Transcript of What's new – number and algebra
Katherin Cartright: Thank you very much. So welcome, everyone, to our third session of Syllabus PLUS for this term, this year, and today's session is on 'What's new: number and algebra'. I'm here with Chris Francis. You can hopefully see her just in the background. And we'll start our session now. So, welcome.
So today's session is on what's new for number and algebra. This is the first one of three, I guess, in our little tiny series within our larger series that's going a little bit deeper into each strand of the K-6 maths syllabus. So today is on number and algebra. Next week is on measurement and geometry. And then the last session, the week after that, is on statistics and probability. So our syllabus states that number is about quantifying and describing the world. That's how we use numbers in everyday. So our students are encouraged to develop number sense, to use mental computation strategies in all stages, and to explore number and prealgebra concepts. So this is sort of the essence of what number and algebra is about. And, as people might know from our current syllabus, number and patterns and algebra were separate strands and they've now been combined back into one as in line with the Australian curriculum. So we haven't lost patterns and algebra, it just exists as a substrand within numbers and algebra.
So if I have a look at the organisation of content, we're used to this kind of concentric circles diagram in primary, and you can see that number and algebra sits down the bottom. So the diagram shows the scope of the strands and the substrands and it also shows that central role of working mathematically. You can also see that our diagram now has the 7-10 substrands. They were previously referred to as 'topics'. They've now taken on board the same terminology that we use about substrands, and I think this is really helpful to see where some of these substrands, or topics, go and develop into Stage 4 and, obviously, then into Stage 5 and into Stage 6. The picture, if you're not familiar with it, is not indicative of the time spent on each of those substrands. We do still see number and algebra as the most important strand, so it probably gets a bit more priority than the other two. But just so you have an understanding of the whole scope, this is a great diagram to have a look at.
So overall, what's new for number and algebra in Early Stage 1? So there's not a whole lot of changes for Early Stage 1 in our new syllabus. Most of the information and the content is the same as what you're currently teaching. But in whole numbers, there's a bit more of a focus on correspondence between collections, so the idea that the students compare how many they have in comparison to how many somebody else has, so maybe the person sitting next to them. So that example there of "I have four counters, you have seven." So it's seeing the difference, whether they're using addition or subtraction or just visually looking at that, that's something that's a little bit new for Early Stage 1. The use of the term "is the same as" to express equality of groups. Now, this isn't particular new. It exists already in Early Stage 1, but it's been moved to whole number. So that used to exist in another substrand, so it's been moved across. And it is also something that continues through our whole syllabus, which we'll have a look at in a moment. But that whole idea around seeing the equal sign as actually like a balance, I guess, thinking that "is the same as", so both sides need to match - not be exactly the same, but that they have the same value. Also in patterns and algebra there's the use of the word 'classify'. They have to be able to classify groups of objects into smaller groups, and they have to be able to explain how they've classified the objects. Now, straightaway when we read this, we had a link straight to 3D objects. So we think that is somewhere that it could match up nicely if you're looking to scope and sequence whole numbers...sorry, patterns and algebra, you could do it with 3D space, because that whole idea of sorting is something that's also mentioned in that strand.
So I mentioned that term "is the same as". So in our session today we're just going to show you a few examples of learning development. So it's not necessarily the learning continuum through the one substrand, but some of these topics or concepts appear again and again throughout the syllabus, and particularly with things that are new, it's really important to see, well, where do they go or where have they come from? Particularly if you're in a different stage than what you're used to, you need to have an understanding of how that concept develops not only within its substrand but across substrands. So I mentioned that in Early Stage 1, it's now in whole numbers, "is the same as". In Stage 1, it now appears in multiplication and division, and it talks about how 3 groups of 4 is the same as 4 plus 4 plus 4, so it's relating grouping strategies with repeated addition. In Stage 2 it appears in addition and subtraction, where we're looking at "is the same as" when you're looking at two sides of an equals symbol. So you've got 32 take away 13 is the same as 30 take away 11. And our students do struggle with that idea of balance on either side of the equals sign. They're so used to seeing it as the answer and not as that balance. So we need to really bring that out in our teaching and show them examples of what we mean when we say "is the same as". In Stage 2 it also appears in multiplication and division. And it appears when we're talking about factoring. So when you see 5 times 8, they can then factor 8 down to 2 times 4 to then help with your working out. If you don't know what 5 times 8 is, you can change it to become 10 times 4, and that might be something that's easier to work out because the student knows, you know, on the decade counting. So there's a really flexible use of this concept within number and algebra as a whole strand. We also see it in Stage 2 in fractions. So it starts to talk about, you know, that 0.1 is the same as 1 tenth. So there's not...it doesn't just sit in one substrand. And then in Stage 3 it actually starts to apply to other areas. So in this example here, it's looking at time, and that 15 minutes is the same as 15 sixtieths of an hour, which is the same as a quarter of an hour. So it's looking at units of time as well, and that equivalent fractions, so there's a whole lot of other concepts that match up together that you could teach around this kind of concept.
So an example from Early Stage 1 - we wanted to give a little example of lessons you could still use. So this is a lesson from our 'Talking about patterns & algebra' CD or ebook, as it now exists. That little image on the left there is actually a hyperlink, so if you want to click on that now or later on, you can, to be taken to where you can download that ebook. So this is a resource that currently fits with our current syllabus, but you can also use it within the new outcomes context. So this activity is all about using the term "is the same as". And the students make posters to describe different ways that we can represent the number four. Now, obviously you can make that a lot more complex for as you go into higher grades, and particularly when we talked about fractions and decimals, you could make a poster or cards that represent those as well. So it's a nice visual activity that helps students understand what that term really means.
So, what's new for Stage 1? So these are some of the key points as an overarching understanding that we want to get across to you today. So there's a real focus on not only looking at standard but nonstandard partitioning, and I've got another example of that in a moment. But that's something that's coming across the whole of the new syllabus. There's also a fair amount of movement of content from patterns and algebra, that substrand, to addition and subtraction. Now, it's not going to make it any more difficult to teach, it's just good to know that it's been moved across. So patterns and algebra definitely already had very strong links to addition and subtraction. It was just in the new syllabus it was found that it fitted better not only, I guess, with the Australian curriculum descriptors, but in a sense of when you're dealing with addition and subtraction, some of those patterns and algebra concepts fit nicely in there about number combinations and counting and finding the difference between numbers. So there's been some movement of content. There's also a bit more specific information on models of division. So we had this in our current syllabus as well, but it's just been separated out a little bit more now so that that idea of sharing and grouping, so seeing those different models of division. We just advise you to have a look at the background Information and language section on this in Stage 1. So it talks about partitive and quotitive as different ways of dividing. So, how many in each group compared to how many groups are there, so it's that difference between the student sharing out from a collection one to one to different people or grouping or counting into groups of, say, 3, so 3, 6, 9. And it depends on what you're doing, which model you're using, with how your number sentence looks. Sorry, I think my boxes have moved a little bit on the screen today, but you can still get the gist of what we're saying there, so that sort of missing element appears in a different spot depending on the kind of division model that you're using. There's also a number of moved content from Early Stage 1 patterns and algebra into Stage 1 patterns and algebra, so they've just sort of put it back into Stage 1. So it's a different focus in Early Stage 1 - more about objects - where the number focus comes in in Stage 1. So again, one of those examples of a learning development, I talked about non-standard forms of partitioning.
So we see it in Stage 1, where they have to be able to use both of the partitioning standards of...standard and non-standard forms. But we're just giving you some examples here of the non-standard partitioning. So 32 can be seen as 32 ones or 2 tens and 12 ones, as well as just the usual 3 tens and 2 ones. So it's just using it for a different purpose. In Stage 1 it also appears in addition and subtraction where they're partitioning numbers to aid with addition and using their place value knowledge to help with that. It also appears in Stage 2 addition and subtraction, where they're again using non-standard forms with 3-digit and 4-digit numbers to aid with addition. So it's just another one of those skills and strategies we want to build within our students, different to "count by one" strategy, or different from just partitioning because it's in tens, hundreds or ones. Also it appears in fractions and decimals, so they also now need to be able to partition decimals up to two decimal places into non-standard forms, so it's seeing how that decimal appears as a whole number and a fraction, or part of a whole number as well. And in Stage 3 it appears in whole numbers as the numbers get larger, of any size. But not only are they asked to do it, to present the numbers in non-standard forms, it's to aid in their mental calculation, so it's actually giving a purpose and a use for doing that. And we really want to encourage that throughout the whole of the syllabus as you present the idea of non-standard partitioning into different...different ways that the students are problem-solving. So it's definitely something that's new and it's a concept that will need to be developed along with standard partitioning.
So this is a little bit more in-depth into the changes. I won't go through this in a large detail during today's session because it's a short session today, but you can go back and have a look at this later or if you came or watched the first of our series, I gave you a document, an Excel spreadsheet, that had a lot more detail around this. This is just highlighting some of the ones that we think are really important. There will be links, as mentioned, in the chat today to the previous sessions, sent out with the link to this session. So you won't miss out if you haven't seen the previous two sessions. So you can just see I've just broken it into the substrands within Stage 1, what it looks like in the current syllabus, and if there's nothing there, it means that that content doesn't exist currently in our syllabus, and then where it appears in the new syllabus. So there are some things in there that are a little bit new for Stage 1. Particularly around equivalent values of money, there's some moved content from patterns and algebra. There's that difference between sharing and grouping. fractions now includes eighths. And there's also a little section there about combinations in addition and subtraction - that it's not just combinations of 10, it's up to and including 9, and from 11 up to and including 12.
And that's where our example comes from for Stage 1. So we've titled it "Friends of everyone!" So we're not just trying to teach friends of 10, we're trying to teach friends of all those numbers all the way through really to 20, but particularly of 6, 7, 8 and 9. So, often we find that we just teach friends of 10 and it's great, the students end up knowing these combinations, but then don't really know what to do with them. Well, what are they used for? So again, this is a little activity from the 'Talking about patterns & algebra' ebook that's really quite useful. So it's just rainbow relationships, and you can see the picture there. They're just making a little coloured rainbow to find the pattern and to look for combinations to 7. But, obviously, you can do it for all of those numbers. And, you know, what do we use them for? Why do we need to know how to partition those sort of numbers? And, you know, I've got that little example there. If I know that 2 plus 5 equals 7, then I want to be able to use this knowledge to solve more difficult problems. So if I had the problem 18 plus 7, we want them to be able to see, "Well, I can break that 7 into the 2 and the 5, and I can then see 18 plus 2 plus the 5, and they might then, you know, add the 18 and 2 to get 20 and then add the 5 on. So another strategy they can use. And there's a little example there that's helping them see that it is the same as. You know, 18 p;us 7 is the same as 18 plus 2 plus 5. So it's just another non-"count by one" strategy that we really want to build up with our students.
So moving on to Stage 2. So as we get into Stage 2 and Stage 3, there is definitely a larger amount of newer content, and particularly then into Stage 3. So, Early Stage 1 and Stage 1 are fairly similar, although it still is good to have a look through and see some of those language differences that come up. We haven't had a huge language focus particularly in today's session, but it's something that you might want to be aware of, some of those language changes. We wanted to bring out some of these key points. So the use of symbols for multiplication and division, it's not introduced until Stage 2. In our current syllabus it's introduced in Stage 1. We just want to make sure that when you introduce the symbols, they're introduced with meaning and they connect to that concrete concept they've developed in Early Stage 1 and Stage 1, and see how the concrete concept links to the symbolic form. So you might have students that already know how to use those symbols earlier, but we want to make sure that the students have a really good grasp of that concept before we start introducing the symbols used to represent what they're doing when they're grouping or they're sharing out their amounts. And money has a stronger focus. I guess money's sort of been something that's...it was a little substrand in our old blue and white syllabus. In our current syllabus, it was sort of, you know, teach money throughout number, and now it's sort of come back in quite strongly into addition and subtraction, particularly with problem-solving around money. So I think that kind of financial mathematics is quite a push for our new syllabus and it is into Stage 4 as well. There's a section on financial maths that's quite strong. There's also a focus on unit fractions, so not only are they learning about halves and quarters and eighths, but they're learning about thirds and fifths. And I think this is probably something that's quite useful for our students to have an understanding of thirds early on, and how they relate to sixths as well. So it's bringing down all of the unit fractions together in the one section in Stage 2. And so we want to make sure that we move from concrete to abstract for our examples, so always beginning with concrete, even into Stage 2 and 3 and 4. There's a lot more explicit visuals in the syllabus to describe these models of fractions. There's no percentages anymore in Stage 2, which I think is a good thing. We need our students to develop and understand fractions particularly, and decimals, before they get on to percentages and see the relationship between the two. So I guess there's no percentages in Stage 2 anymore so you can focus more on fractions and their relationship to decimals. Previously, students were just asked to model fractions, and now they're being asked to sort of up the ante and identify and describe them, so there's a link to communicating the working mathematically outcome there, where we really want them to be able to talk about what they're doing and use it in a flexible way. There's also a focus on reasoning within patterns and algebra, so they're making generalisations and they're investigating odd and even numbers. So there's a lot of reasoning that's going to come out when you're teaching patterns and algebra in Stage 2.
So our example of learning development here is that idea of using concrete materials or visuals to aid in understanding fractions. So I mentioned there was a lot more pictures or visuals in the syllabus, and here are just a few of them. So in Stage 1 they're asked to record the halves using drawings. In Stage 1 in fractions and decimals they're also asked to use concrete materials to model half, quarter or an eighth of a whole object. In Stage 2 they're asked to model the fractions and to model the objects, shapes and collections using concrete materials and diagrams. So it's really asking them to get that visual representation first before they actually do anything with the fractions. And then in Stage 3, they're asked to express mixed numerals as improper fractions through the use of diagrams and number lines. And it's leading to a mental strategy, so that idea of being able to visualise first helps them understand, "I'm making a picture in my mind when I'm trying to do this in my head." So I think that's really, really important and I love that they're using more and more diagrams for fractions, because it's definitely an area that our students struggle with.
So what's new for number and algebra overall in Stage 2 you can see from the current syllabus and the new syllabus. They now use 5 digit numbers. That partitioning is strong again in Stage 2. They're calculating equivalent amounts of money. That focus on fractions and unit fractions. Percentages have moved. And they're asked to make these generalisations about number patterns, including multiples and odd and even numbers.
There's also a new focus on using the area model as one way of showing a visual representation of how to multiply two numbers together. So if this is something new to you, it's really good to start investigating, "Well, what is it? How do I do it? And how do I use it with my students?" So it's basically just...it helps link to that algorithm. So I think we don't want to jump too quickly. The algorithm doesn't appear until, I guess, the later part, so the part 2 in Stage 2. And there's a reason for that - we really want students to understand the concept and the place value of those numbers. So, often when students jump into, like, vertical algorithms, they start seeing them as single digits again and start having that conversation around, "Oh, I just put the zero here," not understanding why they're doing it. So if you want more information on that area model, I've created a little "how to" video. That's not a hyperlink there, just a reminder that you can download my video from the file pod at the end of the session today.
Stage 3 - as I mentioned, large amount of content change for Stage 3. There's been movement of a few concepts from Stage 4, OK? The Australian Curriculum wanted to have those as a focus for Stage 3, and so we had to move content down to Stage 3 from Stage 4, remembering that they still deal with this again in Stage 4. They're not going to have a gap that they're not going to fill ever, so any of these topics are touched again in Stage 4, so things like factors, order of operations, some work with decimals, and the cartesian plane. So there's also a large number of key ideas and concepts in fractions and decimals, so again that focus on fractions and decimals. And there's also a bit of a focus around modelling and representing strategies when they're solving fraction problems. And as I mentioned before, percentages have moved up to Stage 3 only.
So there's some information there. There's quite a lot of this information. I've got two slides representing information for Stage 3 just to give you a fuller understanding of what's new and what's not. There's some terminology as well around integers, and some more partitioning that's happening, factors, highest common factors and lowest common multiples. And I think these kind of focuses are a really nice opportunity to link in with your feeder high school, or if you are a K–12 school, if you're a central school, get in touch with those people that are teaching the upper grades and maybe get a few lessons from them about how they effectively teach these topics within their classrooms.
Our example for Stage 3 is about solving word problems involving addition and subtraction of fractions. So if people haven't visited the resources website as part of our new NSW syllabuses for the Australian curriculum site through the intranet, that little link in the bottom right there, 'Teaching fractions: a primary concern', is a little building-capacity resource that Peter Gould created. It's a fantastic ebook. You can click on the link there. And this just really shows...it's a really nice activity that's a part of this ebook that talks about a student having to...they're given a half of a whole object and they're asked to create a sixth of the whole, so not a sixth of the half they've got, but a sixth of the whole portion. And these kind of activities are really good for strengthening their understanding of fractions and having students develop a flexible understanding, so not just filling in the fraction wall and seeing equivalent fractions, but really using it in a problem-solving approach. I've also got a little image at the top there of the 'fractions' book, the teaching pikelets and lamingtons for fractions. That's quite an old book - we've had it for a while. Most schools probably have it in your school library. I don't think you can buy it anymore under DET Sales, but if you can't find it, if you click on that little book, hopefully, fingers crossed, it'll link to a couple of the activities that come with that book in the little CD. That's through the mathematics programming support website. If you google that, you'll get to a fractions section as well. But yeah, if you google 'Fractions, pikelets and lamingtons', it will probably get you to there as well, but some of those things do exist behind the intranet firewall, so you'll probably have to use your login to get to those. But some great resources already getting developed for fractions.
So a little bit more new content for Stage 3. I did say there was a fair amount there. So they're also using the area model in Stage 3. They're using the term 'quotient'. Some people might already use that term with their Stage 3 students, but it's now within our syllabus. Speed - it used to be taught in measurement. It's now come into number. Order of operations, which I'll go into in a moment. The percentages, the fractions focus, the link with decimals, and that cartesian number plane, which is just pretty much a number plane. It's just the name of it that's used that might scare people a little bit.
So order of operations - we just wanted to talk a little bit about this for a moment. We really want it to be more of an investigation. So we need to explain and let the students find out why we need them and when do we use them. So the brackets are really just helping to organise our thinking and strategies, and it relates to problem solving and real life problems. And it aids in our understanding of, "Well, what do I do first in my problem?" We really want to encourage you to stay away from the acronyms, like BODMAS or PEMDAS. They can be confusing for a lot of students and we don't want to focus on a procedure or a formula, we want to focus on the meaning behind it. So, for example, in a multi step problem like this one, "I buy six goldfish, they're $10 each, two water plants costing $4 each. What is the total cost?" So if you just wrote it as the number sentence you actually wouldn't get the correct answer. You need the brackets to understand, "What do I need to do first?" And it's communicating meaning. So if I was going to give the working out to my friend to do, I need to sort of help them to know what to do first, so the brackets help me do that. And I think that's really important, to give a purpose to using the parentheses. So if you haven't seen 'Red dragonfly mathematics challenge', you can click on that link. There's also an app for this book. You can still buy it for $22. It was through DET Sales, but I think MANSW are still selling that book. But you can get about 20 or so pages of it for free with the PDF that that's a hyperlink to there. So it has a couple of activities on there about using brackets. And I'd pretty much just maybe not even give all of that information at the top to the students about the using the grouping symbols, and just see what they come up with first with trying to solve that problem, and then they'll start to get to the point where they say, "Well, actually, I didn't do it in that order that it's written, I did this bit first" or "I did this bit and then I did that and then I added this." So that whole conversation and reasoning will actually bring about the need to use grouping symbols. So there are some really good explanations for it in our syllabus as well, the new syllabus. And...and I think it's something that's going to actually provide a bit of structure for our students when they're doing multi-step problems and just getting them to organise their information as sort of that prealgebra, prehigh school level.
Just another couple of resources. We don't have time today to go into cartesian planes, but there is a Syllabus bites on the cartesian coordinates system. So you can go in there and find that. I've put the hyperlink to that, attached to that image. There's also a great little GeoGebra activity called 'Coordinates', a little applet that Nagla found for me. So if you haven't already downloaded GeoGebra, you'll need to download that before you can download this document. I've got a document in the file pod of how to download GeoGebra, or a link to that, and I've also got that 'Coordinates' in the file pod as well. So although we didn't have time today to go through cartesian plane, I don't think it's too far beyond what people already understand. You know, we already use coordinates in position with positive integers, and now we're just going into negative. And for our students in Stage 3, it's just learning about plotting the points on those four quadrants, so we're not going into that concept in a huge depth.
We just wanted to remind you that when you teach a new concept, because there's a whole lot of new concepts here, we want you to use a familiar model, and to teach a new model, we want you to use a familiar concept. So I talked about that area model for multiplication. You want to teach it maybe with some really simple numbers to begin with, something as simple as, you know, 15 x 10, just so they can explore using the new model or the new strategy without having the numbers too hard. So it's really important that you start with something that's familiar and build on it for them. Don't start with, you know, new, hard content and here's a new model of the strategies to solve that problem. So it's just something to remind you of.
These are our contact details. Please, if you have any further questions around number and algebra, we are happy to answer them. Although you might not get an answer immediately, we will hopefully answer the questions for you. I hope today's session has been useful.
I'm just going to click on the conclusion pod for you so that you can see the files down there. Yes, there are four files in the four files pod today. And just a reminder that our next session is next Thursday, 21 November, and we'll be going through Measurement and Geometry next Thursday. So, thank you for joining in today. We really appreciate that you're interested in the TPL around the new syllabus. We hope today's session has been helpful.