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.
Here's a Wordle - I still like using Wordles in my presentations, I like a visual format - and this is just sort of explaining and showing you the wording just from the syllabus around problem solving itself. And I think it's really, you know, interesting what it brings up about problem solving, and sometimes we might have a particular idea of what problem solving is, about strategies students use, about guess-and-check or about working backwards, but really it focuses there on solutions to situations and that they're formulating solutions to these problems. The answers are in there as well, but it's that...communicating is there as well which is one of our other Working Mathematically components, and it's also about students making choices about the approaches that they take to solving problems. And it also has in there about reasoning and "Is my answer reasonable?" and really something to think within themselves about their learning as well. So it's really important that we get that over-arching picture of what problem solving really is and maybe just widen our scope of what we already come with, a pre-idea of what it is, and it's always good to go back to the syllabus again to look at what it says about problem solving.
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.
So today I'm going to go into my lesson. It's called Splitting Shapes. It does draw on an activity from the 'Red dragonfly mathematics challenge' resource, which, as I've mentioned many times before, if you google that title you'll still find the online version of it in Curriculum Support website. There's also a free app that you can download for your iPads and I do still have copies of that book as well. And we are looking to make more of it readily available online in the future. So I use that along with another activity of my own to divide the lesson. The lesson will be in the file pod today along with the notebook that I'm about to open into. So it's about splitting shapes. So now's the time for you to have your Brenex paper ready and you might also want to have someone near the computer for the chat because I'm going to be asking questions. So if you just bear with me for a moment, I'm going to open up my SMART Notebook.
This is where my two computer screens are hopefully going to come in handy today. Just bear with me. Sorry about that problem today.
So we've had some shapes already that people thought that they could make, which was wonderful. I'm just going to scroll back up here and have a look. So irregular and regular hexagon and pentagons, and that's a really nice focus on language as well - what do students understand about 'regular' and 'irregular'? So are there any shapes that you think might be difficult to make by folding the square? And when I talk about folding it, I don't want any little overlappy bits hanging out the side. In particular, we're looking at making other 2-D shapes using folding. Circle obviously. Anything else? Oval. What about any of our shapes that have straight sides? Are there any shapes you think might be hard? Maybe a trapezium might be a little bit more difficult. Keep going. We could get into a few different ones there. That's great. So we'll come back to visit that kind of questioning once students have a go. So I would probably show the students how I could fold it in half to make a rectangle. OK, you can't see me on my screen at the moment, but I have a rectangle sitting next to me, and so I would be getting the students to have a look at what I did and show them about aligning, which, again, is linking to measurement within my 2-D space lesson. And then I would say, "OK, now I'm going to give you some time to have a go." So I'm going to do that right now. Have a go, have a fold with the people there, and once you've actually made a particular shape, can you then type it in the chat pod for me? A rectangle, yeah. OK, what type of triangle, Megan, did you make? And a trapezium - is it a regular trapezium or is it an irregular trapezium? A kite, yes. I will just get people to stop typing for a moment. I want to have a little discussion about that word 'diamond'. OK, if that's a square on its side, it's still a square. The word that we use in Australia is a 'rhombus', and it has to have corners, or vertices, that are not right angles. So those people that say they've made a rhombus, does your rhombus have angles that are other than 90 degrees? Thank you, Megan, you made an isosceles triangle. Is it also a right-angled triangle - is it a right-angled isosceles triangle? Or did you work out how to make an isosceles triangle without right angles? I've seen someone use the word 'oblong'. I do love the word 'oblong'. It's sad that we've sort of let it go past. Without right angles - excellent. Because that's one of the challenges that I would bring to the students, is looking at different ways that we can make shapes that aren't necessarily the ones we particularly imagined the first time. Parallelogram, which is, obviously, a rectangle's a parallelogram. I know that I found it difficult to make shapes that didn't have right angles. The trapezium was fine, but then, making a parallelogram I found a little bit tougher. And along with the rhombus, I found that quite difficult. So I've had students have a go at making these, share them, possibly working with themselves first then with a partner. I think the good thing about using paper is that if they make a mistake they can either refold the one they've got or they can throw that away and start a new one, and it lets them explore the different things they can make. So I might ask them questions about how did they check? How WOULD you check to make sure that your shape was a kite? Let's say you chose a kite, how are you going to check to make sure it's actually a kite? Just to answer Lee's question, a rhombus IS a parallelogram - all opposite sides are parallel. A parallelogram is, like, a category for many of our quadrilaterals. Obviously a trapezium is not a parallelogram because not all sets of opposite sides are parallel. Lovely - "Our rhombus has two acute angles and two obtuse angles." Lovely. And it's a shame that, you know, Adobe Connect is really great for doing this kind of interactivity but I would still love to see some people's videos and images of what they're making, so feel free to take pictures of them and email them to me. I would love that as part of our lesson today. OK, so I've had students talk about well, maybe we could use length to check that our shapes were either regular or irregular, and knowing some of those properties of our two-dimensional shapes is going to come in very handy, so there's definitely some prior knowledge our students are going to need. So I'd also then have the students attempt to draw the folds they made, draw through the steps that they took to actually fold it to make that other shape. I'm not going to ask you to do that with your grid paper today - your grid paper is for a different purpose. But I would have them have a look at that. And then I'll show you what mine looks like.
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 so, we want to allow the students to either work individually, by themselves, or with a partner or even in a small group to work out the problem. So I'll give you a few minutes now to have a go and see if you can split any or all of those into four identical shapes. And just note that wording for 'identical shapes', not identical areas. I need some working-out music. No, they're not allowed to overlap. And if you have the 'Red dragonfly' book at your school, you are not allowed to cheat. Just give me a yes if anyone has got an answer to any of those four shapes. I've got one, top left. Top left as well. Those bottom ones are a little bit more difficult. Yeah, they have to be identical shapes, so, yes, they have to be exactly the same. Other things with students, you might even want to let them have scissors or other resources. They're most welcome to chop it up to see if they can make it a little bit like a puzzle, I guess. That might be able to help them. They might have drawn a different diagram to help them. Did anyone have a strategy for working out their answer that they want to share? "Can you cut and replace a part?" You can't move them, no. You should be able to do this by drawing lines and you'll be able to see. Ah! "Count the squares and divide by four." Thanks, Diane Read. We divide the squares by four. So either using counting or an area method, you're using some of your number and pattern strategies to work out at least, well, I now know I need a shape that has, let's say, three squares in it. Which, when I look at those shapes at the bottom, is probably going to include half squares. Having that advice, does it help any other people now to solve any of those problems? Yes. And that's the idea. We're all learning together, we're sharing our strategies, giving a little bit more information, trialling it. You might want to give the students one answer at a time and then let them see if that helps. When you had the answer to that top-left one, I know it's a little bit hard, but can someone describe the shape that each of their four identical pieces is? It's an L. It's an L. It's an L. What about the top-right one? Did anyone solve that one and can tell me what shape that one is? Anyone for the top right? So depending on how your students are going, you might want to give them a hint that that top-left one, the shape that you then made four of, looks a lot like the shape that the whole area actually is. So I won't push you any longer because I know I've got a few more slides to get through back in my presentation, but I will show you the answers because I know that you're dying to find out. If you are watching, this is a recording. Please feel free to pause for a bit longer and have a go, but today I need to get through my session in my half an hour.
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.
I mentioned before about googleable versus non-googleable questions. This is something that hit me after seeing Ewan McIntosh and I think I saw Jennifer York on today, who brought him out for one of the ICT conferences, and I still go and read his research and the work he does around problem seeking as well, which I've got a slide about next. But this idea of googleable versus non-googleable questions - problem solving is embedded really deeply in mathematics and we do often ask questions that are easily answerable by doing a Google search, but it's part of the process for developing problem-solving skills. But it does get to a point where we actually want to push students past that, asking questions that aren't as easy to search through and find an answer quickly. We want to start fostering meaningful problem posing from our students as well. And I think that can definitely be utilised in primary school as well as secondary. So it's something to look into. The link is there if you've never been there. And they do a lot of work around design thinking and I think it's really inspirational. I could listen to him all day probably. If you were up at EduTECH in Brisbane, I think it was last week, he was up there as well. A lot of people in the pod are saying how much they like him as well.
Question stems. If you're not familiar with using question stems in your classroom, for me, questioning is where this all lies. For me to use the questions, for the students to use the questions. Sorry, the man's name is Ewan McIntosh. There's a slide coming up that explains who he is. So these sort of question stems, there's some signs in the file pod that you can access today - how did you know? You either finish that question off or the students finish that question off. You know, is there another way to solve this problem? How did you work it out? And it actually links to something that I found through Ewan's site about thinking cubes, where they're like dice that the students roll, either one or two of them together, and it helps them to pose the problems that they'll then solve. So you can actually buy them, these thinking cubes, but you can make your own. And there's actually... I've put a document in the file pod today that you can make your own from. So up to you if you want to spend the money to buy the commercially made ones. But, again, great way to just start thinking happening in the classroom.
So here's Ewan's video here - this is one of his TED Talks he did, I think it was 2011. I'm going to say that. I haven't got time for you to run through it today. It's only eight minutes long. There's another longer one - an hour one - it's really interesting. But he talks about problem finders and seeking the actual problems, not actually solving them, and that's really interesting, so I do recommend you have a click on that and watch that. It's hyperlinked there and there's also the link on the right there. So I find him really interesting in seeing where problem solving can go.
Final comments. So how much of that learning work are you doing as the teacher compared to your students? So how much of that research around the problems are you doing before you even get to the classroom? So it's just something to think about. And that whole idea is, do we need to present all the content in the same way? People are quite familiar with flipping the classroom. Is there things that the students can do and then bring their questions about it back into the classroom? Thinking about that and making those connections so that we see this fluidity between problem solving and real world and their lives themselves. So that's something we really want to bring out with our students.
That's it for today and for this term. I am running another series next term. I know you might be Adobe'd out, but I'm just trying to provide you with some resources. One of them next term will be on financial mathematics and probably another one will be on integrating something like maths and science and tech - I'm going to get together with Tanya Coli and do a presentation together, so... And, remember, if you have any other suggestions, it's great if you can complete the online evaluations either if your school scheduled the My PL event or if you enrolled as the contact person in mine, I really appreciate your feedback and suggestions for future sessions, if there's things you'd like me to re-explain or that you're still struggling through. I'm just going to... Also, I think there's one more slide there.
Just a reminder that we have that 'Mathematical Bridge'. That image is now actually hyperlinked to the website where the newsletter can be accessed, so I finally got them up and downloadable from a website - in Curriculum Support. So click on that link now before I go to my conclusion page. Because it won't be there in a moment. But it is in the presentation if you download it from the file pod today.
Going to my conclusion... [INAUDIBLE] ..slide today, as I mentioned, next session is on 22 July. There's all the files in the pod today. There's quite a number of them, so you might need to scroll down and save them. Remember, they probably open up behind this window if you're trying to save them off onto your computer. And, yes, you do have to save them one at a time. But thank you for listening. Thank you so much for participating in the lesson today. I really appreciate that. I hope you enjoyed doing something a little bit more interactive today as part of the lesson, as part of the session. And I hope that you have a lovely break over the holidays, and thank you very much for listening. I'll be on for a little bit longer if you've got any questions.