7 May 2024

Discover what learning trajectories are and how they can support you to help students develop their mathematics abilities. Joining us this episode from the University of Denver are Doug Clements, Professor of Early Childhood Learning and Julie Sarama, Professor of Innovative Technologies. They were the keynote speakers at this year’s Numeracy Summit in Adelaide.

Show Notes

• 2024 Numeracy Summit plink presentations
• Learning trajectories

Transcript

Dale Atkinson: [00:00:00] Hello and welcome to Teach, a podcast about teaching and learning in South Australia. My name is Dale Atkinson from South Australia's Department for Education. And today we are at the Adelaide Convention Centre where we've just wrapped up the Numeracy Summit. And I'm joined by two of our keynote speakers: Doug Clements who's the university professor at the Kennedy Endowed Chair in Early Childhood Learning, and Executive Director of Marsico Institute for Early Learning, University of Denver. That is a mouthful, Doug. That's a lot. And I'm also joined by Julia Sarama, who is the Kennedy Endowed Chair in Innovative Learning Technologies and a professor at the University of Denver. Thank you very much for joining us too.

First of all, you've been here to talk to us about learning and teaching with learning trajectories. Can you tell us a little bit about what learning trajectories are?

Julie Sarama: Wow, that's a big question. My daughter who works in communication, our daughter who works in communication, told us it's a horrible name because it's not that difficult a concept, but it's a pretty complicated name. But pretty [00:01:00] much what it is, is a three part thing.

And one part is a sequence of goals, like a curriculum, like you have here. And then, so we have a goal, a learning goal about math that we want to keep in our mind. And then we have an understanding of children's development that will help kids meet that goal, that kids will go through to meet that goal. And then we have the third part, which is all the kinds of interactions, instruction, activities, environmental things that'll help move kids from one level to another. That's the easy way.

Doug can probably give a little more background as to what those levels are in terms of, it's not just saying kids can add two-digit numbers and then add three digit numbers. It's not that kind of thing. It's a little bit more complex than that.

Doug Clements: So, what the levels are, are qualitatively different ways of thinking about a problem [00:02:00] and about how you would address that problem.

So, yeah, it includes incremental growth, such as two-to-three-digit number addition and stuff. Or for younger kids, you know, count to 5 before you count to 6. But the main thing is the conception, procedures, strategies, and the like that constitute a level of thinking so kids will learn numbers 1 to 5 in a very different way than they learn numbers from 6 to 10.

Their understanding of it, they're often called the "intuitive numbers", you know, because fingers in one hand and experiences like that, they're often very intuitive. Kids can do more with those numbers far before they can do larger numbers. And it's not just that the numbers are larger, but a different way of thinking about it.

So we try to capture those kind of qualitative differences in something that [00:03:00] Australian researchers call "growth points", and those are broad levels. We often have levels in-between those growth points because we're searching for those that are most instructionally relevant to teachers.

Kids are at this level, where do I go next? That's the developmental progression understanding. And then, how do I get there? That's the teaching strategies that Julie explained.

Dale Atkinson: How do educators access this and what's the step change in their teaching of mathematics that we're looking for if they engage with this as a way of approaching the kids?

Julie Sarama: Well, I'd say the first thing is we really try to turn what is typical mathematics teaching on its head. The way I was taught to teach math was to think about what your goal is, turn it into an objective, and then come up with a good lesson that helped kids meet that objective. Not thinking very much about who the children were in front of me, but really thinking about what was the content, right?

So, [00:04:00] taking an example say from kindergarten. If I was trying to teach kids to add numbers like 10 + 3, 10 + 4, 10 + 5, what I'm doing is thinking: 'How am I going to go in and make this relevant to this group of kids?' I know nothing about those group of kids. I'm 100% focused on the content and how I'm going to break down this content, maybe model it, maybe do some fun activities or whatever to teach that content.

We're doing kind of the opposite. We're starting with where the children are. So we have a goal, okay yeah, I want kids to learn addition. But the next step is figuring out what the children know and use my understanding of children's development to know what do I do next. And so I'm really going to differentiate my instruction based on what my children do.

And that's just such a flip. And so when we hear teachers say, "Alright, I tried that, you know. I tried doing 10 + 4 and they just don't get it", or, "I have my stars who get it, [00:05:00] but I have some kids who don't get it. What do I do?" My very next question is always, what do they get? What do they understand? What are they doing?

Well, they're still counting on their fingers. Okay, what are they doing when they're counting on their fingers? What is their idea of addition? Because there's no child that knows nothing. There's always some strength that they're bringing to the table that we can use to build from.

And although I taught, I thought successfully, for a long time doing what I said in the beginning, I've been saying for a while now that it's like giving directions without finding out where the person is, right? It's like somebody calling you up and saying, "I'm a little lost. I'm trying to get to your university. Can you give me directions?"

And me saying, "okay, get to my house first and then follow me and I'll take you to the university". I mean, the very first question you say is: 'where are you', right? 'Where are you?' And then you try to figure out how to get them there. And that's our approach. And there's lots of ways for teachers to access information on it.

Doug Clements: That's so good. And to just randomly think [00:06:00] about the 10 + 3, 10 + 4, what do kids know about those numbers that are going to be the sum? So, many kids have no idea that 13 means 10 + 3. Our language, and most Romance languages, hide the fact with words like 15, and 11, and 12, and, and the like.

So you might first say when they count those, what do they know about those and everything? Let's see, let's talk to them about it. So, 11, what does that mean? You know, and this kind of stuff can build that understanding as opposed to saying, "let's do a worksheet. I'll put 10 + 3, 10 + 4, 10 + 5 on the worksheet and have them fill in those answers".

If you do that, actually, kids will get all the answers right because they'll copy the 1 and copy the 3 from the written problems with no understanding of what that means quantitatively and no way around the difficult [00:07:00] language that we have. That's 11 and 12, 13, et cetera.

Julie Sarama: That actually made me think also of if kids do the worksheet say that's, again, a standard way. 'I'm going to do a worksheet. Maybe it's not the best way to teach that'.

But you're like, 'Okay, now they're going to practice it. They're going to have a bunch of these to do'. For the children who get them all right, get the gold star, that's it. They're just like, 'Great. They know it. I'm happy'. And again, with the learning trajectories approach, when I say, "what do they know?", it's also going to be, "they know this plus more".

So we're going to meet them where they are and challenge them. So it's differentiation at both ends. We're not only focused on children who might be struggling with a concept, we're very much trying to include kids who can sometimes sit in math class for a year, honestly, all of kindergarten, maybe all of first grade, and really never be learning anything new, and feel pretty good because they're always getting that little gold star.

Doug Clements: It reminds me of the true story of an interview of a first-grade kid who went up to his teacher in England with a paper that had something like: 10 + 3, 10 + [00:08:00] 4, 10 + 7, repeated and repeated and he did 3 and he turned it in. She said, "What about the rest of them?" And he just looked at her and said, "How many do I have to do to show you that I understand this?"

But normally we'd have the kids finish that paper regardless of whether they understood it before they started the paper, and then go to the back room and play or something. No future challenge. And like Julie said, the research is very clear that kindergarten in the United States, and I would wager in Australia and many other countries, is a place where we teach kids what they already know. And, unfortunately, our view of teaching is, of mathematics especially, is often, "well, it's good practice for them".

Is it really, you know? Or is it just a lack of new challenges that'll keep kids both interested in mathematics this year and the next. And, you know, our responsibility to make sure those kids have something interesting to learn.

Dale Atkinson: I think there's a, [00:09:00] perhaps a degree of comfort and ease in handing the worksheet over for an educator. It's not a simple thing to do. But it is, it seems like the worksheet as an approach asks less of the educator in terms of their confidence to do the thing. Can you talk a little bit about how we encourage educators to take that step and move into the space that you're describing?

Julie Sarama: I try to encourage teachers to engage as professionals in trying to take a scientific approach truly, right? In trying to understand where their children are developmentally and appeal to their sense of professionalism. When we conduct professional development, we talk about being in a safe space. We understand that many people who are teaching young children may not have grown up with a really good math attitude themselves.

They might not feel very comfortable with math. And so we say, 'we're going to make mistakes, we're going to try things out, we're going to do it in front of each other, and it's just going to be good'.

And I kind of say, "look, if you're going to go and learn how to drive a car, and you went to a classroom [00:10:00] only, and you sat quietly, and then you got behind the car, it would be bad, right?" so, "you're going to drive the car on the road by yourself" is the same thing as, "you're going to be in the classroom with the kids".

This is your chance to work with an instructor and "try driving" where we "still have brakes on the side" and we can help you if you make a mistake. So we have to engage and we have to try.

And so a lot of our professional development is very much hands on. It's not teachers sitting and listening. It's them trying things out, and, in role playing and pretending. There are different levels of thinking. You know, we play charades. What if you can, you know, everyone knows a level of thinking and teachers can ask them questions and figure out what level of thinking they are to really learn it.

But I guess that the other part is rather than teachers coming to something and saying, "Look, I've done this math teaching this way for a long time and I don't want to give it up. This is what feels very comfortable for me. And almost a sense of loss over being told what I've done for a really long time, all of a sudden you're telling me, isn't good enough. [00:11:00] You know, I've always played the dot game that kids do for six months and they really aren't learning anything, but it's a fun game and they like it".

Rather than saying, 'You know, you have to give that up', we try to say, 'Keep focusing on the kids', right? So our questions are always with teachers. What are your children thinking? What level do you see your children at? Only you know. You're the teacher. You are the professional. You have the power to differentiate activities. No one can tell you. A calendar can't tell you how high to count. The number of children can't tell you what your, you know, for attendance can't decide all your arithmetic problems. You're the instructor. You can make those decisions and try to empower them.

So it's about really urging them to be scientists, to be professionals, to make those decisions. Because, in the end, we all want children to learn. I would say that not one teacher we've worked with doesn't care about children learning math. They really do want to learn. They just don't want to quite give up what they're comfortable with, like you said.

Dale Atkinson: So how can teachers access and start using these learning trajectories within their own work?

Doug Clements: We have a [00:12:00] website called 'learningtrajectories.org' the name of it is: 'Learning & Teaching with Learning Trajectories'. But that's the address.

It's based on about a quarter century of work in developing the learning trajectories, validating the learning trajectories, and then making what Julie just said about professional development work for teachers and giving them a resource. For instance, lays out learning trajectories for counting or the like, and teachers can hear about the goal which is often more than they think. Teachers of very young children will often say, "I teach them rote counting. That's the most important thing. They count to five and then they count to 10".

And we try to expand that saying, "first of all, even for verbal counting, we don't call it 'rote counting'", we're not arguing with the teachers, "use our phrase".

But we are trying to make them understand that, for instance, when kids learn to just count to [00:13:00] 6 or 10 or something, they're just verbally counting, sure, but they know that 9 is a big number compared to 2. For instance, they might be able to recite the whole alphabet, but they don't think that "G" is really big and "B" is very small.

So they understand something about sizes or quantity is increasing. That's a relative term, a relative understanding of counting that's mathematically coherent and interesting. When they start counting, and maybe you want to tell the story real quick of the child counting and getting stuck at "39" for a second.

Julie Sarama: Right, it was just that a child goes: "39..." and then says, "What comes after 3? Oh, I know, 4". And then says, "Um... Okay, 40".

And so that kind of was just a good illustration of a child who understands that "1, 2, 3, [00:14:00] 4" is directly mapped onto "10, 20, 30, 40" and so she was into 39, which means that she knew what came after 3, which was 4, she got 40, which told her what happened next. Which is really cool and show that it's not rote.

Doug Clements: Patterning, in the numerical sense, rather than just "red, red, blue, red, red, blue", kind of patterning, sequential patterning. And it's also structure. She understands mapping the structure of single numbers onto the structure of decades.

Julie Sarama: Right.

Doug Clements: It's amazing. And that's coming from kids who are, again, four or five years of age, compared with kids who are taught more like it is a rote process. I don't know research in Australia on this, but I do know in the United States, at least a little while ago when this research was conducted, it wasn't till fourth grade that the majority of kids knew that 50 had something to do with 5. Fourth grade.

Because if your mindset is: 'this is rote', the way you're teaching it [00:15:00] and the way you lack discussions, math talks with kids, can convince the kids that it's just memorization all over again. And they miss that, the beauty of the structure and pattern.

Julie Sarama: So the 'Learning Trajectories' website usually has a video that will talk something like Doug did about how important counting is. And then it'll have all the levels of kids counting from, hopefully, birth through third grade. And videos of what you can kind of look for and a good written description. And honestly, the good written description, sometimes when I try this out with groups of teachers, the written description can be more helpful for people because it really puts the different things, the different parts of the understanding there.

And then below that are instruction activities, kinds of things, your routines you can add to your day that will help children achieve that level. So that's all there on the website. We have several topics. We have lots of resources for parents, teachers, people who [00:16:00] are engaging with the website in a sort of a professional learning community because it can be overwhelming.

Sometimes I say it's like a recipe website where it's like everything is on there and you look at it and you could be like, 'I have no idea what to make for dinner'. But if you can go in there and say, 'Alright, I have chicken. Give me some chicken recipes’. And then you're like, 'I still don't know', you need a way in.

And sometimes professional learning communities can give you that way in. We can look at what we're planning to do. We're just like, 'okay, the next couple of weeks we're going to be working on, keep going with the same, we'll say geometry. And I think we're going to be learning about shapes. Let's look at about the levels that we think our kids are going to be. Do a little bit of reading and watching those videos and seeing if any of those activities make sense'.

Dale Atkinson: Well, if you're interested in what you've heard here today, speaking with Professor Clements and Professor Sarama, the information on the learning activities, how to engage, support and extend each learner on the 'Learning & Teaching' and Learning Trajectories tool, the website link is below in the show notes.

Also, you can access the [00:17:00] Numeracy Summit presentations on plink, where you can see Professor Clements and Professor Sarama giving their presentation. Doug, Julie, thank you very much for your time.

Julie Sarama: You're welcome.

Doug Clements: Thank you.