# Transcript of Working mathematically—problem solving

Katherin Cartwright: Well, good afternoon, everyone, and welcome to our last Syllabus PLUS for Term 2. I know you might be able to still hear the music now but when I move to presentation mode that will go away. So thanks for joining in today. Again I'm by myself monitoring the chat pod. I am doing a bit of an interactive session today, so hopefully I can answer your questions in the chat room as well. So I've got two screens working today, so you might see me turn away a few times later on when I have the video camera on again. So just bear with me, particularly if you're watching this as a recorded session. And just a reminder that today's session will be recorded and that you'll be able to assess the recording through the website and also I'll be sending it to those people who are enrolled, through the link. So, yes, the music will disappear when I move on to the presentation screen, which is going to happen now. So welcome again.

Just a reminder for today as well that I'm doing a little bit of an interactive session, so if you don't have any Brenex squares or square paper and grid paper with you already, I suggest you send a runner to a classroom because we'll be doing a lesson today in the middle of my presentation. So hopefully people have some resources with them ready to have a go at a problem-solving lesson today. So today's focus is on problem solving as one of the Working Mathematically components of the new syllabus.

So the takeaways from the syllabus around problem solving is that students need to be able to formulate and solve problems. So it's not just solving problems or solving word problems but the students themselves are formulating the problems, and that also involves a lot of questioning both from the student and from the teacher. They're also required to design investigations. So they can do a bit of their own project-based learning. It doesn't just mean that they can have a choice of the investigations that you provide, but they can design some of their own, maybe something of interest to them around mathematics that they'd like to investigate. And later on, I'll talk a little bit about the types of questions that are useful for investigations, googleable versus non-googleable questions - if you've ever read anything about that, it's really interesting. And also, they're required to apply the strategies to seek the solutions. So it's really bringing in all those skills. We're trying to bring them up and understanding around from something like acting it out and guess-and-check, all the way through to creating a list and seeing patterns and simplifying problems. So they really need to have that toolbox ready to help them find the solutions to the problems that they're setting or that you're setting for them.

So it's also good to note that at times we might focus on a particular component, like problem solving, but often the components overlap, so it's going to overlap with communicating, it's going to overlap with reasoning. It's hard to do one without the other, really. So they might not all be applicable to all the content, but they definitely indicate the breadth of the actions that teachers need to emphasise. So not every outcome has all of the three Working Mathematically outcomes attached, it'll bring out ones that are more so relevant for that content. Not to say that if it doesn't have problem solving you can't then apply that knowledge in a problem-solving fashion, but it's just to let you know that that's the ones that come out the most from that section of content.

The value of problem solving - there's many values of problem solving but one of them is that problem solving resembles what mathematicians do. So our students in primary school are not necessarily high-level mathematicians into pure mathematics, and it's still very real life and contextual for them, but it's definitely about finding those solutions that might not be immediately obvious. So, how often do we ask in the classroom questions that as soon as we've got the correct answer we just move on from? How often do we probe for further questions or further strategies to find the question or to find the answer to the problem we're setting? So it's just that shift in how we operate in the classroom, so that problem solving is just a part of how we teach, not particularly just something we do on a Friday afternoon. So it really needs to be embedded in the way that you're presenting the content to your students. So we want students to have that opportunity to explore their mathematical understanding and to apply skills to construct new knowledge, so it's that from the known to the unknown and they're building on something they've already learnt before.

This is where my two computer screens are hopefully going to come in handy today. Just bear with me. Sorry about that problem today.

OK, so these are a couple of the shapes that I made, so you talk to students about drawing and here you're going to start looking at length and area as well at the same time as splitting and combining shapes. And even just talking through that that dotted line means that's where I folded my shape to make some different shapes there. You can see you just move across from left to right to make the shapes. Obviously I didn't get to fit in my last picture of what my isosceles triangle would look like at the end - I ran out of room on my grid paper. But I think this kind of drawing of their thinking and processes is a really important part of problem solving, reasoning and communicating their ideas. You could even extend this activity where the students could draw a set of drawings and then see if their partner could actually make the shape with the Brenex squares - I think that would be quite useful. And even just having those further questions about, "How did you start? What was the first thing you did?" Did most people fold their page in half first? Is that true of the people listening today? Did you start by folding your page either in half diagonally or in half to make two rectangles? Yes. OK, so you're using prior knowledge. I know I can make other shapes out of a square by folding it first in half. And then how technical did you get about the length of sides? Were you happy just to make irregular shapes or were you trying to make sure that the length and the opposite side length and angles would be equal? So you could definitely do a lot of angle work with this lesson as well. So you might want to stop here - this might be enough of a lesson because it could take you quite a while to get through this. It would depend on the prior knowledge that your students have and where they're at in their understanding. For me, I'm going to extend it that little bit further and go into now a task that's from the 'Red dragonfly' mathematics book.

And so, this is the activity, it's number 36. It's about splitting them into four. So it provides you with these four figures, and the idea is that the students work at trying to break those figures into four identical shapes within each of those. Now, this activity's quite difficult for a lot of students in particular. These are sort of aimed at Stage 2 for me, so I have sort of added an extra step in between. Obviously the answers are available in the book and also online if you want to have a look at those - this is one of the free activities that you can download. So what I did is I actually drew it larger on the grid paper to help with the process, because if I go back for a second…

You can see that it's quite small, and as part of the answer that it provides later on, one of the strategies students might use is to actually make the images bigger. Still the same dimensions but making them a little bit larger.

So up to you how you want to present it to your students, but I did this. So, now here's your chance. Get out your grid paper for me. Again, you might want to change this activity where you actually give the shapes already drawn on the grid paper. I prefer to make the students draw them themselves. That's a task in itself.

And as you can see there, each of the four identical shapes are exactly the same as the actual shape itself. So I think it's very interesting. Now, that might not be all the possibilities. You might want to have that discussion with your students in your classroom. So, then you could get your students to design their own versions of these or work backwards. I would also then have a lesson wrap-up where the students complete a class journal and we talk about what we did today, what they had to do and what they found out, OK? So that's just a little bit of an idea of a lesson and how I would use it in the classroom. I'm just going to go back to my presentation now.

So it's really important that I didn't start the lesson by saying, "Oh, today we're doing problem solving." I didn't present it that way for a reason, because it's just the way I'm teaching, it's just seeing that inquiry process as the way I'm teaching. It's not just a tool that we use on a certain day, it's all the time.

And when I look at that splitting shapes lesson as part of my scope and sequence, if I'm thinking about 2-D space as the topic I'm looking at maybe over that five weeks that I'm working through, I can see all of these other areas, all these other substrands and their key ideas, which are in those outside circles, that connect, that would then help me teach some lessons maybe prior to this lesson or teach some follow-up lessons or try and see those connections. There's even connections with whole numbers which I haven't put on there as well because people just counted the number of squares in their shape. Lots of work with fractions, particularly with the Brenex paper activity part of that lesson today. So there's lots there for students to work through. So I hope that just gave you a little bit of a look into what a lesson could look like that involves problem solving as the way that you're working through the content together, and that hands-on, concrete nature to build something from the known to the unknown. And we obviously did a lot of communicating there. Always, we're looking at reasonings and explaining what we were doing. So they don't really sit separately, but that's just showing how problem solving can work in your classroom.

So in the file pod today, I've just got some signs that I use. A lot of them are based on some Blake Education resources around different types of problem-solving strategies. If you're not familiar with the list there, they're the ones we generally look for and use within our syllabus. It's sort of in a hierarchical fashion. Students in Kindergarten use objects or act it out as the first kind of problem-solving strategies they use. And we work through those to create that toolbox for our students.

I've also got the links there to Newman's - Newman's are a great way to problem-solve in the classroom. If you're not familiar with Newman's, there's a little link down there to where it is on the Curriculum Support website. I've got a problem-solving proforma in there. It's based on one that Diane McPhail and I used to use that Diane created. I've just moved it around a bit. But my little numbers on that problem-solving sheet relate to the Newman's questions so that I have an understanding of where it is my students are having trouble with. Is it about their reading the question? Is it about their comprehension? Is it the way they transform the literacy side of the question into mathematics? Is it the process, the actual doing of the mathematical process? Or is it that they just didn't know how to answer the question appropriately? So that's in the file pod as well today.

There's also think boards - the Harrington think board. If people haven't seen these before, they're a great thing to use. With K-2 in particular, I use these because that Newman's one is sometimes a little bit more difficult for them to work through. So it's just having four quadrants. You can make it with a sheet of paper folded in four - you don't have to use the proforma - where it has, like, a story, it has pictures, there's a link to a number sentence, and then they can actually make it with objects. I make these up onto A3 size and laminate them and the students actually work on top of them as a mat. But as you can see, I've got a couple of examples there where you start out with the problem and they solve the rest of the areas. Or you can start with the number sentence and they can fill in the story. So great little things to get problem solving in your classroom.