Transcript of Working mathematically—communicating
KATHERIN CARTWRIGHT Welcome to Syllabus PLUS, 'Working Mathematically communicating'. If you can hear the music over the top of me, that will disappear once I start the presentation. You might be able to see today I have Kerrie Spencer behind me, and if I move a bit further, you can see Yvonne Hughes behind me. They're helping out with the session today. They're both on the chat and they're also going to be having a go at some of the activities we do today, as are you. So hopefully you've got your pens and paper ready and some Post-it or markers or highlighters with you as well, if you're with a group. If you're by yourself, you might be doing some chatting alone today. But thanks for joining us. I'll start onto the presentation now. Just a reminder this session is being recorded, both the presentation and the Chat, and will be available on our Curriculum Support website from tomorrow. So, thanks for joining us, and it's great to see in the poll questions that some people are here for the first time. So, welcome, and I hope you enjoy today's session. So, as I mentioned, today's session is about Working Mathematically - Communicating. It was the only Working Mathematically area I hadn't covered yet in the series, over the whole 4 Series. We've already done Problem Solving and Reasoning, so I thought it would be a nice way to end, would be to look at communicating. So if we look at the 'Principles and Standards for School Mathematics'... If you've not been to the National Council of Teachers of Mathematics website, it's a great place to go, and they have a lot of great resources, but the way they talk about communication in mathematics is that it's a way of sharing ideas and clarifying understanding. And we know that in our new syllabus, that that idea of understanding and being able to be fluent with our mathematical thinking and the way that we understand concepts is very vital for our students. So, it's nice to see communicating in that way, and those ideas become something that we can reflect upon and refine and amend. That's really important as well, that we can develop those understandings as we go through. And I think sometimes by talking through what you're working out, you start to amend on the spot as well, so it's really important that you've got somebody else to discuss with. There's also...this is from a capacity building series from Ontario. And today we're going to actually use a fair bit of information from this paper that's in the file pod today at the end of the session. And they talk about communication in mathematics, that it's not just about answering the question using words, numbers, pictures or symbols. And I think even for me, that's something that I've always done. Like, I see communicating as something where, "Oh, great, I've got the students to answer and talk about the strategy they're using," and then I move on. But what they sort of focus on is going beyond just using it as how to answer a question or explain a strategy. They talk about it in the idea that these forms of communication they've selected and applied in order that we can have a mathematical argument, so that we can be precise about why we use our strategies, we can be more precise, and it's about it being persuasive. It's moving away from just using them as, like, a descriptive narrative. And that sort of blew my mind. I thought that was really interesting, that I think we do use it as a bit of a narrative, so the students have a bit of a talk about the strategy they're using, and then we go on to something else. We don't really have a purpose for, What do we do once they've shared that information?" And what this is saying is that...well, it's so that we can be more precise about why we use these strategies, and that they're almost to persuade other people of why you would use my strategy in this particular point in time. And there's definitely times we want students to use particular strategies, and we're going to have a couple of activities today that hopefully will enhance that. But for me, this is a bit of a mind shift, was getting beyond seeing communication as the answer and more about how we actually use the communication. So, from our syllabus, we know that our students need to be able to use communication, both written, orally and also in graphical forms, and that includes diagrams, drawings, informal and formal methods. And we want them to use the appropriate language and terminology. And we also want them to be able to, I guess, organise their information using some of those symbols and conventions. And if we look in our syllabus for written, there's plenty of opportunities for students to partake in this around using symbols, notation and conventions. And I don't think it just means the operator symbols - I've got that example there from Stage 2 - but it's even pictures or symbols in graphs, so how we're using those, and also those conventions like the introduction of order of operation that's now happening in Stage 3. So, that's a convention that we use so that everyone understands it. And there is different ways of seeing that convention across the world, but in Australia we try and have a set way, so that everyone understands how to read our working out and how to follow the steps that we've taken. When we're talking about students expressing their ideas orally, I mentioned before we want them to use the appropriate language and terminology. So in Early Stage 1, it's that "is the same as" to express equality of groups. That's a really appropriate language we want our students to come to understand and use. In Stage 1, in Addition and Subtraction, we ask them to use specific words for the operations in preparation for understanding the symbols and notations for those, and the same with multiplication/division, and also being able to use some of those more generic terms that aren't specifically mathematics terminology that are really important in mathematics, like "twice as much" or "half as much". So they're wording that we want our students to be using. When it comes to graphical, expressing their ideas graphically, we want them to be using tables, diagrams and graphs. That could be about tallies to graph the results of chance experiments. We've also got it in data, where they're using and creating picture graphs of different data displays to express what they've found out from their investigation. And also in Fractions and Decimals, we want them to actually be able to use diagrams and number lines to show how we maybe add and subtract fractions, so that will lead to an understanding of a mental strategy, so that's that focusing on the visual that diagrams and graphs and tables provide us to deepen our understanding before we move to that abstract understanding. And I think that idea of the visuals that students see in their head is really important to develop their understanding of the concept in its entirety. So, students, for them to communicate, it's about organising and consolidating their thinking. And that organising is really important, and that's where we sort of see the idea that mathematics is about patterns and relationships really coming out. Because the more you can organise your working out, then the easier it is for students to see the patterns and then be able to make generalisations about them. We want them to be able to communicate to their peers and to teachers and also to others in the outside world. We want them to be able to analyse and evaluate, and that's where we're looking at those higher-order thinking strategies that go beyond just regurgitating information or what they think we want them to say. And then we also want them to just use the language of mathematics to express what they're thinking and the ideas that they have. So, today we're going to do a couple of tasks. And I mentioned that these are from Ontario in Canada, and the reading I got them from is in the pod. And if you want to go to the website as well about it, it's called LearnTeachLead. And this is where a lot of Canada have all of their professional learning for their teachers. It's very easy to use, and you can access a lot of it without being, obviously, a member of their education department, which I did. And there's a couple of other links in this page as well to other places I have seen these sort of strategies used. And so, these might be familiar to you already. They were, in general, new to a lot of us here, and they've been around for a while. But I think they really pinpoint that idea of moving past just having students respond with their strategies and what are we actually going to use that communicating for. And so, the first one is called a Gallery Walk. So, it's basically an interactive discussion technique that gets students out of their chairs and actively moving around the classroom, looking at other students' work and commenting on other students' strategies. And the purpose of that is so that they can engage in a range of those solutions and see how their peers work and their peers solve the problem, to then improve their own strategies. And I thought that was really a lovely way of working with it. Now, I will say that... I'll go back for a second. This Gallery Walk, obviously today we're not in a classroom all together where we can pin up work all around the walls. And as I mentioned before, you might be solo today, but if you're with a group, that's excellent. You still probably won't have time to put them up on the walls, but you could just hand them around. When you get to the tasks, you might just want to pass your paper around for people to put the Post-its or comments. And I think that will work fine for today. So it's obviously...if you Google things like 'Gallery Walk', they're used a lot with literacy lessons or English lessons as well, and you can manipulate them a bit, but in its true sense, they actually have the students walking around the room, hence why it's called the Gallery Walk. I'm just going to move to a video now and then I'll come back to the presentation, because I want you just to get an idea of how it kind of works. And this is from that LearnTeachLead website and this is just one of the videos. There's many that you can go and watch, and I do recommend you go to the website at a later date and watch some more of these videos. But this is just a bit of a taster
VIDEO NARRATOR John Van de Walle offers the idea that the beginning of a math lesson has the purposes of mentally preparing students for learning by activating their prior knowledge, ensuring the students understand the lesson terms and what is expected of them in their work without being told how to do it. In this class, students are working towards an understanding of multiplication and understanding where they will eventually view multiplication as much more than simply repeated addition. During their last lesson, the students completed a task that asked them to determine the number of pieces that are required to cover every square on a chequerboard. Visit 'Analysing Yesterday's Lesson' for more information. Today, as students enter, they are asked to do a Gallery Walk of their work from the chequerboard problem and to select and label strategies of their peers that make sense to them.
GIRL You can count by a number and then if you can't keep counting on, you can just stop and then you can just count by ones.
GIRL What could this be called?
TEACHER If your math partner isn't here yet, you can talk to someone else. Make sure you and whoever you're talking to agree. [INAUDIBLE]
TEACHER I'm hearing partners ask another partner a question - "Why do you think it's that?"
GIRL They only got to 40 so far and they don't know where to go next, so that's counting by ones. [INAUDIBLE]
BOY And they keep counting up by fours in each row. [INAUDIBLE]
TEACHER And why did you choose that one to comment on?
GIRL Some strategies aren't easy for you, so you can just use another number strategy and then continue on it.
TEACHER I agree, mathematicians persevere, don't they?
GIRL Then you can get a different answer when you don't...if you don't know how to count by that. [INAUDIBLE]
TEACHER OK, boys and girls, make sure all your morning jobs are done and then you can sit on the carpet beside your...beside your math partner. Mathematicians, what was your job this morning when you came in? And, Remy, you were starting to speak.
REMY We had to...
TEACHER And talk to all of your friends.
REMY We had to pick what strategy made most sense to us, and we had to write why it made more sense to us.
TEACHER OK. And why do you think...why do you think I would ask you to look at a friend's strategy and try to make sense of it?
REMY Because you want us to learn what other...what other friends learned.
TEACHER And I'd like you to talk about that with your partner. Why did I ask you to do that?
KATHERIN CARTWRIGHT So, hopefully everyone can hear that, and I do apologise for leaving my microphone open. I was just seeing if both Kerrie and Yvonne wanted to hear it because they obviously can't hear it when I've got the headphones in. But they have seen it before. So that's just a taster. And obviously this is something that happens a lot in these classrooms. They've obviously built up that environment with their students around how to feedback to each other as students, and also as the teacher, around the strategies. And those are things you'd need to build up in your classroom, being able to talk with students about, you know, what they noticed or what they were wondering about other people's strategies, so that when they write things on there, they're quite constructive. But you can definitely see how they're learning from each other. And those students then take back their work and have a look at the comments other people made before they come and share with each other. So that's, sort of, a little bit of an intro of what a Gallery Walk can look like in your class. And that's the link to the video there. So, we're going to try it today. So, as I mentioned, we probably won't get to walk around, but what I want you to do is I want you to answer the task I've got here. It is a past NAPLAN task. And my learning goal for today, if I was using this in my classroom, would be to understand the connection between finding a fraction of a collection and the operation of division, so how do fractions and divisions kind of have a relationship there. So, what I want you to do is I want you to, as an individual - because there might not be many of you at your school at the moment - so develop one solution to the problem on some paper, and then I want you to take turns looking at each other's solutions and either recording comments on their worksheet or using the Post-its to record your suggestions on theirs, or any questions you have around their strategy, OK? Then I want you to rotate and look at the next person's solution to the problem. And then I want you to get to the point where you get yours back and can have a look at what other people wrote on your...on your strategy. So hopefully you've got some pens and paper ready. So, I'm going to give you a few minutes to do that and then I'm going to call you back together and go from there. OK? OK, you might still be working through that task. Michelle Paton, I can see you've got your hand up. Do you want to type your question in the pod because it's easier for me to do that than to give you control of the microphone today. Did you have a question? Also, while she... I can see that she's typing. So, while she's typing, so, obviously, if you working it as a Gallery Walk, you'd be working on this as a small group, and you'd put it up on the wall and then the students could move around and make their suggestions or comments, or even note if they thought it was a strategy that was efficient or not, and if they know other strategies as well. And we were having a lovely conversation in here, and what Yvonne brought up, which is a really important point, is that, you know, what is the prior knowledge or prior learning that has to have occurred before students can access this task today? And obviously when I would do this with my students, we would have probably already done some work around fractions. And you can see that we're harping back to that 'Making Connections' idea. And so the idea is is that these students are actually learning from each other. So it's not just about, "Oh, someone's sharing the way that they've solved the problem," which is normally how we report back. We normally just get everyone back together and we say, "Oh, how did you work it out?" We want them to actually go and read other people's strategies. And what Kerrie mentioned in here as well is that that sort of changes the way she wrote down her working out, knowing that someone else was going to have to read it and try and understand it. And I found that too. It made me try and be more particular and maybe set it out a bit nicer and with steps, so if someone else was trying to see how I solved the problem, they could see my working out. So, do anyone have any comments about the types of strategies you saw? What kind of strategies were people using to solve this problem? And you can type that in the chat. OK, so 12 x 3/4 x 2. And you'll find that some students don't get to the 2 bit at the end. They'll get the cost of one child and put that in the box and not realise it was asking for two children. You might have drawn a picture to represent the fractions, yes. Yes, finding quarters first. Just how many 1/4... And that's a really important thing, and I'm going to get you do another task in a minute that looks at that kind of idea of finding the unit fraction. And that's going much higher into Stage 3 and then beyond when we start to look at ratios, but that strategy is really important for students to understand, "Well, once I know the base rate, I can then work out other amounts." Some of us divided by 4, by halving twice. Yes, that's a lovely strategy as well. 3/4 x 2, and then 1 1/2, yes. So you know that two lots of 3/4 gives you 1 1/2 and then you can times it and work out the answer to that, yes. And obviously it would be nicer if I could get everyone just writing this up on the whiteboard than having to write it as a sentence, but you're doing very well there explaining how you worked things out. OK, so you can see, and a lot of them are quite similar but they're just a different way. And sometimes what you'll find is that students might have solved the answer, but then when they go to write or communicate their working out, they don't actually write what they did, they write what they expect you or what they think you want to see. And so, when you get this opportunity to go around and make comments, they might see those other strategies and go, "Oh, that's actually what I did." Or they might help them for the next time they're trying to work it out. And this is obviously also a great time for the teacher to roam around the room and look for the misunderstandings and look for the gaps in these students' understanding around fractions, around division, around operations, and then that can be like a formative assessment to lead their next lot of understanding and their next lot of teaching and learning. Thank you for all those comments and thank you for participating today. I really appreciate it. Here's just two pages of a snapshot of what I did. The one with the picture of the quarters, the little window, is how I first worked it out because I'm a very visual learner, so I knew that when I do the 3/4, that's $3 per quarter, and then I knew what 3/4 were. And again, see, my talking about my strategies is much clearer than, say, what Kerrie says, that if you go and actually just read that on a wall, you might not actually understand the thought process. So you really want them to start thinking about, "Well, what am I doing?" And when you look at that one on the top right that I did, which is, I guess, the pure maths, the quick version of writing that, it depends what your purpose is. So, was my purpose in this activity to get an answer or was my purpose to see different working-out strategies. And if my students can do that, 3/4 x 12 - they know how to do that to find out the answer - if they know how to do some of those operations, can they explain it then? Can they do a written explanation of how you do that? Or could they teach that to someone else who doesn't understand how to use fractions to help them with this division problem? So you can see...and that's not all the different ways you can do it either. And someone else has made a comment in there as well. You've been talking about how... showing our thinking. We just need to give an answer compared to explaining our thinking. Yeah, and I think this is quite a simple thing. Like, you don't have to do it as perfect as the article talks about, Gallery Walks. But I really like the fact that they're commenting and being peer assessors of each other. And that's gaining them...developing their knowledge as well. And that they then report back on having...and they get to see what everyone thought, immediately, about their strategy. And you'll see...and some kids might use more of a repeated addition and that would be fine to work it out. But if I go back and my learning intention is about seeing the connection between fractions and the operation of division, if my students only know how to multiply or use the repeated addition to solve it, they're not actually getting to my learning intention. So it might be a workable strategy but does it meet my learning intention? So I really like the way this works. So it's gauging a students' understanding and their use of vocabulary. And you might even see some of the symbolic notation they use and is that working for them. So after that Gallery Walk has happened, they share their strategies like we've just been doing and then that provides the teacher an opportunity to summarise the mathematical ideas that relate to the learning goal, which I think is really the important part of that. It gives them that immediate audience, they're motivated, and they're learning new ideas. The other aspect of it... They have three in this article. They're the ones called 'Bansho' which is a Japanese word for 'whiteboard' or 'writing things on a board', which you can also read about. But I'm not going into that one because I've only got half an hour today. But if Math Congress... Now, obviously this is American because they don't use the word 'maths' or 'mathematics', but you can change that if you want. But it's called Math Congress. And it has some similar ideas to the Gallery Walk, but with this one we're wanting to really reason about a few big mathematical ideas, so it's really important that we don't go outside the box. We want to rein the students in and just focus on maybe one or two solutions. And it's about those solutions that, again, meet our learning intention. So, the students will work out the problem themselves, and they might have a go at it by themselves or in a small group again, but when it comes to sharing, the teacher only picks a group, one or two, to share where it matches the learning intention. So where the last one, everyone kind of got to share their strategies and we got to talk about what was efficient about it, in this one it's just picking out the ones that absolutely match the learning intention. So, all the students could obviously give their responses, and that's what we generally do in the classroom, but how do we use those responses they're giving us? Do we pick them specifically for a purpose? And this is where the teacher needs to have an idea of how the learning intention matches the strategies you're looking for in the classroom. And we want our students to also see any commonalities. So when they look at the strategies, when you have that discussion with the students, can you get to the point where you say, "Well, what was similar about these strategies?" or, "What was different?" or, "What can we learn from this?" So I'm going to get you to have a go at this Math Congress one. So, again, my learning goal is a little bit like the other one. We're still looking at almost like that. It's the precursor to rates and ratios, really. So we're looking at that a unit price can be found using division and that also multiplication or division can be used to find equivalent pricing. So that's sort of what my learning goals and intentions are as part of this learning. And this is the task. It's from the reading that's in the pod today. So I'll read it out for you. There's a cat food problem. It says, "Kittens have to eat a special kind of cat food. There are two stores that sell this kind of cat food. The cans are the same size and the same brand. Which one is the better deal? And show your work." So for today's purposes, I want you to explore this strategy and try solving this problem yourself in two different ways. OK, so this time you're not going to put Post-it Notes and write comments on other people's solutions. You're just going to solve the problem yourself. And see if you can solve it in two different ways. OK, off you go. Pens and paper out again. So don't swap your solutions around. Just keep your own one or the one you worked on with your buddy, or you can work as a team. But see if you can work out two different ways to solve the problem. I'll give you a couple of minutes to do that. Thanks for hanging on. I can already see people are starting to write their little strategies in the chat pod for me. Thanks, Michelle, for getting that started. I know it's 4:00. I know it's actually 4:01 now. But I'm going to keep going because I want you to get the full experience of this today. So, I can see that Michelle's group there, they worked out what it was for 10 cans. They can see which one's going to be cheaper, if it's Bob's or Maria's. Are there any other ways that people, sort of, solved that problem? So they were looking for a common. They went and found a common amount between the two. Long division. Someone used long division. [INAUDIBLE]
KATHERIN CARTWRIGHT Actually, we just had a little conversation about HMS 'Bring Down'. I have a fond memory of that from Year 5 mathematics. I always remember that, but a different topic. So, someone worked out the individual price using division. Repeated halving. There's another strategy right there. And so, if I was in the classroom as a teacher, I would be going around and seeing the different strategies that my students were using to solve this problem. And when I get to the point where I want to have that Math Congress, where I want to bring the students all back together to have a discussion, I'm only going to choose the strategies that fit my learning goal, OK? So, I can see other people there rounded the number of cans up to 100. Don't know why, but it worked. That's fine, as long as it works. As long as it works every time, and that's one of the questions you might want to ask your students - if it works every time. You could find the first common denominator. Lovely. 60 cans. I can see some really good thinking going on today for the afternoon. I'm very impressed. All of those people that will end up watching us by recording will get all the answers that are provided for them. But hopefully they'll stop along the way and have a go at these tasks themselves. OK, so using a calculator, compare those fractional divisions and looking about the division with the denominator. So, lots of different ways you can solve this problem. Looking at mine, I know it's a little bit fuzzy on the left-hand side there, but I did the division. But I sort of went from the other side - I also used a bit of addition at the same time because I knew that, "Oh, it's about 1," so I know that there's $12, that gave me a remainder of 3. I then kept breaking that down until I got to my 125, and I did the same thing with the 23 divided by 20. But again, it comes back to if someone else was reading this, I would probably have to write that in a better, step-appropriate way so that you could understand what I actually did. I put that one on the right there because you are going to have students that don't understand the problem at all. Even when you do know what prior knowledge and prior learning your students have had, you will get students that just take those numbers away because they see the numbers, they want to do something with them and it works for them, and they might think that they both have the same...that they're both the same value if they take it away from each one. And that's the whole idea of this Math Congress, that instead of just having random students share their strategies now as sort of like a reflection session, which is also good, it's going a bit deeper. It's saying, "OK, now I am going to find a strategy that matches my learning goal, and I am going to go with that one and explain it to students and share that with everybody." In the article, when you have a look at it, these are the different solutions to the cat food problem. Now, obviously this is going into Stage 4, where we're looking at rates and ratios, but it's really important to see that these are the different types of strategies that we want the students to be using, and therefore they're the ones I'm going to pick out to share to the whole class. And we saw, I think, 3 out of the 4 of those. Someone used repeated halving. Someone used going with straight division to find what one can would be. And someone found a common core of 60 cans as well, just like the common whole example here. And what those lead to when you start using the division is that you can start to create that ratio table over there, which is about being able to utilise the information you have to find out how much other cans would be worth. And that's where it's going. But you can see that you're trying to find the strategies you want to share with the students. So, for the Math Congress, the teacher will select the specific strategy, they'll bring that group up to share it. And they might ask questions like - once they've got the two groups to have a go - How is this strategy similar or different from the first one? Will the strategy always work? How do you know? When will it not work? Why not?" It's just continuing with that questioning that we're sort of trying to encourage to focus on the communicating being rich. OK? So it's not about showing every solution. It might be about the ideas, sharing the ideas that have to build on the learning goal that leads to the next lot of learning. We want to see if they can generalise their strategies, possibly sequence them for discussion so that it can scaffold the next lot of learning. So that's why we're using a Math Congress in the way it's being used for communicating. You also need to think about preparing yourself mathematically, you know, especially if maths isn't your strong point and you know you've still got to present that to your students. So, you know, you're going to ask these sort of questions about, well, what mathematics is evident in what they're communicating. What mathematical language should they be using? What mathematical connections are we discerning between the different solutions? So, trying to get some commonality between what the students are communicating back to you about the strategies they're using. I'm not sure if I shared this one before, but I'll put it up again anyway. I got it from Anne Prescott from UTS, and it's a really lovely little poster about higher-order questions for higher-order thinking. And it's got those "who, what, when, where, why, how" questions and then the "is, did, can, would, will, might", and you could put 'could' in there as well. So things like, "When will this strategy work? When might it not work? Where would you see someone use this strategy? What can you determine would happen...should happen next?" So, those little coloured boxes in the middle is just where you'd plot different questions for your students to answer. But I just even like the idea that I can start any of my questioning with any combination of those words. And as you get to the higher ends of both of those arrows, you're going to get into more complex questions you're asking in the classroom to facilitate that communicating. So I hope that's sort of helpful. I know I went overtime today. I hope it's been helpful for you. They're my details, and Kerrie's details as well, if you want to contact us about anything to do with Mathematics. That is the final session for our Syllabus PLUS Series 4. I do have something else in my mind for the pipeline for Term 4 that I'm going to talk to Kerrie and Yvonne about, so stay tuned. I will send out emails to people that have participated before about something new I'm thinking about undertaking. I hope that was helpful. I'm going to go to the conclusions slide now. There's a link to that website there. There's a couple of documents for you to read, and the presentation from today in the file pod. Thanks for hanging around a bit longer than usual. I hope you found today helpful to do with communication and it really spurs you on to go that little step further than just seeing communicating as giving answers. But let's start using what the students are communicating so they can learn from each other as well as from us. So, thanks for participating. I hope lots of you go out and have a go at Gallery Walk or Math Congress tomorrow in your classroom. And please go on and evaluate this session on your MyPL if your school set it up, or if you're the contact person in the session that I set up. And as I mentioned before, this recording will be available from tomorrow. So, thank you and good afternoon. I'll be on for a couple more minutes if there's any more questions in the chat pod. Thanks.