Transcript of Making connections—part A
Katherin Cartwright: Well, hello, everyone. Good afternoon. Wonderful that you could join us again for Syllabus PLUS K–6 Mathematics. I'm here today with Nagla. She's behind me there. You can see her on the chat today so if you have any questions, Nagla will be happy to help you out today. Just a reminder that today's session is being recorded and the recording link will be sent out to your contact person tomorrow so that you can share it with other members from your staff who possibly couldn't make it today or if you're going to use it in future for teacher professional learning. So welcome.
So today's session is on making connections and it's Part A. So, yes, there is a Part B and that's in two weeks' time. So make sure that if you are going to use these for professional learning that you watch them in order for these particular ones because they do build on each other.
So making connections. So our topic is broken into two sessions and it's really focusing on how mathematics is interrelated—so within the subject of mathematics, what goes together and what is built on one another. And we really want our students to see that maths are not just individual building blocks, which is an important aspect, but they're interconnected pieces. So I've sort of used that puzzle piece today as my little image to get you to see how they actually relate to one another and I think that's really important for our students to see how it connects together. So today I'll mainly be dealing with number and algebra links with itself and also with measurement and geometry, and next week I'll deal with links to statistics and probability and where the strands link to one another when it's not a number strand.
So when we come to programming for making connections we have a policy for assessing and reporting, which at the moment is under review, which people will get the opportunity to comment on as well. But at the moment, it states that our programs are based on the key ideas related to the syllabus outcomes and that it needs to build on what our students can do, and that advice is still current. We really still want to focus on those key ideas to understand the concepts that we can break our syllabus down into. So that 'Continuum of Key Ideas'—it's a hyperlink there—is on our Board of Studies website. As people will know, the key ideas do not sit within our syllabus anymore for the new syllabus because they're not content, nor are they outcomes, but they're a great guide, particularly for programming and scoping and sequencing our lessons, so it's really important that you still access those key ideas. Nagla's been working really hard on getting a poster ready for that as well for schools that will, hopefully, be out soon.
So we have a course available called 'A process for programming a unit of learning in Mathematics K–10'—there's one for each of the Phase 1 syllabuses—and we have a diagram in there that talks about the process we go through when we're programming and that idea of selecting key ideas and concepts is really high up on our process. So there's more that comes below that—selecting content—but this is just a snapshot of what happens at the beginning. So making sure we understand what the concept is, we want the students to actually learn about, is one of the first things we do and it links to that quality teaching question of, ‘What do I want my students to learn?’
And that's just a slide there with the link at the bottom to that course if you want to undertake that course. It's registered hours. It's five registered hours that you can do as a staff or as a single teacher. It is better if you do it as a group because there are a few tasks in there that you could work on together. But that gives you a better understanding of a process for programming within mathematics specifically. It's new.
So when we talk about a scope and sequence, our Board of Studies—which if people aren't aware is now called BOSTES, which stands for Board of Studies, Teaching and Educational Standards—they make a comment about ... that it's in a sequence. So the things that we're teaching our students need to come in a particular order and the emphasis needs to be given on certain areas more than others, so we do encourage teachers to spend more time on number and algebra than the other areas of our syllabus, and that there will be adjustments that are required based on the needs of your students. So if you're using a whole school scope and sequence, you still need to adapt it for your students.
In our maths programming support on our website, we give advice around scope and sequence for each stage and the programming is across all strands and it's in a five-week block and that's something that we still recommend—that idea of sort of chunking out your term into a five-week section and within that, you might want to have two or three consecutive lessons per substrand. So that's a recommendation. It's not mandatory. But I guess what we're trying to move away from is just those one-off lessons. I think when we used to have the blue and white syllabus before our current one, because we had number and measurement and space, we'd do three days of number, a day of measurement, a day of space and it all fitted in quite nicely into a week, but what it meant is that the students only got one lesson with which to understand a concept and there was no depth. So when we're talking about retention of knowledge, it really didn't work for students. So we're trying to move away a little bit from that one-off lesson and then particularly if you happen to miss it because of a swimming carnival or something that's happening in your school, that topic might not get revisited for quite a while. So we're trying to sort of recommend that we have at least two or three consecutive lessons—as in consecutive day after day—where they can build a little bit of a depth of knowledge around that topic. And, look, they might overlap with other substrands as well.
And you can see that here. I saw that Tracy's on from Beverly Hills North. I did some work with them last year around scope and sequences and we did a bit of work there on a five-week cycle and trying to nut out what that might look like, so that's just a little example of what a five-week scope and sequence may look like in your school. They are different in every school, though, as well.
So why five weeks? So I think the move to that was so you're not trying to stick to a rigid weekly focus. I think that puts a lot of pressure on teachers and you feel like you're always playing catch-up and you never actually get...get to what you really want to teach and you might not get through the concepts, so you feel like you're chasing your tail. So we want to try and move away from that and it lets you spend more time on a concept if you need to. So when you've got five weeks to play around with, you might decide, ‘Oh, I'll actually spend an extra day on that number concept instead of just looking at it for two or three days.’ It also allows you to possibly plan your scope and sequence together as a larger group, but then when it comes to your teaching your classroom you've got a bit more control over how that lesson is taught or how those group of lessons is taught depending on the needs of your students. And we also want to make sure that we're revisiting number concepts and other concepts as well, but particularly number concepts quite regularly throughout the year so if you have, like, a five-week block there's a good chance you'll get to revisit those number concepts maybe twice a term and that helps you to build on that prior knowledge. So we don't want to do something for a whole term and then not deal with it for another term or two because the students won't be able to recall that information readily.
So I mentioned that some of that advice came from our 'Mathematics K–6 Programming Support' website and you can still find it either through googling or through 'Curriculum Support', but it's sort of in an archived mode at the moment. We don't have a new web presence at the moment, but the advice that's given in there around scope and sequences is for our current syllabus, so I've updated those for the new syllabus and they're in the file pod at the end of our session today. So you can download those, and I am hoping to replace them up on that website soon so that if you do happen to go there, the information will be up to date for the new syllabus. I don't know if you can read that, but the contact details on that still says Bernard Tola and I know he hasn't been in here for a while, so we need to update those as well. There's also a link within that site to the 'Programming Support' with those little colourful buttons there. Now, they are based on our current syllabus and the resources that we use with our current syllabus, but they're still really, really useful and I know in the past sessions I've recommended a few activities from some of those resources that we are looking to update, but still go there and have a look around. So you can still find your way to the 'Curriculum Support' website. Yeah, googling is probably one of the easiest ways to do it, but both of those images are hyperlinked today.
So those scope and sequences that are broken into five-week segments for each stage—so as I mentioned, they're in the file pod today—that's just a little snapshot of what one might look like. That's for Early Stage 1 and that's the first five weeks of Term 1. We just give you the substrand headings and then you can decide what key ideas out of those substrands you'd want to deal with within that time frame. Now, to give you a little bit more assistance, I've also created a 'Linking number concepts across the strands' document. That's also in the pod today. And so what I've done there is I've got the outcomes down the left-hand side, you've then got your number and algebra key ideas from the 'Key Ideas Continuum' in the middle section, and on the right I've put in there links to other substrands. So it's obviously not an exhaustive list, but they're just some suggestions of ways that you could interrelate those number and algebra concepts to other substrands, not only so that you can see the relationships for students, but it also means that any sort of assessments that you want to be running, either weekly, fortnightly, five-weekly, termly, you can probably cover more than one outcome in those assessments. So you're not just looking at whole numbers. You might be looking at time and whole numbers together because there's some relationships that they have together.
So when we talk about making connections to develop our maths program, we want our students to have a thorough understanding of number concepts and a lot of the other concepts that we deal with in mathematics, build on those number concepts. So if they don't have number sense and that whole idea of understanding place value in multiplication and division and the four operations, they're not going to be able to do some more difficult maths that's required of them or they may be using inefficient strategies, which is why we look at the numeracy continuum as well to see if we can bring those students up to speed that are not quite there. There's also a real push to look at data and graphs in all syllabus areas and also in the real world to provide some context. If you're teaching data, it might not even be in your maths timeslot. It might be part of a HSIE or science unit. So that's a really good way to get across to other KLAs with your mathematics as well and really link to that prior knowledge. Like, make it clear, you know—‘Hey, we're doing science today. But you know what? The skills you're using are mathematical skills.’ And they need to understand that those things, you know, cross-pollinate.
So here's a little example about making connections with basic knowledge of operations. So some people might have already seen this before. It's a countdown game. It's like the 'Letters and Numbers' game from the SBS show. If you're a bit of a maths or English nerd, you might have seen this before and it's a really great way for students to explore their knowledge of operations. So basically you create the number. You then have to use those six cards to make the number 680, in this case. You can use multiplication, division, subtraction or addition and you can only use those cards once. So if you cover up the answer I've written below and have a go yourselves, feel free. So I found one way that you could get there, but you could always find more ways and that's part of the activity you do with your students, is getting them to explore, ‘How else could I make that number as well?’ And you can keep playing that game and it'll keep resetting and has an infinite number of options that you can try. So it's really good for using basic knowledge around operations and seeing numbers and their strategies as flexible.
Making connections and that idea of integrating data, there's so much out there. There's a wealth of data resources that go beyond our syllabus. So there I've got the Weatherzone app from my phone. It has some lovely graphs on there that are quite complex for our students to understand, but it's great to have a look at them and delve into, ‘What do all those numbers mean? How is the graph being used? What does it tell me?’ so that they can start to work out what graphs might be appropriate for them to use for specific purposes. There's also some data there I grabbed from the Sochi Olympics website on the gold medal match for ice hockey, and I found that quite a complex graph. There's a lot of information there about time and timelines and also two-way tables. So it's interesting for students to explore how data is used in the real world, not just in mathematics.
So some other links for scope and sequencing is integrating patterns and algebra with number. Now, in our new syllabus, they're together. They're back together, but they've still got quite specific purposes and they interrelate really well, so they need to be taught together, but there's also some aspects of patterns and algebra that you'd want to draw out and link to other areas of mathematics as well. You know, there's a lot of things to do in patterns and algebra to do with coordinate geometry now and so that's another aspect of patterns and algebra where it doesn't only link to number as well. We also want our students to understand inverse relationships—really, really important for both addition and subtraction and multiplication and division. We want our students to be able to see the links between multiplication and area and we also want to see ... Help them to understand where fractions link with measurement. So when you're scope and sequencing, putting these things next to one another or teaching them together at the same time, really means that you're giving those kids a real depth of the concept.
So making connections with inverse operations—something as basic as using the turnaround facts and using coat hangers. If you've not used those before in the classroom, it's a really good way of helping students understand the communicative properties of numbers. So you could have 7 + 3 there or you could turn the coat hanger around and have 3 + 7. You could even look at it with subtraction as well. So you could make it differently, and you can also use that for multiplication and division. So it's just a nice visual and helps students to see some of those properties of numbers that are explained in our syllabus. So really simple and easy to use. A lot of K–2 teachers already use that in the classroom. I've seen it in many classrooms that I've visited. Also, looking at multiplication and seeing how it relates to division, arrays are one of the most brilliant teaching tools you could use in your classroom. They also can go into fractions as well looking at halves and quarters and fifths and thirds of numbers. So I find them really useful, a great visual for students to build that understanding of area and they can see where the rows and columns are and they can relate those facts together. So this is an activity that Diane McPhail used to use from my old region at South Western Sydney and I've used it ever since—love it. And those activities are in the file pod today as well if you've not seen that before. You can just get the kids to fill them in like that or you can chop them up and get them to put them back together to see if they understand what that array looks like in two different visual representations.
Another way of looking at inverse operations is KenKen. This is from a lesson that Michelle Tregoning from Fairfield Public School did when we were doing a project on rich maths lessons in 2011 and it's a great use of the KenKen. She really took it as a problem-solving approach and talked with the students about, you know, ‘Why does this number have to be number 2? And what other possibilities could any of those other numbers be?’ If you don't understand what KenKen is, go online, have a Google search for it. They're great little games a little bit like Sudoku, I guess. You can get quite addicted to them. There is an app. You can download an app for your iPad or phones as well. But I think one of the most brilliant things that Michelle did with this is she actually blew up that KenKen onto a really large sheet of paper—old-school I know, but paper still works—and then she had Post-it notes. So the kids used Post-it notes to put the numbers on where they thought they went so that during their thinking process they could move the numbers. So they hadn't written them in and had to rub them out. They really got to use some reasoning skills to complete that activity. So another good use of that flexibility of operations in the classroom.
Making connections with multiplication and area. So these are just some snapshots from Google Earth—a great way to find images of arrays. So many. You could find them in your local area, internationally. I think the one on the bottom left is something I got from in London because I've lived in London and I knew the houses would look like that so it was a great place to go for images. So it's a great way of bringing in some of those outside connections of where we would use multiplication and area at the same time and those cars are even in rows and columns which makes it even easier to see the relationship there. So another good link to make when you're planning your lessons.
Some other links within our scope and sequences is fraction concepts with chance. Now, I'll deal with that a little bit further next week. I'm not going into statistics and probability today. Fractions to practical examples is really important in measurement and not just in Kindergarten, but all the way through primary school. And also teaching fraction concepts and the understanding of decimals as separate strands in Stages 1 and 2. So don't relate them together straightaway. Kids need to have ... Our students need to have a really strong understanding of fractions before we introduce how they relate to decimals because if you think about where decimals sit as well on our place value part of our continuum, that understanding comes much later than probably their understanding of basic fractions. So that whole idea of the decimal place system and system place value and seeing the multitudes of 10 and being able to move numbers flexibly around the decimal is something that comes a little bit later. So keep them separate to begin with and then start to teach them as related concepts later into Stages 2 and 3.
So some of those practical examples for fractions—obviously NAPLAN is a wonderful source of practical examples for fractions. You could obviously do any of those activities as a physical activity in your classroom, not just using the NAPLAN resources there. And also I've just got a recipe there from taste.com and it's got fractions on it and you could make the recipe with your class or you could talk about how you could double that recipe or if you had to halve that recipe, what would you do with those items or even just, you know, if you didn't have a cup measure, how could you work out the quantities for some of those ingredients. So there's lots of great examples of how you can use practical examples for fractions.
Some of the links in our scope and sequences is building measurement concepts in sequence. So length before area and then area before volume. That's really, really important. And also using those measurement activities to provide a context for decimals and percentages, particularly decimals. We do try and move away from money as the focus. It does have a link there. Obviously, it's how a type of decimal is used in the real world, but it only goes to two decimal places and when you look at the coins, 50 cents doesn't look like half of $1 whereas when you use measurement activities, half a metre looks like half of a metre so you can get the visuals working a lot clearer when you use measurement.
That sequence when we're looking at making connections in measurement and geometry, you can see that length is one dimension, then the area is two dimensions and then the volume is three dimensions of that object and having students see how those are connected to each other is really important, OK? So we look at that three-dimensional object—the rectangular prism. We can measure its length, we can measure its surface area even just on one side and then we can also measure its volume. So they're seeing how rows develop into rows and columns that then develops into rows and columns in layers. And that word 'layers' is mentioned in our new syllabus and it's specifically there to move away from just teaching a formula, Oh, you just do length times breadth times height,’ which, yes, you need to eventually understand that that's the formula for working it out, but when you're just starting, you need to understand that, ‘I find the area for volume and then I times it by the layers. I add the layers on top.’ Because they're going to start with counting by ones, they'll then move to a repeated addition, maybe even the rows, and then they'll start to multiply. So our students will develop through these strategies at different rates.
So there's our puzzle again, our jigsaw puzzle. I wanted to show you how some of these concepts relate together. And, obviously, as I mentioned, there's that document that I've added in about linking across substrands that will help you see the content and the key ideas in these concepts as well.
So I've just made up some of these mind maps, or concept maps, to give you an indication of how those all link together. So if I look at multiplication and division as the focus—that's my focus outcome or my focus substrand—there's then all these other areas that connect in to multiplication and division and even across some of those as well. Again, this is not an exhaustive list. It's just an idea to get you going. And now look, this is a great activity to do with your staff. Give them a blank one of these and a number or algebra substrand in the middle and see what links they can find in their particular stage specifically because some of them change as we go from Early Stage 1 through to Stage 3. So you can see there's some links there, particularly strong links with patterns and algebra, obviously with multiplication into area, addition and subtraction and obviously whole numbers is going to be throughout all of these number and algebra substrands.
So we're talking about operations with whole numbers, skip counting, that rows and columns structure, finding areas of shapes, and that links with factors and highest common multiples, lowest ... Sorry—lowest common multiples, highest common factors and area and volume examples. So there's a lot of connections there between those four operations.
Here's another set. So we're looking at whole numbers that link with patterns and algebra, time and data.
So we have whole numbers in the middle there. They link with time, to do with date and ordinals and duration. We've got a link with data. Obviously, if you're doing data displays, you're doing a one-to-one or a one-to-many count and tally marks. There's links, obviously, to patterns and algebra and multiplication. And whole numbers links into all of the areas of measurement as well where you're converting units or you're measuring or you're counting units. So all of those strategies they're using in whole numbers is also going to be applicable in the other substrands.
So we're looking at one-to-one correspondence, counting by ones, twos, fives, tens, counting hours, ordering numbers and creating patterns.
Another one is looking at how fractions and decimals links to measurement and 2D space and time.
There we go—fractions and decimals in the middle and then we can see how it relates to time. So, obviously, when I'm talking about time with the decimals it's not quite a decimal point, but it's still something that you could talk about with your students about how else decimals are used in the real world, but definitely the language of fractions, so looking at 'quarter to', 'half past'. The same thing with 2D when we now have that focus on 'rotational'—so about 'half' and 'quarter' and 'full' turns. We see it in volume where we use the language of 'half full' and all of that fractional language that goes with volume, particularly capacity. There's also links with length when you're using and recording decimal notations. So fractions links across many, many aspects of our maths syllabus.
And there's a whole lot of them there. So I mentioned a fair few of those as well. And, yeah, using decimals to record measurement and that idea of when they're actually using it when they're converting, that's something they really need to grasp by the end of Stage 3 because if you're asking them problem-solving questions where they need to do some kind of conversion if it's between grams and kilograms or metres and centimetres, it's not something they're going to be able to understand by re-reading the question or slowing it down or looking for key ideas or concepts in the question. You either know how to convert or you don't so they really need to have a good understanding of that decimal place value system.
Another one is patterns and algebra where it links to other aspects of number.
So if I place patterns and algebra in the middle and look at all the places it connects to as well, we've got back into fractions and decimals with patterns with fractions. We're looking at numbers with whole numbers that also goes into integers now—that negative numbers—and links to coordinates. We've got whole numbers and looking at skip counting, counting forwards and ordering numbers and some other areas there as well. And in addition and subtraction and multiplication and division, there's a whole lot of key ideas now about looking for missing elements in number sentences that involve either addition and subtraction or multiplication and division so you're going to see that come up in the content in our syllabus, so just be aware that there are links between all of those. So I probably could have put a red line between multiplication and addition and subtraction as well. So you'll notice that a lot of these concepts are quite separate in our syllabus because we want to put emphasis on them, but as they come through primary school into Stage 3 particularly, getting ready for Stage 4, a lot of these concepts start to join back up again and their interrelatedness is really important. 'Interrelatedness'—is that a word? I'm going to use it. So you just need to be mindful of that, that although you might have taught multiplication and division separate to addition and subtraction, once we get to something, like, say, looking at order of operations in Stage 3, they need to understand how all of it works together.
So looking at even geometric patterns and using rotation and side properties, numerical patterns, just forwards and backwards counting, those missing elements that I just mentioned, they're all important connections to make from patterns and algebra.
Another great feature of the new NSW syllabuses online is that you can search for a topic or a concept. So I searched for 'graphs' in the science syllabus and straightaway it came up with a little link there to 'Working Scientifically' where the students are having to construct and use a range of representations and it says, ‘Column, picture, line and divided bar graphs.’ Now, I know that in Stage 3 we no longer teach divided bar graphs. It's moved to Stage 4 where it sits with sector graphs. So although we're not teaching it in mathematics, it's still going to be something they need to be aware of when we're working in science so that's that link across KLAs. We need to maybe teach that as part of science and explore it as part of how they might represent their work although it's not something we're going to assess them mathematically on. So it's really important to use this kind of search engine to help you out so go in and have a go. Even if you search for something as simple as 'Addition', you might be able to find some other connections as well, but there's definitely a lot of connections within data and statistics particularly into Stage 3 and 4 more so than other stages, but you will find some other connections as well.
So just some final thoughts from me for today. Making connections focuses on developing mathematical thinking, so that's what we're trying to get our students to get to that point of. Last week or two weeks ago we talked about reasoning, so that kind of mathematical thinking. We want them to build conceptual understanding. We want them to see the relationships between that conceptual understanding. And we want them to develop multiple strategies that they can use across substrands and really across KLAs as well. That's something that is really important.
I thank people for attending today. I really appreciate that you come along and that you give me feedback on these sessions. Remember to go online and complete your evaluation on your local event. Our next session is in two weeks.
A reminder again—they're my details in here at State Office. Again, I've put the link to the newsletter, our new newsletter 'The Mathematical Bridge' in the pod that I'm now going to move to on my 'Conclusion' template. So it's there.
And a reminder that, to do with that, if you are the contact person for your school we automatically put you on our mailing list for that 'Mathematical Bridge' newsletter and Nagla's been working very hard to get her section of it ready so I'm now the one that's dragging the chain that needs to add to that so we can get our next newsletter out soon. So there you go. There's the files in the file pod. There's quite a few. You may need to scroll down today to get all of them. All the scope and sequences are there and some other bits and pieces. I hope that that's been helpful today to help you understand some of those connections that we need to be making when we're planning our programming and scope and sequencing. Remember that next week—sorry—the next session in two weeks' time is a follow-on from this. It's Part B, and we will be going into that statistics and probability side and a little bit about how some of those non-number strands link to one another. So thank you for joining us today. I will be on here if you need any more questions answered or you can email me if you've got questions as well. So thank you.