4 September 2025

Professor Caroline Cohrssen and Dr. Danielle Harris share their insights into enhancing mathematical understanding in early childhood and secondary education, emphasising the importance of purposeful communication and responsive engagement in early years to foster mathematical thinking beyond basic counting and shapes and visualisation strategies in secondary education to encourage students to articulate their problem-solving skills through developing their spatial skills. They also highlight the role of family involvement and the significance of recognising mathematical play in everyday activities and stress the need for a continuum in education, bridging early childhood and formal schooling, to build confidence and capabilities in mathematics.

Show Notes

• Numeracy Summit 2025 presentations
• Caroline Cohrssen - Purposeful talk to support early mathematical learning
• Danielle Harris - The power of visualisation in secondary mathematics

Transcript

Dale Atkinson: 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're talking numeracy on the back of the 2025 Numeracy Summit. We've got Professor Caroline Cohrssen and Dr. Danielle Harris. Professor Cohrssen is the professor and deputy head of the School of Education at the University of New England, and Danielle is a post-doctoral fellow at the STEM Education Research Center at the University of Canberra.

Thank you to both of you for joining us. Today we're talking about the development of conceptual mathematical understanding that supports the strategy for public education. And we're talking to Caroline because she gave a presentation called 'Purposeful talk to support early mathematical thinking', which targets preschool leaders and educators. Caroline, what is mathematics when it comes to those within the early years? What are we trying to go after?

Prof. Caroline Cohrssen: It's not always entirely clear to teachers and educators. The early years learning framework, and I'm going to use their definition, defines mathematics and numeracy as broadly including understandings about numbers, patterns, measurement, time, spatial awareness and chance and data, as well as mathematical thinking, reasoning and counting. So that's quite a lot, and we often find that early childhood teachers will be focusing primarily on numbering counting and shape identification. And there's actually a lot more to it that we can be doing than just those sorts of two strands.

Dale Atkinson: Now, when you talk about mathematical learning through purposeful talk, what does that mean? When the young people are actually learning how to talk themselves, how do we differentiate or how do we combine those two concepts?

Prof. Caroline Cohrssen: First of all, by the time they're in an early childhood classroom, typically they're quite articulate themselves. Rather than learning to talk, to be honest, but what we are talking about there in terms of intentionality on the part of the teacher or the educator. Would be being really purposeful in terms of what and how we communicate with children. So the term that we would use in a teaching environment would be responsive engagement.

So, we are wanting to be listening and watching and being attuned to the child, and then responding in a contingent way that consolidates what children know and extends their thinking a little bit. Part of that process is also being attuned to what children, sort of the skills and capabilities that they bring with them, which are obviously gonna be emphasised through the teacher talk, but reflecting in the child, reflecting in what they're bringing from the home environment. So, it's a wraparound focus. Which really demonstrates why it's really important to be working with families when you're teaching very young children.

Dale Atkinson: That seems to be a consistent theme, particularly well across all education, but particularly in those early years where the family dynamic is so important. You talk about being able to recognise when children are engaging in mathematical play, what does mathematical play look like in the early years?

Prof. Caroline Cohrssen: I would probably say that mathematical play is basically any kind of play, and that what makes it mathematical is that the lens that the teacher's bringing to it. So, if the teacher's looking to see children spontaneously organising or categorising things or putting rocks in a row according to their shape, or I'm setting the table in a play corner with a cup and a plate and a spoon at each seat. All of these types of behaviours that children are doing spontaneously, if we bring a mathematical lens to it, we may be seeing children categorising objects according to size or shape, or colour. You know, it depends on what we are looking for.

Dale Atkinson: What's the mindset that the educator should be using when working with young people and thinking about the mathematical concepts?

Prof. Caroline Cohrssen: You know, we talk in education about pedagogical content knowledge, and basically what that's talking about is knowing the subject that you're teaching and knowing how to teach it. Anticipating what may be difficult for children to understand, and so pre-empting that by providing them supports beforehand or alongside. I guess, but also understanding the longitudinal and latitudinal connections. So how does mathematical thinking within a play-based context align with other types of thinking children are doing?

If children are demonstrating a particular mathematical capability in the moment, what can we assume precedes that and what is likely to follow from it? And so a lot of the work that I do is focusing on learning progressions or learning trajectories, call them what you will, and this is also direction that South Australia is going as well with early childhood, is looking at those progressions in children's learning.

Dale Atkinson: What does a quality interaction look like between an educator and a young person?

Prof. Caroline Cohrssen: First of all, I would say quality interaction is one that is warm and responsive to the child. I think that's the threshold condition that we need, and that again, probably applies across any phase of the education process. I think a high-quality interaction also is one that, as I said earlier, is attuned to what the child is demonstrating in some way that they know and are interested in. It's one that provides feedback to the child, so it's going to be affirming what the child is demonstrating they know, and probably giving them that bit of stretch, which we might call scaffolding.

There's gonna be back and forth exchange between the teacher and the child with that contingent responding and elaboration of the ideas that children are sharing. An important part of this interaction involves teachers asking children to explain their thinking, because we are not just wanting to tap into what the child can do. We're interested in their thinking that got them to that point, that they may be describing or showing us. And I think also one of the things that's important to remember, particularly with very young children is that their responses may not be verbal. They may be pointing or showing, or you know from my own research, dancing or climbing with very young children, there's gonna be a lot of that embodied demonstration of understanding.

And I say very young children, I also mean their children that are learning to speak English. So, children who are speaking English as an additional language or dialect may not have the vocab to explain their thinking, but that doesn't mean that they don't have the conceptual understanding. So, part of a high-quality interaction between an educator and a child is one that is going to demonstrate that we can kind of look through the language to see what the child is demonstrating they understand, and then perhaps responding in a way that models the language that goes with that behaviour or that demonstration of thinking.

Dale Atkinson: So, you've gotta be so deeply observant and responsive in that setting. Now you also talk a little bit, and in your presentation, links are available in the show notes, about a purposeful pause. What does that look like in a play-based environment? And how does it support mathematical thinking?

Prof. Caroline Cohrssen: Purposeful pauses are based on the notion of wait time that has been researched quite a lot back in the seventies and more recently.

Basically, what it does is create space in an interaction, in a dialogue between an educator and a child, a teacher and a child. That provides an opportunity for the child to be thinking. Provides an opportunity for the teacher to be thinking so that the response that the teacher is providing to the child is perhaps more evidence-based in that it's attuned to what the child has demonstrated but also creates a space for the child to think and respond without there being pressure. So, it's just slowing the pace of the interaction down a little bit so that it doesn't feel like some kind of an interrogation. It's a more responsive and attuned way of engaging with the child, that takes the pressure off a little bit.

Dale Atkinson: I think taking the pressure off, particularly when it comes to mathematics, is such an important thing to do. We've had podcasts previously where we've spoken about math anxiety and the need to overcome that. I might bring Danielle Harris in at this point. The point at which I think math anxiety often kicks in is in the secondary years.

Danielle, you gave a presentation around visualisation strategies for students and spatial skills and how they can improve mathematical performance.

What are visualisation strategies, and how do they work in this context?

Dr. Danielle Harris: Visualisation, I believe, and a lot of the literature says is quite broad about the processes that we use when we're thinking mathematically, how do we visualise, how do we gesture, how do we draw, how do we represent what we're thinking? But it's also the product. You know, how do we write out our answers? Are we just putting in a number? Are we putting in a form or are we able to demonstrate a more broad kind of structural understanding of what we're working through? Visualisation is very, very big. It's part of everything that we do, and it offers some really unique opportunities to think about the processes in mathematics.

Dale Atkinson: So what are some of the practical ways that schools can embed some of those visualisation strategies in, in young people?

Dr. Danielle Harris: A big focus, I think as we move towards secondary school is, let's get to the answer, let's get to a solution. I love that we talk a lot more these days about show you're working out and that's part of the scoring. Because, you know, actually being able to demonstrate your thinking (illegible). I think the more we can, even these ideas that Professor Cohrssen is talking about reflective processes with students, what are you thinking, getting them to share their ideas. How are other people solving problems? How are they visualising, how are they representing on a paper or on a whiteboard?

So the more that schools and teachers can vote these kinds of processes and not just go straight to the solutions. And you know, we live in a world where there's so much available online to us and so many digital resources, but can we dial back a little bit and go back to the drawing on paper? We have a really great example of isometric drawing we did with year sevens and eights, and it's a really challenging task. But once the students were asked these ideas, they could actually start to understand the structure of shapes and the relations between shapes.

Dale Atkinson: Now, Professor Cohrssen, it strikes me that some of what Danielle speaking about there, and trying to activate within the kids are things that inherently they have within them when they're engaging in mathematical concepts in the early years, would that be fair to say?

Prof. Caroline Cohrssen: It definitely would. One of the things that struck me while Danielle was speaking as well, was that there's recent research that's come out of France that is showing that the gap between boys’ and girls’ achievement in math is starting as early as the first year of school. So that, that is really interesting to me because children are born with the same capacity for mathematical thinking. So what is it that's happening in the school space that we are seeing that shift coming so early? What can we do to prevent that or pre-empt that from happening? Think it is a really interesting question and that notion of visualisation as well, we know that spatial thinking is something that is malleable.

The metaphor I use when I'm working with early childhood teachers like Play-Doh, we can squash it and shape it in different ways. So, what does that mean then for us as early childhood teachers in terms of the way that we support children's mathematical thinking in their spatial skills, to equip them with this sort of broad capability that will support things like understanding how many there are just by looking at, at a set in it. The notion of quantity, the, you know, all types of mathematical thinking. What do we need to be doing better in the early years of school to prevent that gap emerging so early, and to support more children progressing sort of with confidence through the school system.

Dale Atkinson: Yeah, I couldn't agree more with that intent, particularly around encouraging people leaning in to mathematics and the French study, which I've spoken about with our Chief Executive, had a look at that around particularly gendered language and the impact that has with educators is something we absolutely are just starting to understand and need to be really very, very conscious of and perhaps do a bit more work around. Danielle, back to you. What resources have you found most effective in developing students' spatial reasoning.

Dr. Danielle Harris: So, I think there's two strains there. There's the physical resources, you know, we, even in high schools, would bring back in the link blocks and the fraction bars, and drawing and getting students out walking around their school environment. So there's physical resources that can be embedded quite easily into the practices that are already happening. But there's also just the recognising that spatial skills are so critical for the development of mathematical knowledge and a recent meta-analysis suggests that the further along students go in their schooling, the more the spatial skills are important, and the more that developing these spatial skills have an impact on their mathematics. They're starting to recognise that, you know, spatial skills are developed in our highly spatial world, as my colleague likes to say, and then, we live in this world, so why can't we bring mathematics out into the world?

But everything we do when we're talking about resources starts by grounding the, learning progression in an experience that the students are very familiar with. So if we're gonna be talking about grids and maps, what maps are the students already using? What awareness do they have of their own environment that we can bring into the mathematics classroom? Even when it comes to number things like equivalence, when you've got an equal site, it's not the end of the operation, it's just part of the calculation. Both sides are the same. What can we substitute numbers with? So critically just bringing the real world into the maths classroom is a really good way to start.

Dale Atkinson: So I might just throw to one final question, which is for both of you, Danielle, what can secondary teachers learn from those in the early years? And Caroline, I'll ask you in reverse, but Danielle, what do you think secondary teachers could learn from educators in the early years?

Dr. Danielle Harris: If there's anything I can take apart from what I've seen with the students, it's just promoting that curiosity and that flexibility that we saw children have in the early years. So same kind of reflected practices, if students are willing to talk about how they solved a problem, or how they are visualising a problem, or where they're stuck on a problem, bring in the purposeful pauses. Give them a chance to actually articulate their thinking and communicate what's going on in their minds. And then maybe we can pick up those little threads and follow it through.

Prof. Caroline Cohrssen: It's really interesting, Dr. Cath Pearn and I analysed some children's drawings that, that were done in the kindergarten phase and found that there were drawings of the route from home to school. It was part of a spatial thinking project, and we found that children, some children in a group were demonstrating capabilities that were up to, I think year three or year four of the Australian curriculum, and they were demonstrating this already when they were four and five years old. So I guess for me, learning progressions is crucial. We need to understand where children are at, where they've come from, and so we know what to do next. But also the idea, I think, and this probably comes from early childhood teachers themselves who may not feel comfortable with teaching mathematics or mathematical ideas.

Children need to understand as well that maths language and maths processes are practical and fun, and relevant. So it's not that, that's the maths table where we do the, the counting and the shapes but we use mathematical thinking, a flexible way to solve everyday problems. And I think when it becomes just part of what we do and part of the way we think, and part of the vocab that we use, there is no special language because it's all happening on a continuum. And what we are doing in the prior to school space is part of what Danielle and her colleagues are doing further along in the education process. We have a paper coming out probably this week looking at engineering as part of STEM and how similar to mathematics engineering could be more visible, so you've got a clearer line of sight.

And I think that's the reason I'm saying this is really for that point. We need this line of sight so that we don't see early childhood, and then school as separate things, but they're both part of a continuum. And I think clearer articulation between the prior to school space, and the formal school education space will probably go a long way to strengthen children's confidence and capabilities.

Dr. Danielle Harris: The spatial skills offer this really great opportunity through the research that we've seen to show strengths and it's very non-verbal, so they're really great pathways across the whole education continuum, I think to give children access and some of our best examples from kids that go, this isn't maths, this is fun. And so they don't realise that while we're developing these critical skills and the teachers are working with them and building this passion, they're actually still doing mathematics.

Prof. Caroline Cohrssen: I loved what you said there Danielle, because one of the things we speak about quite a lot in early childhood as well is recognising that children have a linguistic repertoire that they bring with them, which may not be English, but it doesn't mean that they lack the understanding or the capability and so part of our role as teachers is to be supporting - recognising, and supporting children's strengths. And I think that probably applies as well, right the way through the education system where you may have children who speak English as an additional language and just providing the sort of parallel talk or the narration almost for what children are doing or showing us, regardless of how old they are, I think is an important part of what we do.

Dale Atkinson: I think we might leave it there with a challenge from Professor Caroline Cohrssen. Thank you very much for your time, and Dr. Danielle Harris, thank you for yours.

Prof. Caroline Cohrssen: Thank you.

Dale Atkinson: Thank you.