Transcript of The numeracy continuum and the syllabus
Katherin Cartwright Well, hello, everyone. Hopefully people can hear me. Good afternoon and welcome to series 3 of Syllabus PLUS Mathematics K-6. So thanks for joining us this afternoon. I do realise that for some people it's been a very long day with the start of NAPLAN, so hopefully you can stay tuned for this afternoon's session. Just a reminder that today's session is being recorded and the link will be shared via email to the contact people who enrolled in my event. They'll also be located on the Curriculum Support website and I'll show that link again in our conclusion page today. So thanks for being on board and I'm going to start the presentation now.
I will also mention that Carol Field is on the chat today, so she'll be able to answer any of your questions for you as I'm going through the presentation. So, today we're looking at the syllabus and the numeracy continuum.
So, linking the continuum to the syllabus - we have to make sure that we keep in mind that we want to keep the focus on the teaching and learning cycle as we're going through any of our focus on both the syllabus and the continuum, and then it's answering those questions about what I want my students to learn, how will I get them there, how do I know when my students get there, and where are my students now. And that continues to cycle round as we develop units of learning for our students. And so, today we're looking at how the syllabus and the numeracy continuum can help us understand our students better.
So the continuum - what is it? Hopefully, people know a little bit about the continuum and have probably varying degrees of understanding around it. So the continuum describes a process of how students move through their efficient strategies at a sophistication level. So it starts out quite basic and they move through the different levels to display their strategies they're using. And it's particularly around number and measurement problems, so it's to solve problems. So we're not so much focusing on the answer but how they got there. This little image as well that I'm using today comes from a document called 'An Overview of the Numeracy Continuum K-10'. That document's in the file pod and I'll also show you the website where it sits as well to help you out. But we use the continuum to assess, track and monitor student progress, to determine 'where to next' for our teaching, to help identify clear learning goals and to strengthen numeracy in all KLAs. And so, I'm going to go through some of those points today in my session. So it's all about a progression.
The continuum deals with some key concepts. So it's about the development of these key concepts in seven aspects, so you're looking at counting sequences, which involves numeral identification and counting as a process, it involves early arithmetical strategies, which is like the precursor to addition and subtraction, so when you're building on those counting skills to develop early, addition and subtraction skills. It then goes into some patterns and number structure. There's also the concept of place value, multiplication and division, fractions, and unit structure of length, area and volume. If you can still... Sorry, I will comment. If you can still hear the background music, you can go back and you can pause the music in the pod or you might have two windows open. Sorry, I'm just adding there. I'm sure Carol's going to answer that question anyway. So it's really important to realise that the continuum is all around its links to number and number skills. It is a numeracy continuum because it's the application of those skills across mathematics, but I will show you a number of areas where you can also link it across KLAs. But it's looking at a development of these concepts for your students from early to latter years.
So the first section of that little picture talked about assessing, tracking and monitoring student progress, and this is how we've been using our numeracy continuum for quite a while and even before it was called the numeracy continuum when it was part of the learning framework in Number that was developed in association with the Count Me In and Count Me In Too programs. So, many of our programs for numeracy use the numeracy continuum to do this, so often people use it at the beginning of the year for initial assessments, diagnostic assessments, and we use it in Best Start, which the numeracy side of that was developed from the SENA 1, which is the Schedule for Early Number Assessment, there's also a SENA 2. We also use it in our Counting On program, we still use it in our TOWN as part of our initial assessment, and we continue to use it for ongoing assessment and monitoring. Now, I'm aware that some people might not have had anything to do with the majority of those programs and maybe the Best Start is the only one that you've seen, so in the file pod today I've included copies and versions of SENA, SENA 2, the Counting On assessment that we use for counting on and for TOWN. So some of them are quite old and still have some references to our current syllabus. I'm in the process of updating a lot of those and I'll speak a little bit more about where we're going with our diagnostic assessments later in the presentation. But please feel free to use them. You can use them in their entirety or take out sections of them if you've not been using any sort of diagnostic assessment to place students. So there's also the PLAN software, which is part of the Best Start, but it's now K-6 and it will be into Year 7 and 8 as well soon. And it's a way just to place your students on that continuum to give you an idea of where to start them off and what's the next step in their learning that they need to understand. I've just got some images there too of data walls. We use data walls for TOWN. Some schools might already be using data walls, so a few people there are answering my question in the chat about, you know, how many schools are already using data walls. So you might be able to use that in your classroom or your school and it's mainly just to get that quick visual of where my students are at and are there any problems students that are maybe not moving at the rate I would like them to, particularly once they get into Stage 2 and 3, and they're not moving at the same rate as the curriculum is moving for them in the stages.
So there's just a picture of what the SENA 2 looks like or what it did look like in its very beginning days. I've got a couple of versions in the pod for you that you can use. So we want to continually ask our students, "How did you do that?" or, "How did you work that out?" So it's very much these assessments to do with the continuum are diagnostic and they focus on how the students answer questions. And hopefully that's ringing bells for you in regards to our syllabus and that strong focus on working mathematically, that strong focus on problem solving and a strong focus on reasoning. We want our students to be able to give us reasons how they came to an answer so that we know they could possibly do it again in a different scenario. So that's really a focus of these diagnostic assessments that we use.
The second thing that we often use our continuum for is identifying clear learning goals. So we can either do that as teachers, so we can create learning intentions based on the continuum, or we can use them for personal learning goals for students. OK, so I've got an example there that we use in TOWN where the question was '48 + 26' and the student to work out the answer of 74 counted on that 26 by 1s. So if this student was in Stage 2 or 3, they really need to start picking up some of these efficient skills a bit quicker. So on the right there, I've got a reference to the continuum "in kid's speak". I have put a copy of that in the file pod today. It's based on some work - I'm pretty sure it's from Nuwarra. I don't know if they're on today. They might be able to recall if it's the one from their school. It's not the absolute be-all, end-all, it doesn't have every single example of what students could do at each of those levels, but it's just a starting place for you if you've never tried to convert the continuum into kids' talk to help them set their own learning goals. So it's in the example of what you can do. And on the continuum, those dot points are also not an exhaustive list of what those students might do at each level. They're just some suggestions of what we've seen that students can do. So for this student, they might be working towards counting by 10s off the decade. So that idea that we're trying to move them from Place Value Level 0 to Place Value Level 1, where they could maybe count on that 26 as two lots of 10 and then count the other 6 as 1s. OK, so you can definitely use it for creating personal learning goals or classroom learning intentions.
We can also strengthen numeracy in all KLAs. We know that numeracy is one of the general capabilities now in our new syllabus. And so we've got that strong emphasis in numeracy in other syllabuses and, particularly in the Phase 1 syllabuses, it's marked with the icon - the calculator icon, which doesn't mean that it's only about using calculators, but it was an image they chose to represent numeracy. And I've got an example there of those basic mathematics, those number sense skills are often utilised in other KLAs. So for science and technology, this is an example from Stage 1 and we're looking at working scientifically and learning about the living world. And in doing this in your classroom and students learning about this, they have to actually use informal measurements to collect and record observations, so it's linking with measurement and with data in fact. And they're having to look at recording change in plant or animals and some of that might be around using formal units, and so straightaway I know there's a link to mathematics and a link to length, and so they're going to have to use some of those skills in a different KLA.
So determining 'where to next' for teaching is also really important. So we don't want to just guess where our students are at and plot out a series of lessons that we're not sure where it's going to match up for students. We want to know that we're catering for the needs of our students. So "How do I move them?" So "What is that next small step in their learning?" So our syllabus provides us with a lot of content that will help us attain those outcomes for our students. But sometimes we want a little bit more. We want to tease that content out a little bit more down to the strategies our students are using, and the student is at centre of the learning. So the syllabus talks about efficient strategies in both Stage 1 and Stage 2 in particular. There's some nice lists there about efficient strategies. I've put a file in the file pod today about different strategies we're trying to encourage our students to use. So the numeracy continuum just explains these strategies in a hierarchy, so it really pulls them apart so that we can see those little steps the students take. Sometimes they go through them very quickly, but sometimes it takes them a long time to move on from something like counting by 1s to actually starting to break a number into its 10s and 1s. It can take a while for our students to develop that. And the continuum is based on examples of students' work and assessments that they've completed, so it's really evidenced-based, the continuum.
So "How do I use the continuum and the syllabus together?" So I can see there that I've used the flow chart and that flow chart comes straight from our course that's on - a process for programming a unit of learning. So even if you don't complete the whole of that course, you can definitely go into there. And I'm pretty sure I've hyperlinked that today but you might have to check that for me. And you can download that flow chart and it just gives you an idea. And it's not necessarily in that order. That's why the arrows go both ways - some of those things I do all at the same time. But it's just to sort of get your mind working around where you might start when you're programming. So before I even get into programming my unit of learning, I need to have my scope and sequence. So I've got a note there on the side that has... When I'm having my scope and sequence, I might have developed that as a whole school or generally as a stage you might develop your scope and sequence, and I would definitely look at the substrands, so things like addition and subtraction, length, angles, those sorts of substrands are in my scope and sequence. And I might even go down to giving... splitting up the key ideas on my scope and sequence so that everyone is looking at the same big ideas, those key ideas, but then how you actually teach that and the content you use and the level to which you go into that content should be based on your students' needs and decided by the teacher. OK? We don't want to get to the point where our scope and sequences divvy up all of the dot points from every content section in our syllabus and there leaves no room for movement or differentiation for the teacher. So that scope and sequence really should be an over-arching statement around the substrands and possibly the key ideas. So this is just an example of how I would now use my continuum to support programming. So I'm looking at Year 2, OK? I have a straight Year 2 for this example. My substrand from my scope and sequence is addition and subtraction. And the key idea that I'm looking at is use and record a range of mental strategies for addition and subtraction using 2-digit numbers. Now, if you've been a part of these Syllabus PLUS series before, you will know that we did a session on key ideas and we now have the posters, the Continuum of Key Ideas poster K-10. Three copies of each were sent out to schools and you can download and print it yourself from that little link where it says 'here'. And also the image of the Continuum of Key Ideas over there on the right, you can click on the link and that will take you to the website where you can download it. We are looking at the possibility of doing another print run and we'll be sending out more copies to schools. It won't be something you'll be able to continue to order more from, only because we don't have the personnel to do that in here at state office. So we're thinking we may just do another print run and send out a whole lot more copies because we have had teachers asking for more copies. So that's where we're at at the moment. But if you want to download and print your own copy, I did it, I printed it as an A3 here at the office, and it was very small but readable. But you might want to print it a little bit bigger than A3. But just a sidenote, that's there for you. So in my process of programming, I've got to the point where I know what my key idea is and now I'm going to move on to choosing the concept, so what is it that I want them to understand. And I know that my numeracy continuum provides for me some of those larger concepts around addition and subtraction and one of those is place value. And so, for me, my students in Year 2 need to understand that numbers are flexible and can be rearranged to 10s and 1s in different ways to solve problems. And that links directly to the 10s and 1s on the numeracy continuum. And so, I've got a little snippet there from the numeracy continuum showing you how it talks about jump-and-split method. And the beauty of the numeracy continuum as well is it gives you a little link back to the syllabus, so if you're wondering, "Oh, where do students learn about 10s and 1s? Where does it really first appear?" We definitely would be continuing to learn about 10s and 1s, but it does first appear in Stage 1 and it also appears in Stage 2. So you can see the links there, the continuum. So, concepts - we want to start with the concrete, then moving to visual and symbolic. And remember that these concepts are understandings we want our students to develop.
So I then look at the content from my syllabus, so I go back into my syllabus and I find the outcome I'm looking at, which is a Stage 1 outcome for addition and subtraction, I look at the dot points that particularly refer to that content. So I've sort of got through that pink section of my flow chart. So I've got my topic, my substrand, I've found my outcome with my key ideas and my concept, and now I want to include a focus on working mathematically, and for this one it's probably going to be reasoning and definitely problem-solving because they'll actually be doing some problem-solving, I would think, as part of my unit of learning for this. So my next step is to think of, like, a pre-assessment, so work out where my students are at. So you might use this for grouping your students or just working out, Well, what sort of support am I going to have to offer my students?" So I would use an assessment task, just one question, and here's my question for today that I would use with my students - show two or more ways to solve 39 + 12. So it's just a quick assessment to see the strategies my students are using currently and where I need to move them to next. When I teach my content from the syllabus, I want to make sure that I'm looking at that numeracy continuum and saying, "OK, a lot of my students still need support in this area." So you could definitely use a question from any of the SENAs or the diagnostic assessments I have in the file pod or you're most welcome to make up your own questions. I just made that one up. It wasn't too difficult. So that would be something I would ask as a pre-assessment for my students.
And so, this would be some of the results. Obviously I've done these, not a student. I didn't guinea pig my daughter for this. I just did it myself. And thought about some of the things that our students may be showing you to solve 39 + 12. So some students might also write, "I did it in my head," and that, to me, gives off some warning bells around the fact that I need to maybe talk with the students about communication and how they can communicate their strategies, and that takes time and practice. You need to actually teach students how to share their strategies. Some students may also just write the algorithm - possibly not year 1 or 2, but into future years if you're using a diagnostic assessment. I always ask them, "Is there another way?" So my question there also says, "I want two or more ways," because our syllabus does state that our students should use a range of strategies. So I've got a number of different ones there. There's one where the student maybe needs to see the second number in its 10s and 1s form to know what to do. There's a jump strategy method I've shown there. There's someone who's counting on that 12 by 1s, so they drew the little sticks for the 12 and they went back and counted them again by 1s. And then there's someone who might show a split strategy, showing how do you break those numbers in their 10s and 1s. So obviously, again, this is not an exhaustive list. This is just showing you some examples of answers you might get for this.
So once I've done that, I want to be able to look at the continuum and say, "OK, well, where were they at?" Before I've even started teaching my unit, I want to place my students on this continuum 'cause this information's going to help me in determining what support or scaffolds some of my students need. So maybe they need to have counters or a hundreds chart or some empty number lines available to them during the lessons I'm now going to teach around addition and subtraction. So it's providing me with more information about the students. It's also going to help me if I want to put my students into groups, OK? So maybe I want to group my students based on their abilities for a couple of my lessons during this unit of learning where they're working with students that are of the same level as them. I might also do it where I mix them all together. And I would also use this information to work out what modifications or lesson variations are required. So I might have a student who is maybe... or a whole lot of my students all do the count by 1s and that's not what I was expecting, I might now need to bring in some other support for these students using some smaller numbers to get some of these skills up and running. So that's how I'm going to use the continuum for my programming.
So here's a question for everyone in the pod. I've seen lots of questions come up in there and I'm sure Carol's doing a fantastic job of answering them. If there are any further questions, I have got time at the end to answer them. So a question for you, though - if a student is using a count-by-ones strategy, so how would we help them get there? Remembering my concept is about understanding that numbers are flexible and can be arranged in 10s and 1s, give me an idea, anyone, if they would like to, on the chat today. What's something you might do to help this student? The next tiny step in their progression. "Number line" - excellent. I would definitely show them, like, a tape measure or a number line that has numbers on it and show them where those other numbers exist. Definitely explain to them and talk them through that 12 is made up of 10 and 2 - yes, lovely idea. Anything else? "Hundreds chart" - yes. Show them where 39 is. And that's where you might bring in some of those ideas like looking at when we count down on our hundreds chart - what's happening? When I count across on a hundreds chart, what's happening? Yes, getting out MAB blocks or Unifix cubes, showing them how to make that construct of 12 as a 10 and two 1s. Yes, "Revisit 10 as a whole." Yes, "Base 10 materials showing 10s and 1s." Yes, lovely ideas. "10 Frames," excellent. And even making a 39 up as three 10 Frames and then a 9 helps them to get to that point where they can see the compensation strategy, where I can just, you know, put one over from the 12 to make it 40 and then add my 11 on. So you can definitely show that through 10 Frames as well. Yes, showing 12 as 10 Unifix blocks joined together... And if anyone has seen Brian Tickle's work, where he puts the Unifix cubes on his fingers and then if you've got a full group - if all your 10 fingers have blocks on them - you then make them into one stick. That's a great idea as well. Thank you. Excellent suggestions there.
OK, what if the students are already able to show efficient strategies? OK, so what if they're already able to show me a jump and a split and something else as well, maybe a compensation, what do I do for these students in my lessons? If I'm looking at understanding how flexible numbers can be rearranged into 10s and 1s, what would I do to differentiate the lesson for these students? Move them into 100s - definitely. You can make it more difficult using larger numbers because that's what the next step on our numeracy continuum is. Anything else? "Subtraction" - definitely. A lot of our students are very competent in counting using different strategies other than counting on by 1s when it's addition. When it's subtraction, things often go a little bit downhill. Definitely want to show them how it works within trading in the sense of being able to move that 1 from the 12 over to the 9 and then add it on the 11. Partitioning - yes, you can go into partitioning as well. "Give them a problem to solve" - thank you, Margo. Excellent. Remember that we really want to try and extend these students laterally. We want them to have a depth of knowledge around problem-solving, communicating, reasoning around addition and subtraction 'cause, yes, we can give them more difficult numbers if it's in the stage, but once we start going beyond the stage, we don't want to get to the point where they're getting bored into stage 3 and 4 because you've already done that work. We want to extend them through problem-solving. Excellent. Yes, "Get the children to make up their own problem." Provide them with an answer, see if they can give you the problems that would get you that answer, in more than one step, not just making the numbers larger. "Make it real-life" - lovely suggestion. Yes, "Four different strategies to solve and then identify, with reasoning, the best strategy" - lovely idea, Josephine, thank you. You guys are on fire today. Excellent responses.
So where else is the continuum link in the syllabus? So that's an example of how it links with addition and subtraction in number, but there's other areas as well. On our continuum, we have measurement as well. The reason we don't have space is because we don't always use a numerical count or that kind of counting or addition strategies in that area, but we definitely use them in measurement. And like I mentioned with the link to the other KLA into science and technology, it's also in length here with measuring. It's Aspect 7 on the continuum. And if we look at the syllabus there with the dot points from the syllabus - I'm sorry if it's a bit blurry, I just did a cut and paste today - they're looking at using units end to end without gaps and overlaps, using uniform, informal units, and these words should be ringing bells to you if you've got a continuum or if you know the continuum because it's the same language as on the continuum that we're looking at.
So you might even want to pre-assess your students to locate them on the continuum when you're teaching length or some other areas of measurement. It's got two portions here. I want to know the strategy they're using to count. 'Cause if I'm measuring the length of something, I want to know if they're counting by 1s. Are they starting at 1? Can they count on if I want to add another length or compare lengths? How about their unit structure of length? Can they measure without gaps or overlaps? Do they know to align the objects together to compare them? I want to know all of this information before I start a unit on length and pitching a lesson too hard if my students aren't there yet. I really want to have an understanding of where my students rank on that continuum.
So I might use an activity even from the 'Teaching Measurement' book as my pre-assessments. This one's about choosing my unit where they use a unit to make a line the same length as they are measuring on the floor. So they need to use a set of identical units and then you can see, well, what are they doing? Do they know to align the units together? Do they know how to record that? Do they know about using parts of units? So that's a great little pre-assessment there that links me back to the continuum as well as my syllabus.
So some other things to help you and support you along your journey and seeing how the continuum links with the syllabus, the Numeracy Continuum website is excellent. Please go there. The continuum on there shows much more links than the poster does. There's one major link on the poster between EAS and Place Value. But there's lots of other linkages like the one I talked about today with measurement and counting strategies and you can see them on the interactive version in here. It's a great resource.
We also have our resources section of the NSW Syllabuses for the Australian Curriculum link that's on your intranet site. I got that out in one mouthful. So if you go to that link and then click on 'resources',
there's a K-10 Maths link and then in there you'll find the PDF about using the numeracy continuum with the new syllabus - an excellent little e-book PDF that you can go through. It won't take you long. I've got that in the pod today as well - in the file pod. I say that and I'm pretty sure I did put it in, but you can find it on the Web as well. And that is where you can still download a printable version of the numeracy continuum as well.
So that's what it looks like when you get to open it up. I've just got a couple of excerpts from there.
It's really excellent. It goes particularly through EAS, the Early Arithmetical Strategies, and Place Value, and it shows you some videos about students at that level, it has the links to the syllabus, it links to the key ideas and it links to related content. Great resource for you to use.
It talks about considering assessing your students when they're at EAS levels. Look at the associated syllabus outcomes and then look at each outcome and plan your teaching program to help students move along the continuum.
Excuse me. Back in here again, instead of clicking on 'resources', click on 'professional learning'.
There's a link in there under the 'other courses' that says 'using the numeracy continuum' - it is coming. I'm in the process of developing this course with Mario Bugeja, who's our 7-10 Numeracy Advisor. It's mainly going to be pointing you to a lot of those resources I'm showing you now, but we're also looking at creating some more diagnostic assessments K-10, particularly stage 3 and 4, to help you and they're related to the new syllabus, so I did mention those assessments, like SENA, like Count Me In Too, like Counting On. They need a bit of a refresher and we want everyone to have access to them, particularly now that pretty much K-8 are going to have access to PLAN software, people are going to start asking - I'm surprised I haven't had more questions about this - Have you got examples of diagnostic assessments that could help us locate our students on the continuum?" So we're trying to provide that for you. I don't have a date of when that will be ready. Hopefully next term.
There's also a Continuum's website. People might not know this exists. If you click on 'curriculum resources', it's in light blue on that tab across the top there in my image, and it's about literacy and numeracy K-10 support.
When you go in there, you click on 'numeracy' and it has a lovely list of things there to help you. There's a 'getting started' section. It has a matrix where you can locate where you are with your understanding of the continuum at the school. It has links back to that using the numeracy continuum e-book I showed you. There's some other professional learning links in there and some frequently asked questions. I think Carol's still the contact on there, but if you want further assistance, either contact Mario or myself around that at the moment. So it just provides you with a 'where to start'. There's some great resources, and I think it's our real first port of call for schools, so please use that.
Just some final comments from me today. The continuum does not replace our syllabus outcomes or content. We use it to help us place students, to change the way we're teaching or to influence what we're teaching them, where we're pitching our lessons at and how we're differentiating what we're teaching. So it provides that further information on our students. The continuum assists us in planning lessons that meet the differing needs of our students. We still assess on syllabus outcomes around the strands. That won't change until a new policy is out and I'd say this will be quite similar for our new policy. The continuum is not a checklist to mark off. Students don't have to mark off every single dot point to move on to the next. You might have students that show flexible or into place-value strategies but still might go back to using a figurative or a counting-on-and-back strategy for something like a subtraction, as we mentioned before. That's OK to move them to their highest point. If it gets too difficult, differentiate the lesson to suit them. So just providing that depth of knowledge for our students.
I hope this has given you some information today around the continuum and the syllabus. I'm happy to continue with the discussion. The next session is in two weeks time on Tuesday, 27 May and it will be around assessment and reporting.
There are my contact details. If you're enjoying the 'Mathematical Bridge', you can click on that 'Curriculum networks - subscribe here' if you aren't on our subscription list and you'll get sent copies of those. We're also in the process of getting a new website up and running just for primary curriculum, where I will be locating a link to those 'Mathematical Bridge' past copies. So just hold on - hopefully by the next time I Adobe Connect with you I'll have the link ready for you to our new website. Our next issue will be in June, that's mainly for stages 3-4.
I'm going to go to my conclusion page now. Just, also, I've got lots of files today. You can see there's plenty of them, so it'll take you a while to download those to your school computer. I've also got in there the flyer for the K-6 Science and Technology sessions, the History sessions and the 7-10 Maths sessions. If you're interested in any of those, the flyers are in the pod there. And all my Syllabus PLUS - I'm just going to scroll down, hopefully you can see me doing that on your screen - that is the link so you can copy and paste that link where all of the recordings sit for series 1 and 2 and where I'll be uploading today's session in the next couple of days. Tomorrow I will send out the link to this recording to everyone who is on my enrolment list as the contact person, but you can definitely go there yourselves and visit there to look at the recordings from previous sessions as well. Went over by four minutes today. I hope that's OK. But thank you for being a part of today's session. If you've got any more questions or want to repeat a question that wasn't answered, just let me know now in the chat pod. Thanks, everyone.