gonna have a go. I just saw him put toast in his mouth. Exactly.

okay, rule number one of a podcast. Don't have food in your mouth. Well,

I was going to sub in and sort of pretend to be in. Anyway, good morning, good morning. Good afternoon, even.

thanks Adam for covering for Andy. So moving right along, we have, I'm very excited. We have a wonderful guest today. Ross Deans. We love Ross Deans. He's from Muscliff Primary. Hello, Ross. How are you?

Yeah, no worries.

Yeah, very well. Yeah. So Ross, you know what? I'm just going to let you take it away. Tell us a little bit about yourself.

Okay, so I've been working at Muscliff Primary for 13 years, leading maths for the last five or six years. We've been working with maths as a problem since 2017 and it's been a real journey for us and certainly changed our practice and enabled our learners to really enjoy the subject, but also to be really successful.

I was going to talk a little bit about conceptual understanding and it's such a big part of master problem approach, mastery approach and I just wanted to not necessarily propose to be an expert on it but just have a bit of a discussion about the importance and how and why we want to develop that level of understanding. And I guess it starts with my own experience of maths and challenges with maths as a child and not being a lover of the subject because I didn't always connect the dots, didn't understand how things were linked together, but was encouraged to learn in a way that was very procedural, very factual, and not very fun or creative.

And as I, when I became a teacher, it probably wasn't the area. I mean, I did a degree in graphic design, you know, and then I spent my time traveling around Latin America and working with animals and then communities. And I didn't see this happening at this career.

But I ended up in it and probably going in maths wasn't the area that I was looking forward to the most. And in fact, I remember my first placement when I was training, I ended up in a year six cohort in a very, very high performing school and it was intimidating and I was having to do my homework each night. So I knew what I was teaching them. But over the years, it's really grown to be a passion and something I love teaching and it's my favorite part of the day.

But also something I love to lead and try to share and develop across school and in any way I can. So when we come to conceptual understanding, I guess the point is that we really allow learners to understand what and why they are doing things. And in fact, recently since suggesting we have a chat about this, I stumbled across a video you've probably all seen, Simon Sinek, who did the start with why video. was a TED talk.

He did that about 15 years ago. Are familiar with the video? Yeah, so, you know, he talks about from a business perspective, but how, you know, to be an inspiring leader, you need to start with why. But it's so far stretching. And in fact, in our staff meeting in school last week, something we're continuing this week as well, we, we, we spoke about that as we're developing something for our teaching and learning, that you really do need to start with why.

And that's with mastery, master problem, developing conceptual understanding that that's what we're doing. Master problem lessons, start with a real life context with a story in the explore task. Exposure is a problem we want to try and get to the bottom of.

And that is a why in a sense. That's a why because you want to solve the problem. But we also want to understand why things work, why we should tackle something in a certain way.

And I think about the damage that shortcuts can have as well when we teach. So it's pretty easy to teach something and say, look, this is the quick way to do it. Off you go. And I could do that. I teach a group of learners that are wonderful, that there's a fair chunk of them that find maths pretty hard. So I'm trying to slow things down for them. I could just say, look, just do it this way. Divide the numerator.

Easy, you can do that. They're only small numbers. You can get it in two minutes. You get every question right and then we can move on to something else tomorrow. But if I haven't taught them the reason why, they're going to forget that trick in a couple of weeks. But it also is not moving their wider learning on at all in any way. In fact, when I was thinking about this, I did do a little bit of looking online and I found someone that spoke about the beauty of maths and you hear that quite a lot.

It's quite powerful actually. I can read a bit of this quote if that's okay to do say from someone called Karen Sloan who works with Carnegie Learning and she spoke about mass being a beautiful creative thought-provoking subject that sets the stage for students to become critical thinkers, problem solvers and leaders of tomorrow. And then she says don't let a reliance on maths shortcuts, tips and tricks rob them of that experience. I think that's quite powerful.

And that's certainly what I felt as a learner. Going back to my days in secondary school, someone who was quite anxious about maths didn't connect the dots in that way, but managed to follow algorithms and did really well when it got to algebra and got shoved in a top set that they didn't feel comfortable in. It killed my love of maths. And I certainly didn't want to do it after school at that stage of my life. And that's not what we want to happen. We want our learners to see it in that way, see the beauty of maths, to make it thought provoking.

To be critical thinkers and to enjoy that exploration. I guess, you know, shortcuts are not going to help them in the long run. They're just not going to lead to anything, not going to lead to any long-term memory. Thinking about some of the teaching and learning work we've done in school about the process of learning. I think about Daniel Willingham, he's done some great work and got that lovely quote, memory is the residue of thought.

Now if we teach a shortcut without any conceptual understanding, there's not a whole lot of thought involved and therefore it's not really going to stick. It's not going to lead to any long-term change. Yes, that's kind of the why I thought it would be a good topic to talk about.

Yeah, think Ross, I wonder what keeps like springing into my mind because I completely agree with you and I think that that idea around, you know, like understanding something deeply, it just makes sense, right? Like it just makes sense and then you can start to relate it to other situations.

But I just, wonder if there's still a sort of a problem in that when we go into lessons as teachers, when there are understandings at a level that allows us to see the concept that underpins it. You know what I mean?

So I saw a problem the other day, and I won't get too sort of mathsy on this, but I saw a problem the other day which was three quarters of four fifths. That was the question. I thought it was a nice problem. A nice problem. in effect, there's four parts. There's three quarters of those parts as three. So the answer is sort of three fifths. If you understand it. I've just rattled that off like I understand it.

But if you saw that and you multiplied it out and then you simplified and you did all these things, what I'm trying to get to is I want your observations on whether or not there's enough emphasis in maths teaching that we have to look as teachers for each lesson. What's the concept that underpins it? How far back does it go? What's the original thinking around this? Is that prevalent, do you think?

I think, looking at your example there as well, some learners I worked with recently, again, dividing or multiplying fractions by whole numbers, pretty easy to teach those tricks. But by really exploring it, I managed to see that there's a fundamental misconception about what fractions even are.

So if you're just plastering over those cracks and trying to steam forwards in the moment, you're not helping them in the long run at all and there's so much learning that's essential that comes before that about the nature of the maths that they've been dealing with.

Yeah, and I wonder about that. just wondered, you know, like the program, our program, they're carefully crafted steps for a very good reason. And again, I think sometimes what I hope is prominent in people's minds is that what you teach, what you're doing today to give a chance to really understand is for the children's future. It's not to get it right today. It's to kind of go, right, this is just a piece.

that allows you in the future to see these pieces. And you're not, you know, like these pieces will come when you're ready for them. But today, it's this piece here, get it really solid, because what's coming next, you'll be able to hang your hat on, you know, like you'd be able to see these things. And I think too, think that sometimes it's just a case of, that's what we're doing this for.

Yes, to develop those attitudes and being metacognitive, becoming a learner, those sorts of things, but also putting in a really fundamental understanding of something, because that fundamental understanding will still be there, not just next year or five years down the line, but 10, 15, 20 years down the line, that fundamental piece at the bottom will still need to be there. It's not like it disappears, like it's so important. And I just sometimes wonder whether or not that is understood and we're doing that enough spending the time to get these pieces in place because they're so important to everything that comes thereafter.

And Adam, yeah, just you talking and Ross talking. Ross talking about his anxiety over maths, you know, when he was younger. And I think as soon as you said that, you know, three quarters of four fifths, I'm thinking fractions immediately in my head. And I'm thinking now, what are those children thinking? What appears visually to them or conceptually when you say that, you know, are they seeing a pizza? Are they seeing?

My guess is they're not immediately seeing a mathematical equation in front of them. They probably have the ability to be more creative with it and thinking about it in a different way, whereas you said is learning for the long term rather than memorizing how to do it.

Well, so just, think that's a really, really interesting thing that you, you kind of brought up there, Robin. So you say three quarters of four fifths, right? So if you say that to someone, what conjures up in their mind immediately, right? I would say that, if they were taught like we were, right?

Cause I think we all had a similar experience, which Ross explained, you know, anxiety is what comes up in most people's minds and that anxiety is probably led by what am I supposed to do now? Is this the, is that when I flip them over or do this like cross multiplication or what's the rule that will apply to this scenario?

Will I remember the right rule? Because I know I can, I can do a bunch of stuff. can multiply numbers, I can divide them. can, you know, I can do all that, but I don't know what to do. I don't know which one, I don't remember which one applies to this situation. That's probably what happens, conjures up in most people's minds.

And, and most likely they'll immediately change the subject and try to talk about yesterday's football match or something, right? Because that's what happens, right? And it happens all the way through. that's, you know, that's an easy way to get people our age, but you know, it just could be as simple as what's, know, what's 12 times 17 to someone who's younger or even what's three times five to an even younger child. They can all conjure up these anxieties if that's experience of mathematics, right?

But I would beg that that someone who, who had a more fortunate mathematics education, like the one that Ross is describing happens at his school, then what will come up is, they'll say, Hmm, you know, what, what, what is it? What is the scenario where this, this kind of arbitrary, you know, grouping of numbers would apply and they may think about pizzas or maybe they'll think about sharing out some drinks or whatever it is, you know.

Hey, I've got, you know, I need to save a fifth of what I've got. And I have three quarters of a bottle. Right. So, you know, that, that, that all of a sudden, you know, they'll think of a scenario where this thing applies, right. Which is, know, obviously way more powerful because, because the way that that flips around is, is then when they are faced with the problem, the real world problem, they know what the mathematics is that applies to that problem. But if I haven't been taught properly, you know, they'll find some more cumbersome way to work out their situation. Right.

And I think that's, that's what it comes up to that. That's what we're talking about is visualization, right? Can you see in your mind's eye what this problem is, is, is showing you. If you can see it, it's so much more powerful. Right. And that's partially what conceptual understanding is all about that generalization, being able to relate it to something that's familiar as well.

Right. All those sort of core competencies that we often talk about. And that's what you got to, that's why you need to teach to the competencies and not to the subject matter, right? The subject matter doesn't matter. Like I see people obsess about curriculum all the time. They're like, we need to have more of this and less of that, and this needs to come first and that, you know, largely that stuff doesn't really matter all that much, right?

If you're teaching to the competencies, because what happens is by the time they get to the age of 10 or 11 or 12, they have kind of this really powerful toolkit, if they've been taught that way, that they can just kind of work it out without somebody telling them what the trick is.

You know, if you've, if you've measured the interior angles of enough polygons, right. And you've generalised kind of what's going on. It doesn't matter if the polygon has nine, nine sides, you'll know what to do. Even if you don't have a protractor.

And that's the point. So you don't need to teach them what, the interior angles of a nine sided shape is because they already know how to work it out. Right. But if all you do is memorize, they'll have to remember it, you know? Yeah.

It's also with the emphasis being just put on the answer rather than the explanation and justification that the demonstrating what the learner knows. Going back to that fraction question, I mean, some people could read off the answer, some people couldn't, but everybody could start to show their understanding and that is going to be more powerful.

Let's think about what you just said there as well, Andy. It sort of made me think about the research around growth mindsets.

There was a particular experiment I remembered reading about a few years ago, can't remember the details exactly, where they were, I guess children who had been, developed that growth mindset that compared to stronger academic peers when presented with a near impossible problem were more resilient because they had that approach to learning.

So they were able to keep going for longer and had a better chance of success because they had, you know, that attitude towards problem solving. And that's what we want for mathematicians really, isn't it? To not just be looking for the procedure, not be looking for the answer, but to demonstrate what they understand and make connections.

Absolutely. And I think that mindset is such a big part of it. And, you know, this is something Tony Gardiner, not without the language of mindsets, but used to talk about a lot, you know, we're, you know, for those of us who, who know Tony Gardiner, we know that the Tony's passion was working with very, you know, call it advanced mathematicians, right. And for him, that didn't mean someone who knew all their times tables.

It meant someone who was able to do those other things that we talked to core competencies. Then he would often say that, you know, the, the children who, who show early success because, because of the way that we've taught mathematics and, know, we're particularly good at remembering a lot of things and, calculating quickly and stuff very rarely amount to very much in mathematics because, you know, what can easily happen?

You can kind of destroy the mindset because it, it, they get a very early confidence and see themselves as high ability children. They label themselves as that. I am a high ability child, right? Or, you know, I'm really good at maths because they can do that. They can succeed at those types of things. But at some point, mathematics is no longer that. What it requires is tenacity and it requires resilience, right?

It requires, because the problems are difficult, you might work on the same problem for years, right? Those early achievers rarely become the ones who have the tenacity and the resilience to persevere through, you know, the kind of really difficult problems that they'll encounter later on. And that's why so many of them don't choose mathematics when they get to university, right?

Because maths to them is like, if I can't succeed immediately, very quickly, then I'm not good. And that's a mindset thing, right? You know?

And we see that in the classroom when there's this high ability learners that crumble when they don't know an answer, because it's uncomfortable territory. And you don't want that to happen for them, obviously.

But when you see somebody who often finds things really challenging and they've just shown that, I liked your tenacity as a description, that they've shown that approach to learning and they've persevered and then they reach that aha moment. It's so powerful. It's a lovely thing to see.

Yeah. And sometimes it might even come back to you days after and say, you know, look, I had this, you know, and, that's actually what advanced children look like. You know, they, they will just stay with an idea and, and ponder on it.

Right. And that's, that's, that's mindset. That's, that's what Carol Dweck was talking about. And I think that that comes from a, I don't expect anything to be, to come to me immediately. Right. I expect that I have to work hard at it. And those are the ones that, that.

that go beyond, over and beyond. It's fascinating, right? So this conceptual understanding, going back to this idea of conceptual understanding, what does that look like in a classroom, Ross?

Like how, you know, cause I think for some people, this is all just a mystery. It's, you know, they have these ideas about, you know, very lofty sort of, you know, there's still some rigor to it. There's still some discipline in teaching to conceptual understanding, right? What does that look like in a classroom?

Well, I mean, as you see within a massive problem lesson, you mentioned earlier that that real life context is really powerful. If not every lesson, vast majority of lessons start with that real life context, don't they? Where you've got something to pin it on, something that makes sense to you, that applies to your life. And then thinking about using manipulatives, concrete, visual abstract representations, you've got a way to demonstrate why.

why it works, why it makes sense, what the connections are. You haven't just got a number or an answer, you've got a way to represent. Therefore, you can explain and you can justify and you can reason and then you can build on that as you move forward. I really like the approach that we've certainly learned to embed using master problem about exploring multiple methods.

even if someone's got it real quick. So, you've solved that really quickly. You've used that way. great. But did you think about trying it like this? And initially, we used to get a real challenge that, well, why would I? I know the answer. And I'm not interested in the answer. I know the answer. I want you just to explore. I want you to see links. And actually, you know the answer to this problem today. But this different approach to calculating or solving a problem that you never thought of might come in handy tomorrow.

It's that approach to actually, you're developing that attitude and that tenacity and that mindset that's setting you up for the future, not for today. Yeah, so visual representations are really handy in addition to any abstract methods that we teach, but using something like a bar model or manipulatives to demonstrate maths in action.

that shows that a concept's really been understood and then it can become more flexible for future use. I think as teachers we need to really model that as well, not just using resources or visual representations, but how to go through that process as a learner to ask those questions, to notice things between them. There's a lot talk about metacognition these days and I think really it's a very powerful thing as a teacher to talk through that thought process.

Whilst modeling, really we know the answers, but we're going to put on that show, aren't we? What can you see? What are the links between these aspects? And what does that tell us? And what if as well? How can we move on from here?

I'd love to talk about metacognition, but I think we're running out of time. Haven't we? I think maybe we'll have to get, get you back Ross and have a, have a chat about metacognition sometime. And yeah, I'd really like to do that.

For sure. Yeah. I'm just so happy that you have come out of the jungle and into the classroom. This is fabulous. I mean, I think you've picked the right calling at the end of the day. So just so glad you could come on and chat with us.

Andy Psarianos (23:35.1)

And the other thing I'd love to talk to you about Ross, I don't know if you know, but I also have a bit of a graphics, graphic design background. it, yeah, yeah.

I didn't know that, okay.

Andy Psarianos (23:35.1)

I, actually what I found out about Singapore maths when I was teaching in teaching graphic design in Singapore. So, so yeah, so that it's a story for another day, but we'd need to come back and talk about that as well. Sometimes the importance of graphic design.

Yeah, yeah, well, it's really helpful with maths, isn't it? Particularly when you're trying to represent a challenging concept, just finding a really succinct way to sum that up is so helpful.

Thanks so much, Ross, for coming.

Thank you. Nice to see you all