Transcript of Key concepts—a language focus
Katherin Cartwright: Well, hello, everyone, and good afternoon. Welcome to our last Syllabus PLUS session for term one. Thanks for joining us, particularly considering it's the last week of school. So, thanks for coming on today. Today I'm joined with Carol Field on the chat, and you can probably also see behind me, if I turn this way, Jane Wallace, who's here with us as well. She gave me some information today that I'm using in my presentation around support for EAL/D students with language as well. So, welcome, everyone. And just a reminder that today's session is being recorded, and will be shared via our website, which I'll mention a little bit later on. So, welcome, everybody.
So, this afternoon's presentation is on key concepts - a language focus. So it sort of builds a little bit on from last week where we looked at key concepts across and within the syllabus. And today we're just looking at language focus, particularly around the new syllabus, and giving you some strategies you can use in your classroom.
So this is something I've used in the past. I find this sign really interesting, and it sort of highlights for me the importance of understanding a mathematical language in society. And it looks a wonderful poster until you do the maths. And 10 minutes a day does not equal an hour a week if I times 10 by seven days. So, you know, it's a great, a great ad campaign, but the mathematics isn't quite there.
Another ad campaign that I noticed not long ago was 'A Metre Matters', and I quite liked this one, because it really showed how we need to have an understanding of mathematical language in the context of real life. And so you need to understand what 'one' and the 'M' means. You need to have an understanding of why they're having that recommendation about 'A Metre Matters', and that that ad campaign was to try to convince the government to change their policy around how far drivers have to drive away from cyclists. So there's a lot of times where we need to use this mathematical language in our real lives.
So, mathematics and the language of mathematics is about developing conceptual understanding and making connections. That's something we've talked about before, about concepts and making connections, and students can do this through language. Mathematics is almost a language to itself sometimes, and the students need to see it, hear it and use it. And it's not just about technical vocab, although we will go through some of that today. But it's about really getting into the depth of discussing around language, which, I guess, is metalanguage as well, and seeing that as a large part of understanding mathematics itself.
So we're going to play a little game, a little activity first up. This is an activity that I borrowed from Annette Gray. She's a fantastic... She's now the literacy advisor. She was the English advisor. And it's a little game you can play with two students in your class. So, one student would get a table like that one on the board, that I've got on the presentation there, and I would have another version that has the missing words, and we need to try and explain the different words to each other. Now, obviously, I'm not going to explain those ones because they're there, but I'm going to choose four words from my sheet and see if you can guess what they are. So you're going to use the chat again today like we did last week. So, my first word I'm going to try and explain to you. And I sort of relate this a bit similar to the game 'Taboo', if you've ever played that. You can... You've got to describe what the word is without using the word. So, when you think you know what my word is, please type it in the chat. OK, so this word is used when you're talking about turning, but instead of turning around one way, you're going in the other direction. And I tend to use it when I might be talking about... Ah! My watch! Excellent! Anticlockwise. Well done. Yay. So, it's person - Andrew. Well done. No gold stars today, sorry. And I can't pass chocolate through the presentation. OK, second word. This is where two lines or sides of a shape meet, and they form an... Ooh, angle! Berry Public School. Thank you very much. Well done. And you can see that some of the similar words that are coming out are also appropriate, and this is where the person that is me, Person A, might need to give some more information. The next one. So, this is a type of line or angle, and it has 180 degrees in it. It's a bit like... Ah, excellent. Straight angle. Thanks, Elizabeth. Saw that. It's the first person that got it right. Very nice. And I'm impressed with everyone's spelling, and they're typing so fast. OK, last one I'm going to give you is, OK, this word means the same as 'rotate'. And we use it when we're giving directions or instructions, maybe when driving. Yes, excellent. Thanks, Helen. Turn. First person to get that right. Well done. So, this just shows you, it's a great little sort of barrier activity to use with students, and you might use it at the end of a topic. So, talk about your understanding around target language. It's probably not something you'd start with, because you want to see what the students have gained around their understanding of this language. And so you're sort of working together to describe, maybe, the meaning of the word. Other relationships. You might use synonyms. You might have to provide examples for clarification. So, it's a great little task you could use with your students in your classroom.
So, there was a national Numeracy Review report done by COAG, the Council of Australian Governments, in 2008. And they talked about language in mathematics, saying it can provide a formidable barrier to both understanding the mathematical concepts and how students access assessment. And this is a really important thing to understand, is that it's a barrier. We're not saying that students aren't going to be able to develop language at some stage, or don't already come with language when we start learning about mathematics, but for some students in particular it can provide a barrier for them actually understanding the concept itself. So we need to make sure that we're being aware of this in our teaching, and that we're providing some different activities to cater for this. The whole article, if you'd like to read it, is actually in the file pod today.
And that little quote itself comes from this article that was in the NSW Institute of Teachers' 'Digest', and that whole document is in the file pod today. And so, an at-school task - let's say you're watching this as a recorded version, you might want to stop the presentation now, and... Or you might have wanted to do this before you actually get to using it in your staff meeting. And the article's actually broken up into sort of four major sections around the role of mathematics in learning, mathematical language, mathematical problem-solving and mathematics and literature. And you might want to send people off in their stage groups to read the article, highlight a main point and then come back and share it with the rest of the group. It's a really nice way to look at readings, and to share what you've learned from reading it. So, it's a fantastic article, and I think it's very easy to read and gives some really good suggestions that you could use in your classrooms, So I recommend you have a look at that.
So, in our mathematics syllabus, that learning across the curriculum, literacy is one of those general capabilities. And we're seeing that literacy and mathematics is about understanding written problems. Sometimes it's about words with specific meanings in a mathematical context that they might have a different meaning in a different context, and often metaphorical language that's used in mathematics as well. So it's not just the words, but the context in which they're used. And the way which language is taught is really important.
So, just on that note of metaphorical language, there's a couple of really great documents I'm going to recommend today from ACARA, and this one is the 'English as an Additional Language or Dialect Teacher Resource'. And it talks in there about mathematical language when it's metaphorical, that when you're doing something like describing numbers like 10 as a container, so it's being used as a metaphor for when you're talking about the chances of one in 10. Or also when you ask questions like "How many fours in 44?" The literal answer is two. Now, that's generally not what you're asking them to tell you, but if students have language as a barrier, they might take it quite literally, even though that's not how it's supposed to be read in mathematics. So these can cause conceptual difficulties for our students, particularly students with... that are EAL/D, but also many of your other students as well. And they don't have that cultural conceptualisation of how we use metaphors in mathematics, so this is just something to be aware of in your classroom.
Another aspect of literacy in mathematics and language from that general capability is talking about mathematical vocabulary, and the conventions we use for communicating maths. That includes its symbols, its structures, as well as verbally. And so, a lot of these conventions may depend on your cultural background. We'll often use a lot of abbreviations, like 3-D, m, km, m to the power of two, and half - one over two. And that's not how I read any of those, but that's what they look like, OK? I need to know that 'm' means 'metres'. I need to know that 3-D is referencing three-dimensional space or an object. And it's important to note that this includes both oral and written language, that it's not just the way students write in mathematics, it's the way they talk and communicate as well.
So, some changes from our syllabus around language features. This is a slide way back from, I think, the first Syllabus PLUS session I did. I just thought I'd bring it back to remind people. There's definitely an increased focus on recording, so students actually writing using correct terminology, and also writing what they're doing to work out problems. There's an increased focus on actually solving word problems. So instead of just saying "solving problems" it says "word problems", so there's this focus now that they're going to have to read a problem and answer questions around that. There's also an increased focus on comparing, modelling, ordering, identifying and using the term. So, often now it will say that students can use the term. For an example, say, the word 'sum', that means when we're adding things together. So, that's quite specific for mathematics. And I think the good thing about the new syllabus is that that language is back in underneath our content dot points, and if it's new language that the students have to be using, it's in bold. So when you see the words that are bold under the heading of 'Language', it means they're new for this stage or this part of that syllabus. So it's helpful that that might be a place that you start. And that also indicates that the words that are not bold should be prerequisite knowledge for that topic. And the glossary's in the back of the paper copy, and there's also a glossary online as well.
So, just a quick touch on assessing. I'm not going to go through this too much today. But just that increase on the focus on word problems... And this is a little quote again from that resource about EAL/D students, is that it's important to identify the language requirements of the task, whether it's an assessment task or a task in the classroom, while still maintaining the integrity of what it is you're actually teaching. And that you're going to need to modify assessment tasks to allow EAL/D students and other students who may be struggling with mathematical language, to understand the content while they're developing their English language skills. So, they're just some really important points to note.
So, last week we talked about some key concepts, and I've greened some of the ones I'm going to talk about the language for today, because I think the progression of language for some of these topics is quite important, because sometimes the language, it actually incorporates the concept itself.
So, if we look at, in 'Whole Number', when we use the word 'number' you can see how it changes from Early Stage 1 through to Stage 4. So in Early Stage 1 we're just referencing it as the word 'number', and then into Stage 1 and 2 we start using the word 'digits' to refer to these numbers. And then in Stage 3 we now actually introduce that word 'integers', which describes both positive and negative whole numbers, and then into Stage 4 as well. So... And I think what's really important to note in Stage 4 there is that when they're talking about operations with integers, they're using mental and written strategies. So even into Stage 4 we're still expecting our students to use mental strategies when solving addition, subtraction, multiplication and division problems. So... And a lot of these mental strategies, for me, it relies on that mental image or that visual image of a number line. And you can see it down the bottom there. That's straight from one of the activities that Nagla has developed that's in our 'Mathematical Bridge' newsletter from last month, or in February I think it was. And so I think it's really important to sort of help students to form that kind of visual when they're talking about positive and negative integers.
So, in two- and three-dimensional space, we're looking at that word 'corner'. So, in Early Stage 1, we might use the word 'corner' to describe something in general. We still use the word 'corner' of the room, obviously. We don't say the 'vertex' of a room. But when we're talking about shapes and objects, we're now, from Stage 1, using the word 'vertex' and 'vertices'. And then in Stage 2 it starts to develop that use of the word to do with angles, and then in Stage 3 we start to compare and sort based on that property, and also we talk about apex as a type of vertex. And then they start using it into Stage 4 around inscribed shapes and into other areas of mathematics as well.
In measurement there's a language progression with the word 'compare'. In Early Stage 1 we're looking at direct comparison, and then we start to compare an order in Stage 1 and Stage 2. Note that in Stage 1 only in length do we introduce formal units, so centimetres and metres. In the other areas, they're not introduced to formal units until Stage 2, so it's important to note that. I'll come back to that bit on displacement in a moment. So, we're also asking them, by the time they get to Stage 3, to convert and look at equivalence between mass and litres, so between mass and volume, and volume and cubic centimetres as well. And this is really important into Stage 4 because this is what they start using to find area and volume and use formulas. So, it's important to note that 'displacement', it's not actually a word that appears in our language either in Stage 2 or Stage 3. We do touch on it in Stage 2 as a concept when we start talking about overflow and seeing changes in water level. We then actually teach this concept of displacement in Stage 3, however we don't actually ask them to be able to say the word 'displacement'. But I think it's important that we do, and that students have a firm understanding of this concept. It's not as important as being able to convert between units, but it's definitely an important concept in developing an understanding of why centimetres and cubic centimetres relate to litres. And there's no actual application of displacement into Stage 4, but they probably need to know it in areas such as science. So there's also that bit of important information around volume and capacity, that they're quite constant - say, when we're looking at kilograms and litres - but volume and mass, they don't have that same constant. So, that whole discussion when you talk about, you know, a...a kilogram of feathers compared to a kilogram of bricks, that kind of idea of volume. Students need to have an understanding of that.
So, in statistics and probability, just the word 'information'. That's what we use to talk about data in Early Stage 1, and then it develops through where into Stage 3 we start to introduce the word 'variables'. Now again, a similar thing with that idea of displacement. We don't... We don't have it in our language that students are expected to use the word 'variables' in Stage 3, but they need to be able to define the word 'variable' in the context of statistics when they get to Stage 4. So, we introduce variables in Stage 3 because we start to introduce two-way tables. So, there's two sources of data, so there's two lots of variables that can be measured. So that's an important thing to note as well.
So, some common misconceptions around the language in mathematics. "Metres squared" versus "square metres". Now, it seems like a very unimportant thing, but it really is very important. So, it makes a difference. So if you said a floor was 12 square metres, you're really talking about that picture on the left there. So you can sort of imagine and fit 12 tiles that are a metre by a metre, so the length could be, for example, 2m x 6m. Whereas if you say something is 12 metres squared, what you're saying is that both length sides are 12 metres, and it's the 12 that's the squared, not the metre that's the squared, if that makes sense. So you might want to read over that one again and have a little play with that or chat about it with your stage group. But it's an important thing to note, and, look, I've made the mistake many times in the classroom by saying the wrong thing, but it does make a difference. I guess I didn't mention it in this PowerPoint today, but it's that same idea that 3-D are objects and 2-D are shapes. We don't call everything 'shapes' together, because it confuses students. It's not, I guess, so much as being incorrect, that it actually just gives them not a good understanding of how the two differ. So that's important to note.
Another misconception for students is '-ty' versus 'teen'. And I can hear lots of kindergarten and Year 1 teachers saying 'TEEN' with that 'N' at the end. I know it's a common thing for our students, particularly students from non-English speaking backgrounds as EAL/D learners. And they get confused with something, for example, 'six-teen' and 'six-ty'. So it's important to know that the teens are the only numbers where we state the 'ones' value first, in English. And we also have those numbers like 10, 11 and 12 that don't fit either category. So we need to do things like matching words with a visual image and the word itself to help students grasp that. And most students, actually, can understand numbers way beyond 20 before they understand those teen numbers. That's another misconception.
Some examples for '-teen' now, I couldn't embed the video today. I'm a mother of small children, so 'Peg + Cat' is a show that my children love. If you've not seen this before, head on over to ABC KIDS iview. But all of these images today in my presentation, by the way, are all hyperlinked, so feel free to click on them at any stage to get taken to where they exist. SO, 'Peg + Cat' is a cute little show about Peg, the girl, and her cat, and they go on these mathematical problem-solving adventures. And you can even see just by the title, they've go the infinity sign, the background of it is grid paper with all this mathematical work rubbed out. So it just sort of brings that mathematical language into everyday occurrence. And this one particular episode called 'The Chicken Problem' actually has these teenagers come around the farm, and they have the number on their shirt, and they talk about teen numbers. And they also talk about groups of five and some other mathematical concepts in that show. So please, use it in the classroom. It's fantastic. Even for your kids that are a bit older, they'll probably find it really sweet. And it actually explains some concepts really nicely.
So, following on from that, if you've never been to PBS KIDS, that image is also hyperlinked. The PBS KIDS Lab has some great activities, interactive learning objects, to use. They have a number of them there that relate to the 'Peg + Cat' show. And they also have some that relate to other shows, like 'Curious George' is on there as well. So, they're really nice little learning objects that you can use in your classrooms. So, those images are all hyperlinked that you can access there.
So, another misconception is often one that's thought of by teachers, is that, "Well, lets just give them blanket numbers. That's got to be easier. That makes them language-free." Something like fractions of decimals. However, it doesn't actually mean that it's language-free at all, because the language needs to be used to instruct the students for meaning, and to give them instructions on what to do with the numbers. So although we think it's actually making it easier, it isn't always doing that at all. So, concrete materials are really useful for providing a context for numbers. And just to note that symbols for mathematics are not universal. So commas are often used as decimal markers, and full stops and commas are often used for the space in a number like 1 000, where we still use a space. So, blanket numbers. I can just see in the chat someone's asked about blanket numbers.
What I'm talking about is something like this. So we talk about blanket number sentences if it's just the numbers, no words whatsoever, just numbers and symbols. Now, this is a bit of a more complex blanket number sentence. And often you might see them in NAPLAN. Not as many anymore. Mainly they're word-based problems. But in the past there had been a lot more just those numbers in a sentence. So, we use this in the TOWN program as an assessment, and many of our students answer this question incorrectly. So now's another time for you to access the chat. What do you think most students answer to this question? And there are a number of different answers that they put. Two. How they might just see the '22' and put '2'. 31. Yes, they might just add the 22 and the 9. 11. 51. Yes, they add it all together.
OK, lots of different responses there, and it's quite varied. Lots of children write '51', whether they count them by ones, like that example did, or they just add them all up together, they just don't see that equals sign in there. And you've got that one on the right there where the child knows 9 + 22 = 31, so 20 + 11 is 31. So that child gets it. They understand that they're using that equals for a specific purpose. So, what function of these symbols needs to be taught for students to engage in this learning? You can put it in the chat if you like. What function of these symbols needs to be taught? And I guess these symbols in particular, meaning they're equals, what needs to be taught for the students to engage in this learning? Yes. Beautiful, my lovely students in my class today. It means "the same as". It means equivalence. It means equality. It's like a balance. I need to see that whatever happens on the one side of the equals happens on the other. Ooh, this sounds like algebraic thinking into Stage 4 and 5. It's a "balance arm". Beautiful, lovely answers. So that's what the students have missed. So just because there's no language in that doesn't mean it's language-free, because the symbols are part of the language. I mentioned that in one of the first slides, that the symbols are also part of the language of mathematics.
Another misconception, sometimes by teachers, is that I can use a key word approach to solving problems in maths, just like I might do for English, where I look for the who, what, when, where, how? And so the problem with this is that we need to have some really strong comprehension skills. We can't just use skimming and summarising in maths. You'll miss out on lots of the meanings of smaller words, like prepositions and conjunctions, words that we would normally rush over when we're looking for understanding in English matter most in mathematics. And the order in which the words appear is also important.
So, some of those smaller words that have many different meanings are there on the screen now. So, I won't go through all of them, but it's important to note that 'and' doesn't always mean addition. So it's great to make those lists and those posters that say "other words for plus" or "other words for equals", but they don't always mean it. It depends the way the words are written what that number means... What that word means. So it's just important to show students a vary... a variety, of examples.
Here's an example from NAPLAN from 2009 where non-technical words mean a lot in mathematics. So it says, "Kate has 11 stickers and Lucy has 16." In this case the 'and' has not mathematical meaning. "John has more stickers than Kate, but not as many as Lucy." More, but not as many. That's a really hard concept to understand, even if English is your first language. So, and that leaves a range of answers. More, but not as many. And then it asks, "How many stickers could John have?" So 'could' is a possible. So there's obviously more than one answer, but they're asked to only choose one answer, so it's quite a complex question. So, all those small words that sometimes we would ignore, or the 'and', students might think, "I'll add them together, and if they did add those numbers together, they're going to choose that 'D' response of 17, and we know that's going to be incorrect, OK? So it's important to note that those non-technical words don't always mean what we think they're going to mean, but they're generally very important.
So, some strategies. Maths journals. Great to use in your classroom. And even just as a whole class. You don't have to get all your students to write a journal. As a whole class it's a really good idea. Plan it for every day. If you get to it three times a week, that's great. Because we want to get the students past that oral description of something, like - Oh, I timesed it by five and I kept getting none or five." Changing that to - "When you multiply any whole number by five the answer will always end in five or zero." That's a very complex sentence, but that's the kind of oral response we're looking for by the time students are moving into Stage 2 and 3.
Another strategy or scaffold is the Frayer Model. Now, this is an English model that we use in mathematics. So, it looks at having a definition, characteristics or other words for it, examples and non-examples. Non-examples are really important, so kids know what it is and also what it isn't. So, making these posters with students around language is really a great idea in your classroom for mathematics.
Another scaffold to support you is using Newman's prompts, So they're the Newman's questions. That little picture at the top is a link to the website. if you've not heard of Newman's or want some more information on Newman's, it's in there. So, there's five different areas your students could be having trouble with. So, reading. So maybe it's just a pure barking at the print, so actually saying the words. Comprehension - what do I have to do? Transformation - turning the words into numbers, or number sentences. The process skills. What am I actually going to do? Do I have the strategies? And the encoding - where do I have to answer the question? Have I answered the question when I write my number on the other side, OK? What is it they were actually asking me to do in the first place? And most of our students have their trouble within that comprehension and transformation stage.
So, learner diversity, so, looking at that EAL/D student learners again, we need to be aware that they are simultaneously learning a new language as well as the understandings of the mathematics syllabus. They need time and support that explicitly addresses their needs, and assessments that account for their developing language. I think that's really important, and all teachers need to be aware of that for all students, but particularly for students that don't have English as their first language or dialect.
There's also some mention around assumed cultural knowledge, and this comes from another document from ACARA where they talk about... These are called annotated content descriptions. So, it doesn't have our outcomes on it, but obviously the descriptors are in our syllabus, so you can definitely find where they've come from. And they have great information about language and cultural considerations, and even some teaching strategies. Some of the important things that I noted out of that was that not all currencies use decimal systems. The calendar that we use in the Western world is not the only calendar that's used. Telling time is constructed differently in different languages, and people follow different versions of time, in the sense that some people are really particular about being on time, and some people, they just see it as more of a general observation. And temperature is not universally measured by Celsius. So, they're just a few small points, but that document is an excellent document. The image up there links straight to where you can download that document from as well.
Just some final comments. You need to provide time in class for discussions around language. So, talking about language is metalanguage. You need to model the language the students want you to use. You should include students in the creation of class posters to provide a print-rich and image-rich environment. And we need to explore word meanings and definitions from their perspective as well, having them create some of those as well, not just from what we tell them it should be.
Just to let you know as well, I think I mentioned last time that the 7-10 Syllabus PLUS are now up on a website. Ours are too. The image there is hyperlinked straight to where all the recordings are. They're all on that page, obviously bar today's one, because I haven't put it up there yet if you're watching this live, but I'll do that in the next couple of days so you can go there to see the ones you've missed. Remember that if you want to register these in the sense of getting MyPL self-identified hours, you can schedule these events yourself at a school using the course code. So, that's exciting. Yay. We now have them on a website somewhere.
Just to let you know that Adobe Connect session Series 3, I'll be starting that in term two, week three. Now, I know that's NAPLAN week. I'm really sorry. But it was the date I had. So if you don't watch it live, it's OK, you can watch the recorded. But it will be Tuesday, 13 May, will be the beginning of Syllabus PLUS for next term as well.
Again, they're my contact details. We are trying to get our 'Mathematical Bridge' newsletter out this week, fingers crossed. If not, it might come out during the holidays. And because you're already a part of my list, you'll get that sent to you as well. So, thank you so much for coming along today. I'm just going to flick to the conclusion template there. And there are lots of files today in the file pod for you to download to help you in your classroom around language. I hope that was clear today and that I've given you some suggestions for how to deal with language in your classroom around the new syllabus, and around the different diversity of learners you have in your classroom as well for mathematics. So, thank you. If you've got any questions, please hang around in