Episode 235: Problem solving is essential in mathematics education

Adam Gifford

Welcome back here we are another episode of the school of school podcast the numbers keep racking up by the way we've done lots of these I was surprised the other day when Nick and I should probably share this number in front of me but whatever it is it's lots and lots and we're back with the regulars Robin and Andy how are you both?

Andy Psarianos

Here we are.

Andy Psarianos

Feeling regular.

School Of School

Great.

Adam Gifford

I'm going to shoot straight on past that. One thing that I just wanted to talk about, we've done a series of quite snappy sections looking at aspects of teaching and learning, everything associated with that, mathematics, how to go about really good teaching and learning, how to support children. One thing that I hear all the time, and I think it needs to be unpicked because we're in danger of just saying it without knowing about it.

Is problem solving. know, like we talk about problem solving is, if you like, the way forward. I've read about it in various different, just generally, and people talking about it. But it's probably worth unpicking a bit more so we get a good sense of what that, what is it? You know, like how does that contribute to learning better? All of the things that are attached to that. So the floor is open.

Andy Psarianos

Yeah, so I guess, so you know, if we're going to talk about problem solving, I think there's two things that we need to refer to without a doubt. One of them is polia, George polia. And the other one is the Cockroft report, which was a paper that was written in England, which was actually called mathematics counts.

Of the committee of inquiry into the teaching of mathematics in schools was written in 1982. So, and, but it's most famously known as the Cockroft Report because the guy in charge of writing is, was called Cockroft. So anyway, so.

Polia kind of defined what we mean by problem solving, especially in mathematics. And the Cockroft report was really a catalyst in education in the 1980s. There was this transformation where all of a sudden everyone started realizing, hey, mathematics is, well, I mean, we already knew this, but mathematics education should be entirely focused on problem solving.

Not just you know calculations and procedures right and that was a shift in the 80s now I mean I think all good teachers already knew that it was about problem-solving but but that's really what what what you know what launched a lot of people into taking problem-solving a lot more seriously in education so that's what it's about that's how it kind of came around and what you'll find nowadays in most in most curriculums this is said so a huge emphasis on problem-solving

So there's a couple of directions we can go. mean, one is why is problem solving important? And the other one is what are we talking about when we're talking about problem solving? So I don't know. mean, Adam, do you wanna just maybe jump in and just try to maybe define what we mean by problem solving?

Adam Gifford

Yeah, maybe using this sort of, I don't know, I'd call it like a non-example in an example. So I think that when people think about maths and they just say, well, two times two equals four, for example, and you just like that rote memorization, the ability to answer basic facts instantaneously is not problem-solving. That is based on memorization. It's understanding something at a very surface level.

Andy Psarianos

Yeah.

Adam Gifford

I think when we present a real problem, half the dog is 50 % heavier than the smaller dog, big dog, 50 % heavier than the smaller dog. When you start to present that, there are so many different facets that need to be understood to bring into it, to relate, to understand. Well, and you talked about polyure and the steps involved that poly suggested in a problem solving approach.

You know, just at the very first instance to understand it. So that on its own is totally different to a lot of our experiences in mathematics, which was just memorize this, memorize this formula at secondary school or memorize this fact at primary school, elementary school. Where does the understanding come into it? And so I think that between those two things, we've got to, they're night and day. They're two totally different disciplines and principles.

Andy Psarianos

Yeah.

Andy Psarianos

That's right. And I think that's, know, like, so what did you use as an example? Two plus two equals four, right? So if you say, okay, two plus two equals four, you know that, right? Fine, but who cares, right? Like, what, this is just nonsense. Like, how does that help anybody, right? It only helps someone if you apply it to some kind of problem, right? Like, because two plus two equals four in itself is just, has no value.

Right? So any mathematical thing that you do ultimately is to solve some type of problem in the real world, right? Like, you know, why is it useful to know that? Well, it's useful to know that because in some circumstances you need to add two and two together and figure out how much that makes. And that's why problem solving is so key. Okay, two plus two is a really simple example, but anything you learn in mathematics, like why would you want to know how to find the area of some weird geometric shape?

Like why would that ever, like who cares, right? Like really, like who cares what the area of, you know, I don't know, Pentagon is. You know, like why? Like why would you ever care? No one cares. It does not, you know, it's just, there's no purpose, right? Unless you have to apply it to something, right? Like maybe you have that.

Robin Potter

You're being tested on it.

Andy Psarianos

You're being tested. Being tested on it isn't a real world application. Sorry everyone, that does not count as real. That's just schooling for schooling sake, right? Yeah. It's like, no, but if you have to, I don't know, this is kind of, you know, like a maybe cliche example, but if you've got a pentagon room, right, and who has a pentagon room? I don't know. Someone might have a pentagon room somewhere.

It is not. It is definitely not.

Schooling for schooling sake, yes.

Pentagon?

Adam Gifford

Hey, the paint got.

Andy Psarianos

The Pentagon, yeah, and you gotta buy floor tiles, right? How many floor tiles do you need to, you know, to do the floors in the Pentagon? The Pentagon's a big building. If you get it wrong, that's a lot of money, right? Right, so yeah, so that's a problem, right? So the problem is we have to buy the right amount of tiles to tile this room. That's a problem. The area of a Pentagon is just

Yeah.

Yeah.

Adam Gifford

Mm-hmm.

Andy Psarianos

knowledge but with no real kind of application. So talking a little bit in circles here but I suppose the point is the only reason you need to know how to work out the area of a pentagon is to solve problems. So really everything in mathematics is about solving problems.

Robin Potter

Right. So even as a young person, you've got a, I don't know, a six-year-old. I mean, they can understand that concept. The fact that you give them a problem to solve and they've got two cars and they've got two more cars and how many cars they have. I mean, it's a simple way to introduce the idea of problem solving without them realizing maybe that they're solving a problem at that point.

Andy Psarianos

is to make it contextual, right? So, you know, if you say 2 plus 2 equals 4 is very abstract idea and the danger is, of course, is that a child may just memorize or not memorize, I hate the word memorize because that's a memorize is an action. Remember that 2 plus 2 equals 4 and have no inkling whatsoever as to what 2 plus 2 equals 4 means.

Adam Gifford

Yeah.

Robin Potter

Yes.

Andy Psarianos

Right? That's entirely possible. Just like a parrot could say two plus two equals four. It could just repeat that phonetically and remember that that's the sounds that you put together but have no concept whatsoever of what that means.

Adam Gifford

I think the other part to it as well is that often in that first example of just instantaneously saying the answer, the answer is almost a success criteria. So in that sort of situation where you're relying on that, when you're relying on memory to answer in some way, shape or form, or you're just applying a formula without knowing it, the outcome, the answer is success, whereas

Like I'll go back that pentagon one for example, there are probably an infinite number of ways that we can find the answer because what if I was to use a square tile and I went round and literally put them all over that. It would take me days I'm assuming, it would take me a very long time but I could use various different things to measure it so there are probably an infinite number of ways to measure it. Problem solving also asks us to be sophisticated enough to think about the process, the process being part of the success criteria.

Andy Psarianos

Mm.

Andy Psarianos

Yeah.

Adam Gifford

We could either count grains of sand at the beach in a certain area, but maybe it's better to find the weight of one grain of sand, weigh a big chunk of it and say that's it rather than counting individual bits of sand. The idea is that it's not good enough to know, okay, I could just count individual grains of sand or I could just apply this formula. It's about being sophisticated enough to say,

but perhaps there's a better way, another way. Would this work in somewhere else? All of those opportunities are kind of lost. If all I say to you is, Robin, just remember this, seven, okay? That's all you need to do. Just remember that. How can you go about, is that the best way of doing it? You've no idea. Because all I have to do is remember seven and I've won, you know? Which is flawed. It's a deeply flawed system.

Andy Psarianos

Yeah, and you know, and I think that so, so the examples we're using are pretty simple, but you know, you got to extend this now to more sophisticated mathematics and, and just cause you know, just to kind of round off this idea of problem solving. So the importance of problem solving is, should be clear, right? It's like mathematics on its own is kind of meaningless, right?

unless there's a problem to solve. So it's really a mechanism for solving problems. So then you need to think about, okay, well, what kind of problems do we run into? Well, there's simple problems. You know, lot of the examples we've given here are really simple problems, but there's also very complex problems that require multiple steps and the use of various different heuristics and all this kind of stuff. And this is kind of where polio starts coming into play. So polio said, okay, look, you know, there's an approach towards all this, right? I mean, the first thing is you need to understand the problem.

You know, like what is it that we're trying to solve? So in our Pentagon, that's a pretty easy, it's like we need to know how many tiles to buy, right? But sometimes it's not as clear. We gotta really understand the problem because there may be multiple steps in its solution. And then we have to devise a plan. So this is kind of what Polia said, right? He had these different steps. And once you've devised a plan, you gotta carry out the plan.

being patient, requires being persistent, you need to check as you go because there may be multiple steps, you need to be able to say, my plan didn't work, I'm gonna back out and try something else. So there's a kind of a process to all that. And you need to look back as well, right? So once you've done it, you need to look back and, know, does my answer make sense? So this is where things like estimation are really important, you know? It's like, well, wait, hold on a second, the number is way, way larger.

you know, than what I anticipated. Did I make a mistake somewhere? And chances are you might've made a mistake somewhere or you might've misunderstood the problem to start off with. But that looking back is also a really critical step in that, right? And then, so that's one element. It's like you need kind of an approach for solving problems and that's part of what we need to teach in mathematics. It's like, okay, you're gonna run into really complicated problems. How do you deal with that?

Right? Because most problems don't present themselves in a really simple fashion. You know? They're more complicated. And then on the flip side of that, or not the flip side of it, but the other thing you need to consider is this idea of heuristics. okay, well different types of problems are have, they require different types of thinking.

to crack them and that's where heuristics come to play. you know, one of the most common heuristics and one that we I think underplay a lot in general terms is the idea of, know, well, I'm just going to guess and then I'm going to check my answer and

You know, so, well, what if it was seven? If it's seven and I run it through the whole thing, does it work out? Nah, the number's too big. Okay, maybe the number's five. Okay, that's too small. All right, maybe it's six, right? That's guess and check. And that's a valid, completely 100 % valid problem-solving heuristic. And the one that we probably naturally gravitate towards in a lot of cases, right?

So that's an example of heuristic, but there's other types of heuristics, like make a list or work backwards or simplify the problem or draw a model, make a model.

All those types of things are all valid heuristics for solving problems, right? So algebraic problems tend to maybe go more towards draw a model as a really really good heuristic for complicated, know, so like the Singapore bar model or the bar model as we often refer to it. That's great for algebraic type problems or you know problems that have particular semantics like, you know, it's a change situation.

This was the initial situation, then that happened, so now there's another situation. So those types of things are very much lend themselves to certain types of heuristics. So you need to learn all the heuristics as well, right? Not all of them, there's endless heuristics, but there's really common ones. Work backwards, make a list, guess and check.

draw a model or make a model, could be a physical model as well, those are all valid heuristics. And that's really what we're talking about in problem solving. that's the entire, that's really, that's the, if you can call problem solving a skill, that's maybe the most important skill that anybody needs to learn when they learn mathematics is how to solve problems. Because that's really what mathematics is for.