Transcript of Key concepts within the syllabus
Katherin Cartwright: Well, good afternoon. And hello, everyone. Hopefully you can see me on there. Thanks again for joining us for Syllabus PLUS Maths K-6. Today's session is on key concepts within the maths syllabus. Thank you to those who are accessing the poll question today. I really appreciate that, and it bodes well for what I'm going to talk about today. So thank you for being a part of that, and just a reminder, I am going to ask a little bit of Q&A today, so please keep by your keyboard and we'll see how we go.
So, today's focus is on key concepts within the mathematics syllabus.
And first of all we really need to start with what is a concept? So, I found this great little picture on Wikipedia, and I loved that it said, "When the mind makes generalisations such as the concept of 'tree', it extracts similarities from numerous examples. The simplification enables higher-level thinking." And I really liked the idea that we might have all these different constructs of what something is, but we sort of slim them down to have the concept as one easy-to-understand idea. And I think this can flow into mathematics quite nicely, because we do have a really big focus on visualising and making sure that students can understand a concept through making pictures in their mind as well. So, I really liked that idea that you might have some different ideas, but it still comes down to one image that represents all of those things.
So, first question up today - what do you imagine when you think of the number 5? So, please, this is not rhetorical questions today. You can add in on the chat what you think. Says from my end "multiple attendees are typing", which is always a good sign. Now, I'm obviously not going to have time today to read through all of these or read them out loud, but hopefully people can watch the chat as well. And just a reminder, if you are watching this as the recorded version, you might want to pause these at any time when we look at the questions, and answer them together in your group. So, I can see lots of different ideas there. And it's interesting that there are some that are common and some that are quite different. And I think it would depend as well what age students you teach, what you might be imagining in your head. So, maybe you had one of these. Maybe you had the numeral. Maybe you had the dot pattern. Maybe you had the hand. Maybe you see 5 as in a number line or as in its place in the hundreds chart. And I think all of those things are correct, like the tree picture, but we all still understand what 5 is, and that it's made up of five ones, that it can be 5, and that we can change it when we need to. I like that someone talked about the 'Five Little Monkeys' rhyme. That's a great one. But we still have this concept of what 5 is.
What a concept isn't is also really important. Concepts are not activities, they're understandings. So we don't want to present our students with activity, activity, activity, activity during one lesson that might be all about addition, but they actually might be about different concepts. We don't want to confuse it for our students. We want them to have some clarity and be able to build these understandings through focusing just on one concept at a time. It doesn't mean you need to slow down your teaching, but you need to make sure that whatever you're doing in your lesson has a purpose and that it links back to the concept you're trying to draw out for students.
So, the Australian Curriculum has proficiency strands, one of which is understanding, which we have taken on as one of our Working Mathematically components in our syllabus. Now, it doesn't have outcomes in our syllabus, it is embedded. But understanding is all about concepts. It's about adapting and transferring of concepts into other areas of mathematics and into other KLAs. It's making those connections between related concepts, which is what our last two sessions have been on, about seeing where we can make links across concepts, and it's also representing concepts in different ways. So just as we all represent 5 in different ways, lots of different concepts can be presented in different ways, and we want our students to have that as part of their thinking mathematically. And that sort of leads to that understanding that it is something that can be developed as the student continues into higher grades of schooling.
I've got some really great resources today in the way of articles or journal entries for you to have a read of later on. These ones, this one in particular, comes from the 'Australian Primary Mathematics Classroom' journal, which is presented by the Australian Association of Maths Teachers. Now, you can subscribe to these journals, or you can pay and download for these articles. I've just given you a screen shot, because I own this article, so I've just put a picture up there. But it's a really great journal. If you're not a member of the MANSW association or the AAMT, I really recommend you do. These resources are really excellent for the classroom and for you to build your knowledge. And this article in particular is about big ideas, so seeing how concepts go deeper than just topics or content descriptors, and how extracting that is really important. But I love this little quote that's in there from Hill & Ball. "How teachers hold knowledge may matter more than how much knowledge they hold." And I really loved that idea that it's not necessarily that we're the brightest person in the room, but that we know how the concepts are put together, and we also then know how to break it down into smaller pieces so that our students can access it. And now I was definitely one of those people that didn't enjoy high school mathematics as much as I should, and I think that made me a better teacher because I wanted to know the why, I wanted to know the how, and it got me down to understanding the base level of where these concepts develop, and so I wanted to be able to explain it to my students in my class.
So, concepts within the syllabus. Each of our strands, our three strands, the concepts within them develop across the stages from early years into later years. And some of them have preceding or prior knowledge that's required. There's another couple of articles there that are really interesting to read. I've attached them in the file pod. Some of them are just extracts as well that you can access. Again, that little AAMT for Australian Association of Maths Teachers, that's a hyperlink, that image up the top there. You can become a member of their group. And the readings and the journals they produce are excellent and are always up to date. And every one at the moment has a little section in there about Australian Curriculum lesson ideas as well, so it's great to get your ideas from all different locations.
So, having this idea that this understanding precedes, the understanding that comes from early years into later years, our knowledge of this progression is paramount, OK? We're the ones that are bringing this knowledge to our students. It's our responsibility to know how it's developed over time. Are these journals... Sorry, I'll answer that poll question because we've got the chat open today, and I'm sort of looking over there as well. It is about mathematics teachers as in secondary teachers, but I find the journals are directed at K-12, absolutely. There's a lot of universities that put information in there. And Nagla's nodding as well in the background, who's on the chat today. It's definitely for all teachers. I am primary trained, and I find them at my level, and they really give you some good ideas for in the classroom. The one I showed you earlier about big ideas in mathematics is a little bit more of a lexically dense journal, but definitely on the general whole, the primary one they produce is excellent. So, they do have a primary one as well as some other journals that they produce specifically for secondary teachers. I'm sure Nagla will put comments in there if she feels she wants to clarify that any more for you.
So, within our syllabus, we have the continuum of learning, which is the continuum of outcomes. And so this shows you that sequence for a particular concept. And some of them, obviously, not necessarily stop when we get to Stage 4 and 5, but they mould into other types of concepts. So you can see there they've got length and area, and as we go through, past Stage 4 into Stage 5, we're mainly looking at area and surface areas, because length is obviously part of an object. You're measuring side lengths of an object. And that's how it develops through into the later years. So having these progressions at the beginning of our syllabus are really important.
And you can see there it's moving from Stage 1 right through to Stage 5.3.
So, these are some of the key concepts that I thought were really fundamental to student success, and some of these I had in the poll question today. And look, this is not an exhaustive list. There are some wonderful readings that you can read as well that give you some more detail into some of these in particular, and add to it as well. These are just some that I found really, really important. That idea of number sense, which encompasses a number of different aspects there around computation and problem solving. Finding patterns and seeing how we use them within number is also vital. Estimating. Now, that can involve numbers and it can also involve sections of measurement. Conservation, as in, you know, the area of my square. I can cut it in half and rearrange the pieces, it still is the same area. But also conservation of numbers. So, we had the number 5 before, but we can make 5 in multiple different ways and it's still 5. Comparing units of measure. Recording versus representing. So, we often ask our students to record what they're doing, but are we asking them what they're doing, or do we want to know what they're thinking? So there's some concepts there we need to develop. Geometric thinking and visualisation. And so you might have thought of some of these as more important at certain stages of students' schooling as well, but they all form part of them developing their mathematical skills. And not all of these concepts are in black and white in the syllabus. You kind of have to search for them or see where they sit. Or they might sit in one area, but they might breach out into another area as well. So they're important key concepts for us to develop. So, obviously, I've only got half an hour today, so I'm going to focus on part-whole relationships and comparison of units of measure. I did a little test with my secondary counterparts. I asked Nagla and Chris what they thought were the top key concepts we wanted students to have by the time they reached Stage 4, and they mentioned exactly the same ones that I did, particularly around number sense, and also this idea of comparison and converting units of measure, and knowing which ones to apply, they found those were really important.
So, part-whole relationships. So, for this progression, these are the key ideas from our 'Key Ideas' document, because I think it shows more readily breaking that whole substrand down into concepts. So, we start there with combining two or more groups. Now, this is in addition and subtraction. And then they start to look at combinations. We start to apply associative property. In Stage 3, there's nothing in there directly around associative property or commutative property, but we know that they use that when they select and apply efficient mental and written strategies. So I've just added those other key ideas at the bottom to show you that there is still a progression through. We don't sort of miss it out in Stage 3. It's just that they're applying their skills now. And then into Stage 4. And you've got those laws there, but it's not that we're teaching them as laws, it's just that the students are able to apply them as part of their flexible number sense.
So, let's start with Early Stage 1. What can you see in the poll? Off you go. Tell me how many dots you see, how you see them. 7. Everyone says 7. Lovely. That's a good response. Some people are saying they might see the 5 and the 2. Some people might say they see the 4 and the 3, a bit more like a domino pattern for the 4. OK, lots of different ways that you can see that. Some people might see that it's got the 10 less than 3. And that's OK to see that from that image. So, "How many more to make 10?" would be a normal question we would ask students. You don't have to answer that one, by the way. There'd be a normal question we ask students to respond to in Kindergarten around looking at combining two or more groups of objects to model addition.
Into Stage 1, when we look at part-whole relationships, the understanding the concept of part-whole involves knowledge of combinations, combinations to 10 and combinations to 20, and being able to flexibly combine and partition numbers for specific uses, which links beautifully to our numeracy continuum. By the end of Year 2, this is our expectation of our students from our syllabus.
So, note that our new syllabus explores both standard and non-standard partitioning, so that's into tens and ones and also into different forms.
So what do we know about the number 9? How can we make 9? Off you go in the pod. Give me some ways we could make 9. Lovely, ah, and beautiful. And I predicted my students well, so that's wonderful. It's really important when you're teaching a concept, it's wonderful when students sort of go off onto some wonderful tangents, but sometimes it's important to just bring them back and keep them on the one concept. So as soon as students in my class said things like "3 x 3" or "10 - 1", or some beautiful children said, "100 - 91", I can see you, smarty pants from Widemere there. Fantastic. We say that they're great, correct answers but in the context of our lesson today, we're looking at combinations of 9. We're looking at numbers that add to make 9. So you need to rein your students in. You want to keep them on board, on the one concept, OK? We don't want to go into more than two numbers, OK? So you might need to set up those parameters. We know we can make 9 with 3, 3 and 3. Lovely answer. Thank you for that. Today I'm just looking at two numbers that add together to make 9. So your clarity of question, or the prompting or the furthering questions you ask your students, is going to be really important to keep your focus on that concept.
In our syllabus in Stage 1 we talk about some different sort of terminology, like friends to 10. Now, I've written the word "to 10", because a lot of people talk about friends OF 10. And it's great to know all of our numbers that add to 10, but we also need to be able to do this for the numbers 6, 7, 8 and 9. And this is the point where our students miss the boat and often don't return to understand combinations. If they don't know these friends to 10 of all of those numbers and then applying them to 20, they won't be able to do it for two-digit, three-digit or four-digit numbers. These are the students that in Stage 3 and Stage 4 are still holding onto a number and then counting by ones the rest, even into two- and three-digit numbers. This is where the warning bells are going to go off for us. We want our students to have this flexible understanding of part-whole relationships at Stage 1, and we need to repeat it into Stage 2 and 3 and beyond. Because this is the foundation of mental computation, their ability to be able to flexibly move numbers around. We also talk about turnaround facts, is what we generally call them in Stage 1, and that's the commutative property that 5 + 4 is the same as 4 + 5. So, looking at those strategies for understanding part-whole relationships as a concept, how do we apply this to 16 + 9? So how do I apply my knowledge of friends to 9 to work out 16 + 9? What do I do? Give me an answer. What am I looking for my students to see the 9 as? Yes, I want them to see the 9 as 4 and 5, because getting myself to a decade makes my addition easier. It's not the only way to solve the problem, but it's a wonderful use of friends to 10. We often teach friends of 10, friends to 10, and leave it at that. Wonderful, great, you know that patterning. It's the application of that that develops this concept. Excellent. Thank you for all those responses in the pod.
How would you add these numbers together? You've got 2, 6, 7, 3, 8, 4 and 1. Gather into tens. Looking for the tens. Finding combinations of tens. Isn't that nice that people are following on from my last slide to help me with my presentation today. Excellent. Find friends of 10. Absolutely. This is what we want our students to be doing. It's that usefulness of that associative property that we're looking for.
Do they see all of that, or do they just try and add them up by ones? Do they all of a sudden put them in a vertical algorithm and spend the next 10 minutes adding them up or counting them on by ones? You want them to start seeing that patterning, but this concept of part-whole relationships, seeing addition as part of this associative process where 7 and 3 can equal 10, 6 and 4 can equal 10, 8 and 2 can equal 10, is really vital for these students. There's a whole list of these sort of strategies in our syllabus on page 129.
So you then might want to use place value for things like the split strategy. So when I see 25 + 18 I know that I can gather my tens and gather my ones and then re-add the numbers again.
A note here that this idea of part-whole relationships is also developed in multiplication-division, and fractions and decimals, for that matter. And patterns and algebra. But it's essential for these students to understand it as they move into Stage 3 and Stage 4. So when we looked back at that progression and we saw those different laws about commutative and associative and distributive laws, we want them to have these skills before they get there. We want them to have played and flexibly used numbers before they get there. So, seeing that 5 x 8 is the same as 8 x 5. And seeing that we can work something out starting with the tens, then the ones. That 7 x 13, we want students to be able to do this in their head, mentally, and one of the quickest ways to do that is to break that 13 into its tens and ones.
And that's applying distributive law. And this is happening in our classrooms, and should happen in our classrooms in Stage 2. So, we don't call it "distributive law". We're not learning about that today. But it's one of the strategies they might apply to work that out.
The area model strategy for mental computation is also in our syllabus. Now there's some great diagrams about how to use that. And that's sort of the basis, the beginning, of distributive law as well, where you've got that 27 x 8 and we break the 27 into the 20 and the 7 to work it out, to make it more manageable for us.
So, how would you solve 7 x 15? You don't have to go by what I said there, but how would you solve that in your head? 7 x 30 and halve, yes. 7 times the 10 and 7 times the 5, yes. And can you see a lot of these are precursors to grouping symbols, to looking at order of operations, to using those grouping symbols to organise our information? We've talked about that in a past session. Lots of different ways we could use it.
So, using that idea of distributive law without calling it that, with our students seeing, "OK, I know 10 times, I know five times." We don't want our students saying, "I don't know 15 times tables." We don't want them to think about it in that sort of times tables form. It's then going from the known to the unknown, OK?
There's a number of different ways. You'll have to excuse my terrible writing on my whiteboard. I had to use a mouse. I didn't have a pen with me the other day. I even had one of the suggestions that "I know, because 5 is half of 10, once I've done the 10 times the 7 I then just do half of the 70 again and add it back on." OK? So lots of different ways that your students might be using to solve these problems, all of which are correct and all of which are looking at developing this part-whole relationships idea.
So, we look at teaching multiples, and we know that we can probably do the zeros to fives not too bad. We can probably get to sixes because they're doubles of 3, and we can probably do eights because they're doubles of four, but how do you teach multiples of 7, or the seven times tables if you want to use that term? How do you teach multiples of 7? Because that's generally where our students get unstuck. OK, you could break it into 3 and 4, I've got a suggestion there. Fives and twos. Five times, add two. Obviously, initially concrete materials. Yes, thank you very much for that comment there. We always start with concrete. Really, really important, no matter how old the students are.
This is an activity from our TOWN program that we use, and the lesson from this is in the file pod today. It's called 'Structuring Sevens'. So we look at these, and we've got these strips. You can actually get them as physical strips as well, not just as images that you use on your interactive white board.
But what multiplication facts does this diagram represent? That's what we normally ask our students. So, what is it? What does that picture represent? What can you see there when you look at those rows? OK, some people are saying they can see five times, they can see two times. Some kids can see the whole picture. Some students might see all of that. You can see that there's five rows of seven there. It's just a way of chunking the information to help them solve the problem. We don't want kids thinking the only way to solve multiples of seven is to start at 7 x 1 and keep adding on 7, OK? It's an inefficient strategy to use.
So in that picture I can see 5 x 5. I can see five rows of two. I can see five rows of seven. As someone also mentioned there, you can flip it, and then you can see two rows of five and seven rows of five, and obviously five rows of five are still there. So, seeing those multiples of seven as 5 + 2 assists with mental computation, because we want our students to have that ability.
OK, the other section I'm going into today is comparison of units of measure, OK? So, this is a really important concept for our students to understand. They start with using comparative language when they're in Kindergarten, and then we move into comparing and ordering using informal units and formal... Of uniform size. Then we start to look at using metres and centimetres, and the conversion comes in into Stage 2 and 3, and by the time they get to Stage 4 they need to be able to convert between metric units of... And I wrote in 'measure' in inverted commas there because they do it for area, for volume, for capacity for a number of different areas, so I just sort of combined them all together for today's purposes of looking at their key ideas.
So, when we're in Early Stage 1, we want students to develop an awareness of the attributive link as comparisons. So how would you describe the word 'long'? Without using the word 'long'. Great. I can see some people also... It's really nice to say what it isn't as well. That's really important when you're looking at meanings of words with students, so they understand it. You can get as technical as you want, but the one thing we need to remember is that when we say something is long it's always in comparison with something else. Even if you don't mention it, even if you're saying, I had a long day today," you sort of ellipse the other comparison. You know that your day is supposed to be a certain length of time, but today it felt long, OK? Or when I say, you know, "That ruler is long," it's got to be longer than something else. So we need our students to understand that the attribute of length, it's based on a comparison, OK? When you're looking at something being long, the length is long, or longer, that comparative language, we need to see it as a comparison of something else. So, when students say things are big, it's because they're, even in their mind, they're visualising something that it's bigger than, OK? And we need to get that out of the students and into their work.
When we're in Stage 1 they start by comparing lengths informally, and then they move to formal units as well. So they're only comparing within the substrand. So they're only comparing length and length. They're not comparing across the substrands yet. And that concept of visualising, which we're not going to get to today, but maybe I might do a session next term on it, visualising the unit and repeating it is really important. So everyone can do this now, wherever you're sitting.
Use your hands or your fingers to show one centimetre. Look, Nagla's doing it over here with me too. I think hers might be a bit more like an inch. Oh, no. OK? Think about how did you know to show a centimetre? Did you think about a little centicube in between your fingers? Or did you just know because you've done it so many times that that's how big a centimetre is?
What about 10cm? Show 10cm. Again, how do you know? What is your justification for why you think that's 10cm? For me it's often that, "Oh, it's probably a third of a 30cm ruler." I know about how big a 30cm ruler is, so I break it down into thirds, OK? Or it might be, you know, not quite your hand, the length of your hand. You might think of that. People are thinking they might have a ruler in their mind.
What about a metre? How big is a metre? OK? How far do you put your hands out when you think about how large a metre is? What do you use? Do you visualise a metre ruler from your classroom? Or do you use other strategies as well? I know that I'm about 156cm tall. I know that my fingertips to fingertips when my hands are out in span is my height. So I kind of need to, just over, you know, two-thirds of that is a metre, so that's sort of how I do it. Or, you know, it's almost like when you do the yards with the material, which they still do in some material shops. So, it's that wealth of knowledge you're bringing, OK? And so, when we're completing tasks like this with our students, do they have that visual size of the unit?
And it works the same for other areas as well, like for volume and area, particularly when you come to litres. Do they know what a litre looks like? You're now going to bring in some looking at that conservation of size as well. And remembering that this kind of task is much harder for volume. And even for adults it's much harder with volume. Can you, you know, think of the size of a cubic metre? Is it easy to work that out?
So, comparisons of units linking that idea to visualisation, that hands-on physical manipulation tasks are extremely important. So, from our 'Teaching Measurement' books, it talks about that when students think of measurement as a process of subdivision, they no longer need to depend on the concrete representation. So can your kids, can your students in your class, know that I can repeatedly count the units, and that's the same as actually partitioning the space back out again? Can they break it down again so they can just repeat that unit in their head visually? Can they do that? And there's a great little activity there from the book about measuring a plastic bottle, about the circumference compared with the height.
In Stage 2 we also start to bring in that idea of converting between measurements, within units of measure. So if you're doing things like comparing three strips of paper, lengths of 1.2m, 90cm and 10,000mm, first, the students are going to need to convert the lengths to the same unit of measure. Do they know how to do that? Have they had experience with that? Do they have an understanding of the place value system, including decimals, that's going to help them understand this, OK? They need to be aware that you can only compare units to measure if they're equal quantity...if they're of equal quantity if they're identical, OK? We can't compare them if they're not identical.
So using that idea of place value, there's another activity from the 'Teaching Measurement' book around looking and drawing rectangles that have certain perimeters. And there's a decimal in there. And then working out to record the different rectangles you could make, OK? So it's giving them experiences with decimals.
So, into Stage 3. Students start to compare units both within the substrand and across, and we do a little bit in Stage 2 as well. But we really get into it in Stage 3 when we start to look at things like displacement. That last point there about comparing and converting between cubic centimetres and millilitres. They're difficult concepts for these students to understand, but the more exposure they've had with that concept of comparing, and that visualising and understanding conservation is going to help them develop through this an ability to convert, OK? We're talking about dimensions and comparing perimeters and areas by Stage 3. They're looking at working out what's the most appropriate. If you give them an object itself, if you give them a drink bottle and you ask them, "What are all the different things that I can compare about this, about measuring this, you know, the interior, exterior volume, the height, the circumference, you know, the area it takes up on a table?" Anything like that, it's really good to have those opportunities for students where they get to determine what's most appropriate.
So, there's then some other activities here about converting between the cubic centimetres and mils. That's a little displacement activity from the 'Teaching Measurement' book, and there's a great follow-up lesson to that called 'What went in', and it's on page 160. If you still don't have access to the 'Teaching Measurement' books, I've put the PDFs on the lobby page of this room. So anytime you want to come back into this room, you can download them straight from the lobby. So, in a couple of days, I'll take this off the last screen and put it back on the lobby page, and you'll be able to access them there.
So just some final comments. Start where your students are at. You need to understand where your students are at on the teaching and learning cycle. You need to think about where they are at understanding a particular concept. Move from the known to the unknown. It's important to remember that all students have gaps and misconceptions. Even the students you think have got it, they're going to have a gap or a misconception somewhere. Don't complicate the concept. Developing conceptual understanding means focusing on one concept at a time, OK? Don't cloud it over, even if they're great ideas the kids are bringing to the lesson. Sometimes it's great to go off on tangents, but if you're trying to do an explicit lesson around a concept, you've got to rein them in. You've got to know what you want to accept as answers to keep their understanding close and tight around that one concept. And conceptual knowledge is transferable. It's not gonna just be an easy path straight on to the next bit. It's going to be used in other areas as well. That's fine, but you need to make sure you're keeping it nice and tight.
Thank you for attending today. Please complete your evaluation online for your school's event. Our next session is in two weeks. It's the last one for this term. Fingers crossed they'll have the adverts out for next term very soon. I'll at least be posting them to people that are the contact people for these sessions.
Just a little heads up. Syllabus PLUS Maths 7-12, Nagla and Chris's Adobe recordings are online. This... That picture there is web-linked, it's hyperlinked. And it's in the old Curriculum Support site. You can still google Curriculum Support and get there. It's through 7-10 professional learning in mathematics. I'm in the process of doing the same thing for the K-6, so keep looking out. I will have it done, hopefully, this week, when I get permission and the information on how to access this website so I can actually update it. Chris has given it to me, but it's pretty complex, so I'm getting there. But please keep a lookout. I do recommend you go and watch some of the 7-12 ones, particular the one that Peter Gould did on mathematical reasoning. It's particularly nice, and it blends in with what we were talking about with reasoning as well.
Just a reminder, they're my contact details. 'Mathematical Bridge', just an FYI as well. Our next issue is out early April. I say early April. We're fingers crossed for the next week or the week after. So, that'll be exciting as well.
Going to the file pod now for people. Remembering that our next session is on Tuesday, 8 April. There's the file pod with a number of different readings today. There's the lesson in there and my PowerPoint presentation. Thanks for partaking today in the chat and having a go with the answers. That's wonderful. Remember, if you do watch the recording, you can scroll back up through the chat if you want to look at what people said. I know I didn't get to all the key concepts. But look, if there's something you're burning that you'd love me to do for next term, just pop it in the chat, because I haven't set those topics yet. The next session is on language focus in mathematics in our syllabus. So, I hope to see you all then. Thank you, and good afternoon.