Teaching methods: Engaging students with problem solving tasks in maths

Thanks for downloading this podcast from Teacher magazine. I’m Dominique Russell.

Research shows that challenging problem solving tasks have a positive impact on student learning, but there is little evidence on student attitudes towards problem solving when it comes to doing this in the maths classroom.

My guest today is education consultant at Love Maths, Michael Minas. He works in primary schools to help improve learning in mathematics through professional development, classroom modelling and work with parents.

His areas of interest include problem solving and student engagement, and during his time working in primary school maths classrooms, he’s noticed anecdotally that students respond really positively to being presented with challenging problem solving tasks.

So, to formally investigate this, he conducted a study to assess student attitudes towards problem solving in maths alongside Dr James Russo from Monash University. This study focused on 52 students in two classrooms – a Year 3 and 4 class and a Year 5 and 6 class – in a primary school in Melbourne. Michael led a number of lessons in each classroom which presented challenging problem solving tasks to students. The classroom teachers observed these lessons, and then led these same tasks with the students. The lesson structure used was launch-explore-discuss/summarise, which Michael will go over in more detail in the episode.

After these lessons, the students completed a questionnaire to assess their opinions on the task. The results found that three-quarters of students reported unambiguously positive attitudes towards problem solving, the others were ambivalent, and no student expressed a negative attitude.

So, if you’re interested in implementing challenging problem solving tasks in your classroom, keep listening to hear Michael explain in detail the structure of these tasks, and what elements students enjoyed most. Let’s jump in.

Dominique Russell: Thanks for joining me Michael. I just thought it would be good to get a bit of background on the work you’re doing at the moment to start things off and why this research was important for you to conduct?

Michael Minas: Yeah I guess a lot of my work at the moment is in classrooms and one of the things that a lot of the schools are interested in is trying to get more problem solving happening in their mathematics classrooms. So a lot of the work that I do is in classrooms modelling problem solving lessons, working with teachers to sort of develop their sort of approach, their level of comfort with that style of teaching.

And so for me this was really interesting because, you know, I know anecdotally through my sort of experience with working with hundreds and hundreds of students, that I can see the positive responses. But, you know, it’s obviously an area that hasn’t had a lot of research done into it, so it’s good to be able to have, you know, the start of looking at it in a more formal way, of how do students actually feel in these types of lessons? What’s the experience like for them?

DR: And so the research obviously looks at two classrooms in particular in a primary school in Melbourne, looking at those middle years in primary school, which like you say, hasn’t really been looked at in much detail in the literature. So can you describe for me a bit about the school context of this particular school that you were doing the research in?

MM: Yeah, so I mean it’s a typical sort of primary school. It wasn’t, it didn’t have sort of anything outstanding in terms of the cohort of students, the size of the school – you know, 300 kids – it was a pretty sort of, demographically, a regular mix of students.

In terms of mathematics, it was philosophically quite a traditional environment for students to work in with a lot of sort of teacher directed work – you know, ‘I’ll show you how to do it, and then you go back to your tables and you reproduce what I’ve put on the board and maybe answer a series of questions using the approach that I’ve shown you’. So this style of lesson and learning was quite different for both the staff and the students at the school.

DR: And so obviously this style of learning that you exposed them to was received very positively from the students involved which we’ll talk about in a bit more detail soon, but I’m interested then in what the students opinions were of maths before this problem solving task was introduced to them. Do you have any concept of how they viewed maths in general? Did they enjoy it or were they enthusiastic about it?

MM: Yeah, so we had a couple things. So, you know, fairly informal, but when I arrived there one of the first things I did was I did some surveys with all of the students from Year 3-6. And the surveys were around their attitude to maths and also their self-perception (so, how they saw themselves as maths learners) and there were some really clear negative trends there.

So that was a starting point, you know, working with the leadership in the school to say, there are some issues here and you know, you can clearly sort of see that there are some issues here from the survey data.

But beyond that, I mean, anecdotally, my very first day at the school, I distinctly remember this. I was walking into one of the rooms at the school and a little Grade 3 girl said to me ‘oh, who are you?’ and I said, you know, ‘I’m Michael, I’m going to be here, I’m going to be working with you guys on maths’ and whatever. And her friend that was sitting with her, who wasn’t part of the conversation, inserted herself into the conversation to say, ‘oh, we hate maths. Both of us hate maths.’ Like really wanted to make a point of letting me know that she hated maths.

So that type of interaction was probably the most memorable, but I had lots of those types of interactions where people said ‘oh, you’re working with maths? Yeah I don’t like learning maths. I’m not interested in maths. I hate maths. Maths is my least favourite subject.’

So I had lots of those interactions with the kids and, you know, with that girl on the very first day and I said to her ‘well, you know, hopefully if we have this same conversation in November, that you will have shifted the way that you see maths. But that’s my job to do that, that’s not your job.’

DR: And so can you talk me through really the structure of these problem solving tasks that you led in the classrooms? Because I know you were leading them for a little while and the classroom teacher was observing the lessons that you were conducting. So what’s the structure of these tasks?

MM: Yeah, so one of the things is that the structure is a very, sort of central feature of this approach. And the idea is that, that structure is meant to be very predictable for both the staff and for the students.

You know, so we’d start with a warm up activity and the central idea of that is that you want that to be an engaging warm up to have the kids starting the lesson with, you know, a lot of energy and enthusiasm. And that would be followed by the launch of the problem. And, ideally, most problems, we want them to be launched with some of narrative link, some sort of connection to the real world. And we want that to be done in a concise way.

So the idea there is it’s not like a mini lesson of ‘let me get up the front and tell you everything I know about division for the next 15 minutes’. The idea is that we’re giving them a task that – maybe the task lends itself to multiplicative thinking and division, but we’re leaving space for the students to approach the task from their own perspective. So that, I will say to teachers ideally I want that launch to be sort of somewhere around the five minute mark. And for a lot of classroom teachers that’s a challenge, that sort of directly conflicts with the way they’re currently taking their maths lessons.

And then by extension, the shorter that launch time is, the more time the students have to be exploring, engaged with the task. So in order for, you know, students to stay working on a task for 35, 40 minutes, the task needs to be challenging, it needs to be cognitively engaging for them.

And so that explore time starts with five minutes of silent, independent work. And it’s really important that it is silent and is independent. And then from there, I’m a big advocate for actively encouraging collaboration in the classroom. So, not just saying ‘if you want to work with someone, you can’, but actively encouraging the kids; say: ‘hey, why don’t you go over and talk to Megan and see what she’s doing because she’s got some similar thoughts to you, but she’s approaching it a bit of a different way’.

And then the lessons will always finish with some sort of summary of what we’ve done and that again is student-centred. So the idea is that we’re (myself or whoever the teacher is) is looking for student examples to sort of showcase at the end of the lesson to say: ‘hey, you know, talk to me Dominique about what you’ve done’, and getting you to explain your thinking, but being really strategic about it who you select. So it’s not like ‘everyone come to the floor, okay, who’d like to share their work?’ and the same three kids put their hands up every day. It’s you as a teacher being really strategic about who you select and why.

DR: And so something that I found really interesting in the report, just as a bit of an example of how this structure plays out, the example of the chessboard tournament problem. So, the problem was launched with a short story about a family holiday and there was a big chessboard where they were staying. So you displayed a photo of one of the children playing on the chess board at the front of the classroom, and then you gave the task, which was if six children wanted to have a round robin tournament, how many games would need to be played? And then you had a prompt, which asked students to draw a diagram to show how they’d work this out, then you also provided some extending prompts. So can you give a quick run through of how that played out in the classroom? And did the students respond well to the extension prompts?

MM: Just to give you a bit of an idea about the narrative side of things – that task was based around a photo that I shared with the kids of my own family when we were on holidays playing chess on a chessboard. And I’ve had some really positive interactions with kids around that, where some kids will come and say to you ‘oh I play chess lots,’ and ‘I’m a big fan of chess’ so it’s building relationships there where they can say, you might share a common interest.

I mean I’ve had the other experience where I’ve been at a school and I’ve told the kids: ‘this giant chess board was at this particular holiday place in Queensland, and then I’ve had a student come back to me like two months later over the summer holidays and saying: ‘guess where I went over summer? We went and stayed at Paradise Resort and we played chess on that chessboard’ and the kid being really excited to share that with you.

So the narrative, that was the true part of the narrative. I mean the made-up part was – it talks about us having a round robin chess tournament. Now, we didn’t have a round robin chess tournament, we were actually there trying to enjoy, we didn’t spend all day at a giant chessboard playing chess.

So I think it’s teachers being able to feel comfortable taking parts of their life – you know, some real-world application – but also feeling free to be able to sort of elaborate, add to it, and make it work for the maths.

The task – yeah, it’s a really fantastic task because it’s got quite a low entry point, in that you could work on that task just sort of saying – you know it’s like the old problem where you say ‘there are eight people in a room and they all shake hands with each other. How many handshakes would there be?’ But it’s much more – I mean, the idea of playing chess against each other, students can visualise that a lot better and can sort of conceptualise it to say: ‘well if Nash plays against Isaiah, and then next Nash would play against Genevieve…’ so they can sort of work through all the combinations of who Nash would need to play.

Nearly every student you give that task to can enter the task and can have some level of success. But at the higher level it’s a very cognitively engaging task. I mean, the extension task is asking for them to basically find a formula of how to work out any triangular number. And so I used that task with Year 3/4 students and I’ve had students I’ve worked with in the Year 3/4 cohort who are able to sort of show you, ‘I can work out any triangular number and this is how you do it’. And they can show you visually how the formula works.

So I think that’s the beauty to this approach to teaching in that you’re really allowing for true differentiation. You’re presenting a task and there’s scope there for students to work there at a number of different levels.

DR: And as we mentioned very briefly before the classroom teachers were observing you first conducting these lessons before conducting the lessons themselves. Why was that important to do than just instructing teachers on how to run this and getting them to launch straight into it. Why was the observation element quite critical?

MM: I’m a big believer of if you want to get change happening within an organisation, it’s important to have buy-in from people. It’s important for people to actually believe that what you’re doing is going to be doing is beneficial. And for teachers, the vast majority of teachers, when they see that something is effective with their own students, you’ve won them over. So if they can see their own students being challenged in a way they previously haven’t been challenged.

I mean I had this experience yesterday when I was at a school in the western suburbs of Melbourne and I was working in a prep classroom and there was a prep student who, you know, traditionally didn’t really have a lot of success in the maths class and then this student produced some work and this classroom teacher was literally speechless. He was just blown away, he was like, ‘I cannot believe that he’s just done that, I’ve never seen him do that before’.

Now when I go back to work with that teacher in a fortnight’s time or whenever I’m back out there to work with them, they’re going to be much more receptive to this approach because they can see that it works.

And I think you’re also setting teachers up for success then. Because if they’ve seen that lesson structure a few times, the idea that it is very repetitive as a structure, it gives them something that they can sort of say, ‘right, now if I’m going to have a go at taking a lesson using this approach, these are the things that I want to do’. And it’s very easy to reproduce because they’ve seen it done a number of times.

So it’s both about supporting the teachers so they can have success, but also about generating that buy-in and I think that that comes – it’s one thing to deliver PD and to say ‘this is great’. It’s another thing for teachers to see it working with their own students.

DR: And so another big part of the study was how you actually measured the attitudes that students held towards these problem solving tasks and they were overwhelmingly positive. You’d mentioned before that this was kind of what you were expecting to happen because anecdotally you knew that students responded really well to these kinds of tasks. But something that I found really interesting was that they really enjoyed the challenging aspect of these problems and also the collaborative nature. So can you talk me through what the students said and wrote in their questionnaires about those two particular aspects?

MM: Yeah. I guess, I mean one thing that did surprise me was – I expected the results to be positive because that’s what I kind of see when I work not just at this school, but at lots of schools – I was surprised in the fact that of all the students that were involved in the study, that there was no one that expressed, like, negative attitude. Which, you know, was quite sort of gobsmacking for me.

But in terms of what they identified that made it enjoyable, engaging for them. Like you said, there was a couple things they touched on. So one was the idea of challenge. And I think this is something that sometimes teachers struggle with, this idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

And I actually think that’s counter to everything we know about humans. If we think about ourselves as adults, if we’re given some sort of routine, mundane task to perform over and over again it’s every chance that we might do it if we have to do it, but we’re not going to enjoy it. But people love a challenge, people love being pushed cognitively and trying to see if they can be the first one to figure things out. I think humans love a challenge and if I enjoy a challenge as a 43-year-old, there’s no reason to think that like a six-year-old or a 12-year-old wouldn’t enjoy a challenge. So that’s come through to me anecdotally, you know, time and time again over the years, so it was good to see that come in through formally in the study that we did.

The other really big – and again, in some cases this really contrasts with the regular classroom practice – this idea of allowing the students to collaborate. And like I said before, not just allowing, but actively encouraging it. I think a lot of classroom teachers are concerned that if they let the kids move around the room and talk to each other, they’re going to lose control and it’s going to descend into chaos. But I think the two ideas that you’ve just asked about are connected. Like if they’re working on something that they think is worthwhile and challenging, they’re much more likely to stay on task.

And again, humans enjoy collaborating. Humans enjoy socialising, talking, sharing ideas. So if that’s the way, if I was to present PD [professional development] at a school and I was to do five hours of me talking and there’d be no opportunity for staff to actively engage and collaborate with each other, I mean, I would never be invited back to the school.

So then the question would be, well why do we get our students to do this? Why is a maths lesson me talking for 20 minutes telling you everything I know about place value, and then you working on a worksheet by yourself for half an hour and not being allowed to talk?

That’s not going to be enjoyable for us as adults. Why would it be enjoyable for an eight-year-old in a Year 2 class? So I think that in some cases the success that we have when we go and work in schools is partly because it’s such a sharp contrast to the regular practice in the school about the way maths is learned. And that if we can make mathematics more social, then we have much more chance of having students being engaged and wanting to learn.

DR: And is part of that as well – like you mentioned before – the fact that they have that five minutes at the beginning to concentrate on the problem as an individual and silently, but then they open up to the collaboration. Is that balance quite good and quite important?

MM: Yeah. It’s really crucial. And I always tell teachers that I’m working with that one’s not more important than the other, that they’re equally important. But if you let kids collaborate straight away then what you might find is that kids will just straight away – say you and I are working together, and you’re a stronger student in terms of your current performance in maths, well I might just be led by you, and you’ll just be telling me, ‘do this, do that’.

Much more likely if I’ve had some time to think and ponder on the task, that A, when I come to you, maybe I’ll have some questions about what I’m doing and you can guide me and direct me, rather than telling me what to do. But, B, there might be the chance that I may choose not to work with you, even thought we may be best friends, because I may see that someone else is approaching the problem with a similar mindset, a similar approach to me. Or I may choose to say in this instance: ‘I’m going to keep doing this by myself because I feel like I’m getting some momentum here. I can see that I’m making some progress’.

So I think that five minutes silent time is really crucial, and then it becomes really crucial (this becomes a classroom management thing) as a teacher, you have to be able to make sure that it is truly five minutes and it is truly silent and it is truly independent and also truly productive. Because it’s no good them just sitting silently looking at the clock, you know, looking at the stop watch counting down before they bang go into talking to each other.

So the way you know that’s productive is when you see the kids are on task. When you see the tops of their heads looking down at their page, and they’re thinking, and they’re gathering materials. And you can tell really clearly as a teacher when that’s not happening.

DR: And so just finally then, I’m thinking now for teachers who are listening to this episode who are thinking they want to implement a similar approach in their maths classroom for students of a similar age, is there anything that we haven’t covered already that would be good to keep in mind? Or perhaps some good first steps to take?

MM: Yeah, look I think that the model that I see that works really well is I think what we spoke about before. Is that you have to have people that are able to model what it should look like to be able to win teachers over, for them to be able to say ‘I can see the benefit of this, I can see how this works’. So whether that be – I mean, I’m definitely not trying to spruik for work – but whether that be internally – you know, like a lot of schools have really great classroom teachers. Some of those people have moved into learning specialist roles.

But whether that be internally with those people, like give them the time to go into other people’s classrooms and to be able to model this type of approach and to show the classroom teachers how it works and to be able to answer those questions. Or whether it be externally, by bringing in consultants who have the skill and expertise to do it, I think that’s really important. I think it’s important that people see it in practice first before they try to do it.

And it’s also really important, as well as seeing lessons, that people have time to then unpack the lesson and talk about it together. So if you’ve got a learning specialist at your school that’s modelling this type of lesson for, say a graduate teacher, there needs to be some time allocated for them to sit. Because the graduate teacher may walk away saying, ‘that was a great lesson’. But the next step is them being able to identify why was it a great lesson? What worked? And what can they do to plan a similar great lesson the following week?

Because, you know, if you just say ‘well, that was a great lesson, but I can’t do that lesson again because my kids have already done it, so where do I go with it?’ Whereas if you can identify and say: ‘oh I see what worked well. The thing that worked well is they were engaged with the problem.’ Why were they engaged in the problem? ‘It had a real world link’. Why was your questioning effective during the lesson? ‘Well, it was because you knew, you had a clear focus of what the content was’. What are we focusing here? What’s the mathematical concepts we’re focusing on?

So, as a classroom teacher you know the right question to ask and the right student at the right time and there’s a lot of work that goes into that, but like I said, it’s definitely something that’s attainable for all classroom teachers with the right support.

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Russo, J., & Minas, M. (2020). Student Attitudes Towards Learning Mathematics Through Challenging, Problem Solving Tasks: “It’s so Hard– in a Good Way”. International Electronic Journal of Elementary Education, 13(2), 215-225. https://doi.org/10.26822/iejee.2021.185.

Michael Minas says he believes sometimes teachers struggle with the idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

Reflect on a recent lesson you taught. How challenged would you say students were? How do you know the level of challenge was appropriate? Do you think you could have challenged students further? Were there opportunities for students to participate in extension tasks?