Michelle Tregoning

Hello everybody and welcome to our latest In conversation with Dr Aylie Davidson. We are joined in this conversation with a whole range of our primary mathematics specialist teachers as well as Aylie and the Mathematics Professional Learning team. We are all over this fantastic state of NSW and I'm joining from Tharawal Land and in welcoming everybody here today, we'd like to acknowledge the traditional custodians of the land and pay our respects to elders past, present and emerging.

Aylie, we're so excited to get a chance to talk with you today. I think one of the reasons that your research really resonates so strongly with so many teachers, Aylie, is because you yourself are an experienced early childhood and primary teacher before moving into research. And we're pretty convinced that experience in being a teacher led to this interest in planning for mathematics, particularly in primary schools and in ways that put students at the centre of our work. And so that it's the students that are at the core of the decision making when we're planning and programming and the students that are with us here and now today who are in our care.

So, thank you so much for being here and that sort of introduction really leads us into our first question, which is in your research work, you talk about student centred planning, and we wondered what does it mean to you when you say that phrase 'student centred planning' and why do you think that distinction matters?

Aylie Davidson

Thanks so much that introduction and thanks for kicking off with such a meaty question. I might go back to some of my experiences. Firstly, as a student of mathematics in primary and secondary school some of those experiences made me love maths and some of them nearly turned me right off it. And thinking about when I went into teaching, particularly after my first year of teaching and getting a little bit of a realistic feel for what it is to be a teacher of mathematics, becoming more aware of I don't want the way I was taught at school necessarily to influence the way I'm currently teaching.

So, I did want to take my positive experiences with mathematics, but being clear of what was it that raised those and induced those negative feelings of mathematics and trying to raise that to my awareness that I'm not going to repeat those actions and approaches with my students.

And so that kind of led me on this journey to becoming a team leader and then a leader of numeracy and then a lead teacher in a primary school, and then ultimately into doing my PhD on teacher planning.

And it was always because I, as a primary school teacher wanted the last few years I was teaching Year 6. And it was really important that those Year 6 students left primary school believing that they could learn mathematics.

And so, when I'm talking about student centred approaches, we look at some of those UNESCO goals, those international goals for education. They develop student agency and equity and inclusion as 2 key goals for education. And so, if we're thinking about agency in terms of mathematics, yes, there's a lot of explicit planning and decision-making being done behind the scenes, but we need to get students to feel ownership over their learning, particularly for mathematics. And in terms of equity, inclusion, we want everybody to finish school feeling included, part of society in the world. But maths, unfortunately, has the reputation of doing the opposite. I'm not smart, I can't do these things, I don't have a maths brain. And so, maths, similar to science, has the reputation of being a bit of an elite sport and so we can overcome these challenges for our teaching, but it requires us to plan and teach in certain ways.

And as part of my work on student centred approaches to mathematics, there are some nice characteristics that are apparent in the literature. When we are taking a student-centred approach to mathematics, we're considering ways that the tasks we choose and the way we enact those tasks in the classroom. Mathematics is a very active process. We don't learn mathematics in a silo, and so thinking about our classroom cultures, we need to provide opportunities for students to cooperate, communicate and negotiate their thinking and learn to listen to one another and learn from each other's explanation.

And so, for me, that is a bit of a foundation of some of the values that underpin the work that I do in terms of mathematics planning.

Michelle

Thank you so much Aylie. We've read in your work, and we've heard you talk before, Aylie, about effective student-centred planning includes looking at the curriculum and syllabus documents alongside research. And we wondered if you could just share with our colleagues across NSW what you found in your research that shows why this approach matters.

Aylie

Now, the curriculum is a great starting point, and it makes sense to start with the curriculum to gauge if I'm teaching grade 4, what does it say to teach fractions in grade 4, what does it say before, and what does it say in grade 5? It's a really helpful starting point.

Now what we want to avoid is for teachers seeing the curriculum as a checklist that we just cover the curriculum. You know Marilyn Burns, she's an international educator. She talks about this, the teacher's role to 'uncover' the curriculum with the students, not just tick it off and cover it. So, one of the challenges we face in teaching is trying to identify the scope of each individual curriculum descriptor and continuing to build our knowledge of the mathematics as well as the pedagogies. If we're only relying on the curriculum and not drawing on other parts of research, we are at risk of not exploring concepts in depth with our students.

So yes, we do want to start with the curriculum because we have an obligation as teachers to, we all have those accountability measures that we need to meet. So, finding ways as part of our planning to match the curriculum together with what the literature says and the mathematic in terms of mathematics and how students learn mathematics, it's important to find those ways to bring those 2 together.

So, for example, you know, it might be in Year 1, the curriculum descriptor talks about connecting 2D shapes and 3D objects. And for some teachers, some typical particular tasks spring to mind.

[On the left side of the screen, the Stage 1 NSW Mathematics K-10 syllabus content information for three-dimensional spatial structure B appears.]

Aylie

And so, we might do those tasks to address that curriculum descriptor if we engage in some reading. So often in initial teacher education courses, we use texts like Helping Children Learn Mathematics, and they're excellent text to use. Or you know, Primary, the Van der Walle textbook, which is the Australian version.

[On the left side of the screen, 2 images of the textbooks front cover appear – these 2 textbooks include 'Helping children learn mathematics' and 'Primary and middle years mathematics – teaching developmentally'.]

Aylie

Now if we have opportunities to think about the ideas in those texts and think about how it connects to the curriculum descriptor, well we'll notice that actually underpinning that one curriculum descriptor of the 2D and 3D shapes in Year 1, there are 3 really big important ideas there. The idea of properties of shape, visualisation and transformation.

But if those words don't appear in the curriculum descriptor, then again, we might miss, you know, exploring these important ideas with our students.

Michelle

Thanks so much, Aylie. So, you're making me think about how that idea of looking both the curriculum and the syllabus alongside research, that it brings a greater dimensionality to understanding of what statements mean when they sit inside a syllabus and a curriculum, and the depth of understanding that is required for students in order to be able to work with that outcome or that achievement standard of the syllabus probably also helps us keep up to date. And the last and one of the most important is that it provides a real opportunity to build the mathematical knowledge of teaching, which as we know from research, then impacts the confidence of our teachers. And that also has a direct responsibility for student learning as well. So, thank you so much for sharing some thoughts around that.

We wondered if you might have or know of a model of framework that we can use to help us in planning for learning. And we ask that sort of tongue in cheek Aylie, because we know you have something that was tested in your PhD.

Aylie

Now it's interesting because when people ask me about a model, I think often what teachers are after is the instructional model.

[On the screen, the Model for Planning in Mathematics (MPM) appears.]

Aylie

And I'll preface this by saying there is no magic bullet to planning mathematics. It does require some hard yakka thinking and doing during planning. There's also no one size fits all approach to curriculum planning.

So, I was saying before one of the things I had to defend was part of my thesis was, well, why can't you just take this model and use it for literacy? What makes this model special for mathematics? And in basic terms, it's because the way we plan mathematics, the concepts, the knowledge, the pedagogies are not necessarily the same. There might be some overlap, but it's not the same way that we plan literacy.

So, a reading workshop is different to a maths lesson and they have some fundamental differences.

And so, when we're approaching any discipline, so like reading, writing or mathematics, there needs to be, you know, some very key differences in how we would go about those.

So, thinking about Charles Lovett always says, 'What's your school vision for mathematics and do the tasks that you use match that school vision?'

[On the Model for Planning in Mathematics (MPM), the box at the top of the diagram is highlighted and reads, ‘School vision for mathematics learning and teaching’.]

Aylie

And I love that and it's resonated with me for so long. And it makes you think because sometimes our aspirations don't necessarily match our intentions and our actions. And this is for various different reasons.

So, before I tangent over that idea, the idea is, is that this model, it's very detailed and it's because planning is very messy, but we need some sort of way to navigate this messiness.

And so, the model is intended to provide some clear structures, particularly for school leaders and teachers who have the opportunity to plan collaboratively. Obviously, this model can be applied if you don't have the opportunity to plan collaboratively, but it does assume that there is an opportunity for collaboration.

[On the Model for Planning in Mathematics (MPM) , several boxes are now highlighted for emphasis. From the top, the first box states, ‘Collaborative planning: development of a learning sequence’.

Below this box are 2 boxes, one box states ‘Identifying relevant curriculum descriptors and proficiencies for the year level being taught, the year level below and above’. The box on the right states, ‘Analysing mathematics education texts to identify the important mathematical ideas, models, representations, skills and language’.

Below these 2 boxes is the largest text box that reads, ‘Initial assessment of students’ current knowledge of mathematics including:

Formal assessment tools such as: Mathematics Online Interview, Scaffolding Numeracy in the Middle Years, or other researched informed assessment tools.

Teacher-developed assessments such as: Rich Assessment Tasks or teacher-developed tests that are analysed against specific statements of student learning’.

There are 4 smaller text boxes below this large box. From left to right it reads, ‘Establishing clear mathematical learning goals’, ‘Selecting, adapting, and sequencing tasks’, ‘Anticipating possible student responses’, and ‘Identifying pedagogies, lesson structures, teacher questioning and resources including concrete materials’.

Finally, these 4 smaller boxes lead into the last larger text box that reads, ‘Identifying indicators of student progress and planning formative assessment methods. This can include the development of rubrics and/or checklists; and selecting tasks and student work samples to be moderated at follow-up meetings’.]

Aylie

But we want to be more productive and effective. So, what are teachers doing? Our planning time is precious. We don't have enough of it, so we need to make the most of it, but we want to be focusing on the things that will make a difference to student learning.

And so, making productive and effective use of that time so that students’ learning and their engagement is maximised.

So and when we talk about engagement, I refer to Catherine Attard's work about the cognitive behavioural and the effective domains of engagement. So not just having fun. So really the model emphasises some key elements to be considered when planning collaboratively and individually.

And it's all drawn from the research literature on mathematics planning and education, and also from my case study and survey research that was part of my PhD. And so again, the model highlights the importance of a school's vision for mathematics.

How does that filter down into the flexibility that we have for our yearly and termly overviews? So not setting things in stone, some classes might go through that sequence of learning faster, some might need more time and that's ok. We need to be responsive to our students. And the thing is ultimately we need to do things well. So, thinking about the curriculum as that climbing rock wall rather than the step ladder to go through.

And so just along the side here.

[The text box on the very left of the instructional model for mathematics is highlighted and reads, ‘Using a range of high-quality mathematics education texts and resources’.]

Aylie

So really importantly, all throughout my research, it became apparent that, you know, what are the types of resources that teachers are using to inform their planning and their teaching? So, we want to ensure that we're using purpose designed mathematics resources. And this can also save us time by avoiding trawling through Google and Pinterest and those resources that we're not able to verify the quality of but there are so many resources that we can access.

So how can we draw on the experience of our colleagues and using a range of high-quality resources?

[The second text box to the very left of the instructional model for mathematics is highlighted and reads, ‘Drawing on experience of self and colleagues, including a support person and/or critical friend’.]

Aylie

So, here's a little bit of a model and you know the summative assessment, I'll talk a little bit about this…

[The bottom text box on the instructional model for mathematics is highlighted and reads, ‘Summative assessment: This includes opportunities after the learning sequence for ongoing exploration of the concepts, practise, and building mathematical connections in other contexts’.]

Aylie

But for me, if we're doing the formative assessment really well…

[The last 3 text boxes within the instructional model for mathematics are highlighted. From top to bottom, they read, ‘Identifying indicators of student progress and planning formative assessment methods. This can include the development of rubrics and/or checklists; and selecting tasks and student work samples to be moderated at follow-up meetings’.

The box in the centre reads, ‘Individual planning: development of individual learning experiences. Planning the teaching and assessment including differentiating for students in a particular class; planning questions that elicit students’ mathematical thinking and discussion; and preparing required lesson materials’.

And the final box at the bottom reads, ‘Classroom teaching: Includes monitoring, selecting, sequencing, and connecting student solutions for whole class discussions and linking to key mathematical ideas. Also includes identifying additional lessons to further support, extend and consolidate students’ thinking and learning during the sequence’.]

Aylie

Do we need to do that post-test or do we have sufficient information? Because we've had some very clear statements of student learning. Do we need to do the summative assessment or is our formative assessment, together with our observations, what we've heard and seen students do in the classroom, sufficient?

And so up the top here in the model, that's where I was talking about analysing the curriculum descriptors together with the mathematics education text. And this enables consistency across the team. We all have a shared understanding of the direction and the key ideas and concepts and pedagogies that we want to inform our sequence.

Then we're well placed to use and to analyse some data because the data will make a bit more sense, whether it's from a standardised assessment, but hopefully it's more informed by a rich assessment task. And then we're placed well to analyse, that rich assessment task.

Because I could say, oh, you know, 'How are the students unitising? How are they recording their thinking? Can they show their thinking in 2 different ways?'

So bringing in the proficiencies as well of, you know, reasoning and understanding and fluency, and how are they demonstrating all that? And so, alongside the model, they're intended to be used with some strategies.

So thinking about clear and simple documentation, how can we use collaborative planning as an opportunity to build and grow teachers' knowledge?

You know, what are teachers doing during collaborative planning? And so thinking about how can we experiment and anticipate student responses and trial these tasks.

[To the left of the instructional model for mathematics, an image of the 6 key strategies for effective planning in mathematics diagram appears.]

Aylie

And through the process of, you know, trialling, examining that we're actually learning about the concepts, we're building our confidence. And we know that confident teachers usually results in more effective teaching and happier students as well. And thinking about time. So, time and flexibility to teach concepts over a sufficient amount of time.

I just wanted to emphasise that, you know, when I've worked with particularly school leaders using this model in a school, it's often used as a bit of a conversation starter or a reflection tool.

So, teachers might sit in a planning meeting, or they might discuss and they might rank the different parts of the model on a scale of 1 to one to 10 or zero to 10, and saying what's working well and what would be even better if...

So, teachers will often identify, we're doing this really well or this could be strengthened. It's usually quite an affirming process, but it's also a nice way to just to start generating some discussion amongst your colleagues about where are the opportunities to enhance our approach to planning and to advance planning.

Michelle

I think that's a great idea and a great model to share with teachers about how they could potentially use that framework as well. I liked how you also said about the idea of doing things well and I feel like that idea of, we learned this from Peter Sullivan who talks about the work of Laurinda Brown and the idea of relentless consistency, which you can, I think apply to almost anything in the teaching and learning of mathematics and probably outside of it as well. And then being able to build really strong foundations for students in something. And the benefit for that inside of mathematics and the nice part is because it's so inherently all connected to each other, you can then use that as the foundation to explore and extend into other aspects of mathematics, really leveraging those connections and those really strong foundations.

And for us in NSW, part of the new syllabus, one of the tools that have come out with the new syllabus has been a document that provides some of the connections that teachers can be looking for across aspects of the syllabus, which is a really nice document for them, especially in part of that when they're working collaboratively with each other.

So, I wondered if I could use that as a segue, Aylie, to come to this idea of collaboration and what you've learnt over time about what can effective collaboration look like in planning and programming for student-centred mathematics?

Aylie

So, what became apparent, there's usually 2 ways that if there are regular opportunities for collaboration, it usually transpires in the all for one and one for all where we all sit down, we've got our Google Doc open, we're all contributing to the planning, and it's a working document. Or we have the divide and conquer approach and we need to do this sometimes because again, our planning time is limited, and maths isn't the only subject we have to plan.

So, the divide and conquer is where we go maths, you know, you're maths, you're inquiry, you're on admin, so forth. Again, it will post some risk because if someone's done the planning for you, you don't have ownership over the planning and you might not understand the planning. So, in terms of, well, why did the teacher select that task or what does this mean?

So, the all for one and one for all is great because it means we're all involved, all on the same page, we're all developing shared understandings. So, what I tried to do with the teams that divide and conquer was if we need to divide and conquer, what can we do to get on the same page? So there has to be some shared conversations prior to the maths planning that will enable teachers and the other teachers to make sense of the planning documentation that they may not have necessarily contributed to.

So, for me, again, going back to it starts with analysing the curriculum descriptors. So, there are some non-negotiables. So, we might do the divide and conquer, but we have to have gotten on the same page.

There has to be some shared discussion around what is our upcoming unit of work?

What are the key ideas we're going to be focusing on?

What has the curriculum have to say?

But what else do we need to know?

Let's unpack this idea and it is worth time spending on these conversations, particularly when we know teachers may experience low confidence in mathematics. Some teachers experience math anxiety and so having these conversations that are led by a critical friend or a team leader or a champion like who's championing the maths in the school, it's a way of building teachers' confidence. So, using that opportunity, that planning conversation, setting aside that 40 minutes to have a really in-depth conversation about the concepts that we're going to be teaching, you know, what might be the misconceptions? Let's have a go at doing a task, let's trial a task, let's anticipate some student responses.

So, we've been trialling a bit of a new approach with some schools where we plan one task, we discuss the task, we discuss the mathematics, and then from there we can develop one or 2 weeks' worth of learning. Now the other flip side to that with effective collaboration is that we don't need to plan the 3 weeks’ worth of learning up front. If we want to be responsive and student-centred, then we might just have a conversation where we discussed the reading. We might also plan the rich assessment task because we're very clear now on the concepts and ideas, the important mathematical ideas.

Usually, we can find a rich assessment task that we can adapt and then the idea is that we would administer that task ideally before teaching it, come back, identify what the key strengths and you know, target areas for the cohort, and then we might plan the first 3 lessons because we've got a bank of high quality resources. And then it might be that's when the divide and conquer will work really well. And so, we're really clear on why we're using some of these tasks and what the intent is behind those tasks. So without having that conversation initially, and then again, just the other point is that it's a working document. So, you might plan 3 or 4 tasks or 2 or 3 tasks and then come back the next week and then plan some subsequent tasks or that can be done simultaneously, because if we're using, we have really effective online collaboration tools like Google Docs or whether you're using Teams but having those shared online documents where everybody can contribute to the planning.

But for me what I found that getting on the same page is that critical aspect of collaboration. And for those of you that are in leadership positions, particularly middle school leaders, we do have to work within the opportunities and the constraints inherent in your own context.

Michelle

Thanks, Aylie. I wonder because in your work and in others' that recognises planning and programming is such a powerful opportunity to support teacher capacity. I remember reading Fullan and Hargraves' work around professional capital. They pick up on Leonie's work around the critical importance of building social capital. So, it's not just building the capacity of one person at a school, but in fact that idea of collective efficacy of the group and your sort of description around collaboration really reminds me of that research. So, we know opportunities for planning are really powerful places to build that collective efficacy of our teaching for our teachers, including supporting their confidence, which then has this really nice benefit of supporting the learning outcomes of the students that we work for.

You've also mentioned before about how skilful effective planning and student-centred planning is and I wonder with that framework that you've shared and other things that you've learned how we could apply that sort of thinking or these sorts of skills and in fact, even if we need to, to units of learning and tasks that we use that are pre-made for us.

Aylie

It's an interesting conundrum, isn't it? Because, you know, as teachers, we are trained professionals. Planning is our core business. Firstly, we need high-quality resources and as professionals we need to use our professional discernment of what is a high-quality resource. Who is writing these high-quality resources?

Are they trained in mathematics? Because task design and task design is not an area that I specialise in. It takes a long time. So, when we look at, so for example, the maths online interview, which I know is used in some systems in NSW, there are a lot of tasks in there that have gone through many different iterations of trial and refinement, testing with students in schools. The numbers, for example, for a particular task are given a lot of consideration too. It's very difficult to design an effective, useful, meaningful maths task.

And so, when I'm working with teachers, my preference is to go, 'Well where are these high-quality resources?', and regardless of the information contained, so even if it is a high-quality resource, I still need to think about that task myself. I'm going to need to do the task, work through the task, understand the task. Yes, we can always try and wing it, but it's never going to be as an effective as if we actually give some time and thought and the task regardless of where it comes from, is always going to have to be adapted. You might be working with students from a population who don't have English as a first language, and so we might need to adapt that resource to create a lot more visual representation to help those students engage with the mathematics.

And so regardless of where the task comes from, assuming that they're coming from high-quality resources, teachers will always need to use their professional discernment and adapt them for what they are and because you might take a predetermined learning sequence and go, 'Oh no, this sequence is not actually meeting my students' needs'.

Thinking about who wrote them, I think is fundamental for me when I'm being discerning over whether I think a resource will make it into my sequence or not. The other thing is I want to be thinking about is the range of experiences in that resource.

Does my pre-made resource allow me to explore a really important idea in many different ways, or is it just like adding fractions on one day, equivalent fractions on the second day and jumping from idea to idea? So, then teachers need to then think carefully and discerning about this resource and going, 'Well, what am I going to need to do to adapt this resource to strengthen it?'.

Michelle

I feel like that's really good advice Aylie. We also make recommendations of if inside of a sequence that says it's using a task from Nrich Mathematics, then click on the link and have a look at the task as it was designed by Nrich Mathematics. You know, to see whether the adaptations around it best suit the context of your class or if there's other things that you would do around that task as well.

In talking about tasks, Aylie, we have one last question for you. You've mentioned that you're not a great task author, but there are some really good task authors out there. So where are some high-quality resources that you often use as a starting point for your planning and programming?

Aylie

If I think about this student-centred approach, firstly I'll go and say that across any sequence of lessons so, some teachers refer to this as a unit of work, but now we're shifting our language to the idea of a learning sequence rather than just a fixed unit. We want to ensure that range of experiences, so you know, Peter Sullivan, he talks about many ways to teach maths well, and that we need a range of learning experiences. And so, we might have active teaching, which is a little bit like explicit teaching, there's a lot of student contributions still involved with the active teaching approach. Then also need to ensure that we're using some interesting games and puzzles as part of our learning sequence.

And that the games are really powerful ways of teaching mathematics. But teachers need to understand the mathematics and the mathematical potential of the game and also ensure that there's a discussion about the mathematics that students learned and the strategies from playing the game.

We also then can have some what if or like imagined representation type questions. they’re like you know your Fermi problems, for example. So, like, you know, how many popcorn kernels can you fit into the classroom? It's a lot of estimation, logical thinking.

And then we also have our open-ended challenging type tasks.

So, do we have a range of learning experiences across a sequence of lessons? And where do we find these quality lessons from? And there's so many freely available ones.

So Nrich, you mentioned, you know, the AAMT-the Australian Association of Mathematics Teachers, MAV, which New South Wales teachers can also access, they have really good resources and looking at Re(solve). So, Re(solve) is a wonderful resource to use from prep all the way up to Year 10.

[The logos for each of the mentioned of reputable and freely available lessons to assist in planning and teaching for mathematics appear on the screen one by one, including Nrich, AAMT – the Australian Association of Mathematics Teachers Inc, MAV – the Mathematical Association of Victoria, and Re(solve) – maths by inquiry.]

Aylie

There's so many high-quality online resources. And then thinking about Dan Finkel's website where he has lots of different types of lessons and games available, huge range. And I can offer teachers to the list of some of these resources as well.

But then also thinking about where do we find lots of these tried and tested resources? So, you know, challenging math tasks, engaging math lessons, you've got the rich assessment task book. There's so many, and all these tasks are being trialled in classrooms with many students, and they've been refined.

And the idea is that teachers find these tasks, trial these tasks in their classroom, but these tasks often find them, and they're presented as a one off. In the spirit of consolidation, and this idea of experiencing the same idea many times in many different ways, it's then the teacher needs to create a task that's a little bit the same and a little bit different.

And so, when we say planning is a skill, the more we do this, it just becomes part of our everyday planning routine. And so being able to find the task, teach that task, but also adapt it to a same-same but different context for the following day is really important as well.

Michelle

Thanks, Aylie. We'll share for our colleagues across NSW, we'll share links related to things that have come up in the conversation with you today that's connected to this ‘In conversation’.

But we just really like to thank you for sharing your experiences and your expertise that's been built over time and really from a great point of curiosity of what is it that we can do to help teachers really embody the aspirations of some really nice documents that we have around being student-centred and student voice and really thinking about the students that are in our care, but also realising the potential of things like planning, which can be really boring and mundane. But when you get to think about that in, you know, with your colleagues, when you're playing with mathematics, when you're thinking about what your students might be doing, when you're getting curious about how you might respond to that and find out more about what they need.

It makes this thing that can seem really arduous and mundane, certainly my experience when I first started teaching, to something that might still be a bit tricky to put down on paper, but it's much more supportive for you and your teaching, for the students that you're planning it for, and a little bit more interesting than what it otherwise can be.

So, thank you so much for sharing your work with us Aylie, we are really grateful for your time. Thanks to everyone that's been out there listening today. We hope this has given you some things to think about and go away and get excited by.

Aylie

Thanks, everyone.

Michelle

Thanks everybody.

[End transcript]