Transcript of Making connections—part B
Katherin Cartwright: Well, good afternoon. Hello, everyone, and welcome to Syllabus PLUS K-6 Mathematics. Katherin Cartwright here with Carol Field in the background on the chat today. Thank you for joining us. This is our second part of 'Working Mathematically - Making Connections Part B'. So welcome.
So as I mentioned, today's session is on 'Making Connections Part B'. If you haven't viewed Part A, that's OK. I'll be sending out the link to the recording of that and today's session as well tomorrow to those people who are the contact person for their school. So if you've missed the previous recordings, that's OK. You can still watch them in your own time.
So this is a slide from last week - just a reminder of what we're talking about with making connections. We really want our students to see how mathematics is interrelated and seeing maths not only as individual building blocks, but as interconnecting pieces. So they rely on each other and some of them become prerequisite skills for more complex concepts as we move through mathematics from Early Stage 1 right through to Stage 3 and beyond. So last week we dealt with number and algebra and measurement and geometry. And today's session, we're looking at number and algebra linked with statistics and probability and also non-number strands and how they link to one another.
So just some starting thoughts. These were my concluding thoughts from last week, just to get us back into making connections. It's about developing mathematical thinking. We want to see those relationships between concepts and we want students to develop multiple strategies to solve problems across substrands, so using those strategies from number into other strands.
So it's about making links without complicating the concept. And so what I mean by that is that you can have multiple links to the one concept and you can see multiple angles of how that concept works, but not creating multiple activities. So we don't want to bombard our lessons with multiple concepts to try and cover them in one or two lessons. We want to really dig down into that one concept, but you can definitely look at it from different perspectives, how you might use it in different contexts both within other areas of mathematics and within other KLAs. So we can look at how multiple outcomes can be assessed through the one concept. So you might note, particularly in Stage 2 and 3 in our new syllabus, you might see a lot more key ideas and a lot more, I guess, content descriptors, but we can definitely blend some of those together and have a blended approach to our mathematics teaching that can be reflected in our assessment processes as well.
So if I want to assess many outcomes through one concept, here's an example in our new syllabus in Stage 2 in 'Three-Dimensional Space', the second part. Students are asked to draw different views of an object constructed from connecting cubes and they need to do their drawing on isometric grid paper. Some people might not have used isometric grid paper for a while. It was in our blue and white syllabus and sort of has dropped off for our current syllabus and has come back as a resource to use in our new syllabus. And I think it's a great resource, isometric grid paper or isometric dot paper. So when I look at this as a content dot point and a concept of being able to not only construct the object out of the cubes, but then be able to draw it from multiple views, this involves a lot of knowledge, skills and understandings from other areas. So you've got whole number. You've got addition. Multiplication. Length - particularly when you're then drawing the object onto the grid paper. Volume - so do you know what the shape looks like and how large it is, "How many of the dot connections do I need to make to represent my model?" 3D space. 2D space when you look at the faces of the three-dimensional object you've constructed. And position - so where you're positioning yourself in the perspective of the object. So there's a whole array of knowledge and skills and understanding that we can assess as part of this one concept. So see if you can make some of those connections so that your assessments are quite in-depth and really see students can use the skills from one area to complete this concept, to be able to have success in this concept.
So these couple of slides are from last week as well - just a reminder about making connections in our programming. The programming and assessing and reporting K-6 policy is under review. We still recommend that we use the key ideas and how they relate to the syllabus outcomes to produce our programs and our scope and sequences. The link there in blue to the 'Continuum of key ideas' takes you straight to the NSW syllabuses for the Australian Curriculum on the BOSTES site, which is the former Board of Studies site. So that's where you can find the 'Continuum of key ideas' that are really vital for creating our programs in mathematics.
So when we scope and sequence, the sequence and the emphasis on particular areas of content and any adjustments required are decisions made at a teacher level or at a school level. So we do encourage teachers to spend more time on number and algebra and if you develop a whole-school scope and sequence, you can adapt it for your students. So that's just a reminder from last week as well.
So 'Links within scope and sequences'. So number and algebra with statistics, which is... Our substrand of data sits underneath statistics. So linking number skills of counting and one-to-one correspondence with data - so during the collection of data, completing the surveys, tallying - and we can also link addition and subtraction operation skills and patterns within that when analysing data sets and making comparisons. So if you're looking at data that you've completed in comparison to another group or another class, you might need to use some of those skills when you're analysing the data.
So there's a little mind map, concept map that sort of shows you the connections between data as a substrand and other substrands within our syllabus. So there's a couple of ones there from number and algebra and there's some there from space and geometry and I think you can see some of those links are going to be familiar for a number of different areas in statistics and probability, so within chance and data. They're always going to link to whole number. A lot of the data we collect and statistics we use are based around numbers. Whether they're whole numbers or fractions or decimals, we're dealing with number when we're talking about numerical data, which is up there with addition and subtraction. So those sort of areas are going to always appear. You can also look at data in the way that you construct the graphs actually physically. So in the early years, we look at 3D space, we look at organising objects in a data display and we also do that with 2D space as well when we represent data in a display, possibly using shapes. And also position when we get into data, we look at scale - that many-to-one as well as a one-to-one scale - and when we're creating two-way tables. So it has a bit of that...coordinates. When you're trying to work out and read both sides of the two-way table, you're using some of those positional skills to do that.
So if people haven't seen this resource, it's the 'Teaching Data Stage 3 - Dot plots' e-book. And I've just put up in the right-hand corner there... Sorry that's not hyperlinked, but just on the intranet site on that right-hand corner is that link to NSW syllabuses for the Australian Curriculum, which is a mouthful, but that's the link to where the professional learning courses and these building capacity resources sit. Now, you can create a tile for that in your portal. Where your tiles are, if you go under 'My Websites', I think it sits under there. I didn't put that link in today, but if you're having trouble you can always email me or ask someone else at your school that might already have that tile. But you can just go to it from the intranet website on the top right - it always sits there. And so this resource is wonderful. It was created to link with our new syllabus and particularly it talks about dot plots because that's a new area for Stage 3. They're not particularly difficult, but it's just another way that we can represent data. So I think that's a really great resource that you might want to have a look for on our website.
On that website under 'Resources' there's also Syllabus Bites. Now, you can find these on TaLe. TaLe is still in existence at the moment, but all the resources from TaLe are now also available through Scootle. Again, you can put a tile to Scootle on your portal page through 'My Websites'. I think it's one of the first ones that comes up because it has 'AC' as the beginning so that it got to the top of the list. So Scootle is just the new repository where they are housing all of our resources from TaLe so you can access it from a couple of different places. But this particular Syllabus Bites is also about dot plots and two-way tables and it just gives you... There's some tabs across the top that you can use and it's just a great little resource to assist you in the classroom and seeing how number and patterns fits in with data.
Some other links that we can see is through number and algebra and probability, which is our chance substrand. So a reminder that chance isn't in number anymore. It's been moved back with data. They sit together underneath statistics and probability in our new syllabus. So we link fractions when we represent chance, so when we talk about 50/50 or half a chance or three-quarters or 75%. Lots of decimals, fractions and percentages and the language involved with understanding fractions is also a lot of the same language we use with chance. We also use the link of the number line with chance of outcomes - that 0-1 number line gets used when we talk about chance. So there's lots of links there between number and probability.
There's my little concept map for chance and you can see where it links again. Again, whole numbers and fractions, as I mentioned, for statistics and probability, is one of the larger areas that the links can be clearly seen. There's also a link straight between chance and data. Now, that comes particularly later on when you start to record your chance experiments, the results from those, into data sets. So it becomes the one concept or a blended concept as we get into Stage 3 and beyond. And that happens with a lot of our concepts, in fact, as a sidenote. Many of them we teach as these foundational concepts and as the students move through their stages of learning, most of these concepts start to converge together and blend, and you can see the application of the skills into different concepts. So I've also put on here science and technology and HSIE, so there are some other KLAs where you can link a number of areas of mathematics, particularly if you're looking at population samples or trying to gather data to use in your experiments. You can use that for chance and also for data representation as well. Possible and impossible events - something to do with the weather is often what we talk about. And science and technology, when you're recording possibilities of chance experiments within science. So there's a number of different links... Science and technology, I should say, for K-6. That's the title of our syllabus. So there are a number of different ways that you can connect those concepts through.
So here's a little activity with a chance number line with a 0-1. You can draw that on the board or on your whiteboard and you have the words and the students need to work out, "Where would I place these words along this number line, knowing that zero is..." That idea of 'No chance' or 'Impossible' and 1 being 'Certain'. So I've got those words on there, but you might want to use different words to describe that.
So using a resource like this, I might ask the students to place the words on their own number line, they compare it to a partner, they need to justify to their partner where they placed the words, then ask that question - "Do you want to make any changes? Has the discussion you've had with your partner mean that you want to change where you've put your words?" And then I'd create a class version of the number line, having students give reasons to support the placement of words. So, again, we're bringing out that working mathematically focus as well. And just asking that question - Is it possible to get total agreement on where those words would go?"
So if I just flick back to that slide, words like 'maybe', OK, you might have a different place that you would place it compared to somebody else, or 'likely'.
So some of them that are more broader terms we might not get agreeance on and that's OK, but it's those conversations you can have with students that get them to start thinking about numerical values and how that relates to the words we use for chance.
There's also another great lesson that I've used in the past where you have some students come up the front and you provide them with number cards and there's the list of the numbers we use and they don't show anyone else the number they've got. You tell the students that all of them, they all have three-digit numbers - so there's no numbers that are beyond 1,000 so they're all three-digit numbers - and you ask the students to stand along the number line. So if your classroom's the number line and one end's 0 and the other end is 1, ask them to stand as to what chance they think of having the highest number. So you've given them that information about they're only three-digit numbers and they are all kind of high numbers, but we're talking about the highest number. So after they've placed themselves and when I've done this lesson before, they generally all bundle up in one end, so after they've done that tell the students that the highest number is 999 and then you'll ask them if anyone would like to move and it's interesting what happens now. I want you to... If you have a go at this in your classroom, do the students with numbers other than 999 - which is only one student - do they all move to the 0? Do they understand that if they don't have the highest number, their chance of having the highest number is now 0 or do they all stay there because they think they have high numbers, not the highest number? So it's a really nice little activity to do with your Stage 3 students in particular. You could try it with Stage 2 as well if you like or even beyond into Stage 4. So I think that whole discussion around chance words and "what our chances are" has a real social context as well and the mathematics behind it is really important to explain to our students. Often watching a game show as well like 'Deal or No Deal' and the way that people choose things depending on what they think their chance is, is quite a nice study to do with your students to see people just using gut feelings or, "Do you think they're actually using the mathematical statistics of possibilities and chance behind that?"
So some other links are between those non-number strands, so between measurement, space and geometry. There's links between angles and two-dimensional space. Obviously, they are now two separate substrands within the same strand. They have their own outcomes now, and they also link with three-dimensional space, so angles with 2D and angles with 3D. And of course, they're all parts of the features and properties of those shapes and objects. There's also linking with measurement with spatial understanding, so seeing that connection with area and 2D and volume and capacity with 3D objects. There's also a link between two-dimensional space and three-dimensional just by itself. And these are really important concepts to make the connections with for your students.
So I can see there 3D space in the middle and some of the other substrands that we can connect it to and where they go. So comparing objects and ordering objects in maths, you need to have an understanding of 3D space and objects. When you're measuring volume, you're using the objects to do that. In data, that picture graph that we mentioned before when we had the link as data in the middle. Positions, so actually moving around your environment and seeing the objects and being able to identify them in the environment. Surface area of 3D objects. And obviously, 2D, looking at cross-sections, nets, different features. So there's lots of connections across strands that don't necessarily involve number.
So even something as simple as making connections with angles with 3D space in the environment. I just found that picture from a creative commons free website and I just drew those lines in on the whiteboard. So you can have the students draw in the angles or look for the angles within those pictures and then they can even measure them and have discussions about the rigidity of those pyramids and why they use them as building blocks, and you could obviously link that as well to other KLAs outside of mathematics. So can they see where angles appear in the real world and how they're used, in this case, to build structures? So that's one way you can connect angles with 3D space.
You can also make connections between area measurement and 2D space. So these two activities are from the 'Teaching Measurement - Early Stage 1 and Stage 1' book and you might have that in your school already and if you don't, I can send... I will. I won't say, "I can," because people will email me. I will send out that with the recording link, the book version or the PDF version of those for you. We've had permission to do that from here as they still relate to the current outcomes, not the new outcomes. At some point, we are looking to update these and have them available online, but at the moment they're still referencing the current syllabus. So both of these activities are using prior knowledge of shapes to assist with solving area tasks. So the first one is about a patchwork quilt. They're using rectangles and it gives you some dimensions there of the area of those shapes that they would use to create a patchwork quilt. There's also the magic tiles task where you're looking at hexagons, trapeziums and triangles using pattern blocks. I've got a little picture down the bottom there of some designs I created that you would use with a task like this where they're looking at calculating the area using those smaller triangles. So some really nice ways in which we use 2D shapes as the basis for understanding and exploring measurement to do with area.
Another fantastic resource is the 'Teaching Space and Geometry K-6' CD. Now, this came out a number of years ago and the only way generally that schools could get it was by attending regional training. The purpose behind that was so that the resource wasn't just sent to a library and accessioned and never seen, but that someone in your school had some knowledge about how to use it in the classroom. So as that training doesn't continue anymore because there's not that regional structure anymore to do that, we currently haven't updated the CD, but I have a fair few leftovers in the office. So don't email me for them. I'm going to be sending them out and you may be a lucky school that gets one. I don't think I have enough for everyone on my enrolment list, but I'll be starting at first in, first served that enrolled for today's session. I can hear the groans now. But I'm sorry. That's the easiest way I could think to do it. So there will be some very lucky schools that get some of those CDs out. And, look, if you don't get one of the CDs sent to you, I can still send you particular lessons. I have this lesson already attached in our pod today and we are looking to update it and having a resource that links to the new syllabus soon. I say 'soon'. But please don't email me too much about that because I'll just be sending that out to the schools that are from my list. So this activity is a great activity - I've used it a number of times - where the students actually are looking at cross-sections. So cross-sections are now in Stage 3. This activity or this lesson is from Stage 2 in the CD. But you can see what they do is they pour the water in - there's a video you can watch about this - and the students then manipulate and move the 3D container to see what kind of sections they could make. So it would be similar to if you were using, like, the fishing line across one made of playdough. And so in this case, they're putting the water in and moving the container to see what different two-dimensional shapes they can make within the cube. So they actually put some more water in before they get to the point where they can make the pentagon as well, that last picture. But a lovely way to explore cross-sections and seeing that connection between 2D and 3D.
There's also a space and geometry progression focus that comes on that CD. So it's not part of our numeracy continuum because our numeracy continuum is based on number. The strategy is used in number and sometimes or generally in space and geometry, we don't use those strategies. We use different strategies. And so you can see the linked components down the side there, and I've been talking particularly about the 3D and 2D one, but you can see how those concepts and those understandings flow from Early Stage 1 through to Stage 3 around identifying and describing features, manipulating the features and then applying the properties of those and then obviously into Stage 4 as well, applying properties continues on. So that's just something that I thought you might be interested in having a look at, so that's that sort of continuum of how those skills and concepts develop through space and geometry.
So making connections between 2D and 3D space - it's great with cross-sections or that exploring sections we had a look at. It's also great with sketching three-dimensional models. It's useful for finding surface area in Stage 4 and it's also useful for making and identifying nets, OK? So seeing how the 2D relates to 3D.
So here's an activity around geometric thinking. So how many nets of a cube are there? Now, some people may have already used this activity with their students. It's been around for a while. So you're going to ask some of those questions of the students. So how many different nets of a cube can you make? What method did you use to work it out? Maybe you start with giving them a pencil and paper. What do they then do? How do they check to see if it was a net? Do they think about cutting it out or do they just visualise it? How do they work with a partner or a small group to investigate this problem? And is there a way to organise the different nets? Was there a systematic approach you took to finding it out? What sort of geometric thinking did you use?
So here's where it sits in our new syllabus. You can see there they're creating nets and they're looking at a different variety of nets for a particular object, which is a cube. And then down the bottom, it talks about that reasoning and communicating of how they work together between two-dimensional and three-dimensional. And I won't make you keep continuing to try and work out how many nets there are. I will provide you with the answer and there it is. I
t's 11 nets of a cube. But it's a great investigation and you could do something similar for another shape as well. I've seen it done with rectangular prisms before, using Brenex paper that's different colours to help them organise their data. So did they start off with looking at four squares in a straight line and then moving the other two along and then to three and then to two, or did they just use a random guess-and-check kind of method? So there's some great discussions you can have with your students.
These are some other interactive learning objects that can be accessed. You can access these through Scootle - probably the quickest way. You can still get to them through the NSW syllabuses for the Australian Curriculum link, but they're just called 'Sites2See' with a '2' in the title - it's down the bottom there - and they're just little interactives you can use. So it shows you the picture and then if you hover over the picture it brings up that yellow line that shows you where the three-dimensional object is or the outline of the three-dimensional object. So you could get students to trace that onto your whiteboard if you want or onto images as well, and just getting them to see the connection between 3D space and the environment.
So we can also see links between position and 3D space and position and time and length when we're calculating distance between points on a map and 3D space with mass, particularly with hefting or using an equal-arm balance, which is referred to in our new syllabus as a pan balance. It's probably just a word they use in some other states in Australia. There's no reason why you can't still reference it as an equal-arm balance, but maybe just talk about the different language with your students.
This is another little learning object that can be accessed through that 'Resources' section and you can probably still find it on TaLe and in Scootle as well. It's called Viewfinder and this is about making 3D space connections with position. So it has a little grid, it has some three-dimensional objects on it and it shows you different views so that you can have a look at what you can see and you can take little pictures of it and discuss about that perspectives. So another little activity that you can use in the classroom with your whole class.
Another way of making connections between length and time and position is scale. So I've just created a little Google Map there. We can pretend that on our school excursion we need to walk from Luna Park to the Opera House. So they need to use the scale on the map. Obviously, in Google Maps, they have American as well as metric scales, so you might need to have a bit of a conversation about that. But see if they can use the scale to work out the distance in metres. Obviously, they may need to convert that to kilometres. And then I gave them the little problem that if it takes us approximately 15 minutes to walk a kilometre, how long will it take us to walk the whole distance? So a little exploration you could do there. Obviously, if you type those things into Google it will give you all of that data, but we want to just let them explore that for themselves. So that's another great way of using maps and technology in your classroom. So that idea of using scale, it appears in a number of different outcomes that I've listed there. It doesn't necessarily particularly say that students can use a many-to-one scale for distance from two destinations, but when you look at all those concepts together, it's a skill that links them together, that links those concepts. So it's important that you do look across the different substrands to see how you can enrich the task for students so that they have a deeper understanding of that concept.
Also, 3D space with mass. OK, this is a little activity from the 'Teaching Measurement' book again. "Does it balance?" Given different mass of different objects, and they have to work it out. And in this task, they're actually having to use objects as the measuring unit as well as finding out the mass of the object. So there's a lot of background knowledge that we need to have for this one.
This is just a quote. I found some research someone was doing into making connections and I thought it was really a great connection to what we've been talking about. So about connecting ideas together, "The more connected an idea is to other ideas, the easier it is to retrieve from our memory." And so we have this understanding when we can make connections between ideas, facts and procedures in mathematics and that whole idea that, "Good ideas are networks." So you can see those little mind maps, they're all in our head and we can bring them back from our memory when we need them so that we can apply them into the wider area of mathematics and into the wider areas of KLAs and into real-world contexts.
Just some final comments from today from looking at making connections. Some of the connections we're making link to prior learning or prior knowledge from a previous stage, so you may need to look back to stages prior to the one that you are teaching to see where these links are. Some concepts require this prior knowledge to actually make the connections. It becomes assumed knowledge, particularly as our students go into higher parts of primary and into secondary school. And making these connections provides a context in which we can teach. So we want to move away from compartmentalising maths. We don't want to say, "Hey, today we're doing our space lesson." We want to focus on the concept or the learning intention for the day and say, "Today, we're learning about the concept of comparing side features of a shape using length." It's a much lengthier title to have to your lesson, but the idea is that we're not sort of saying, "Today we're doing space," "Today we're doing measurement," "Today we're doing number." We want to take that away and just look at those concepts so the students can sort of break down those walls and see how maths relates to itself in other areas as well. So as I said before, we might start with some single concepts in very early years, but very quickly we want them to see those connections. They need to apply those skills in other strands.
Thanks for your attention today. I appreciate that you come on and that you use these recordings in your schools for professional learning. Please remember to complete the online evaluation that would have been set up as part of your school scheduling of the event. Our next session is in two weeks' time.
A reminder that those are my details, that we have 'The Mathematical Bridge' newsletter, which is in the files pod today, which I'll go to now in my 'Conclusion' template.
So you can see the files there from today. Some of them are from last week. I left them in so that you can still access them in case you weren't on last week and the presentation from today is in there as well. And I did take a sneaky look at the chat during my presentation today so thank you to those schools that say, "I already have the CD." That's very nice of you to admit that and not take a second one and I'll send it out to the other schools because I can keep that chat. So thank you for doing that. You can, obviously, probably save them onto your server anyway if you only have one copy. So thank you again. Thanks for coming and listening for the second part of 'Making Connections'. We're back again in two weeks' time. So thank you and I'll turn the recording off now, but if there's any more questions, please feel free to stay on the chat.