Carolyn Matthews

Welcome everyone to today's workshop on understanding effective teaching approaches in mathematics. My name is Carolyn Matthews and along with my colleague Nadia Walker, we are the Primary Curriculum Mathematics Advisers.

Today we'll be walking you through the workshop and your school's facilitator will also be leading activities and coordinating discussion topics throughout the session.

I'd like to start this workshop by acknowledging Country. We recognise the ongoing Custodians of the lands and the waterways where we work and where we live. We pay respect to elders past and present as ongoing teachers of knowledge, songlines and stories, and we strive to ensure that every Aboriginal and Torres Strait Islander learner in New South Wales achieves their potential through education.

Nadia Walker

Prior to today's workshop, your facilitator would've given you a participant workbook. This workbook is provided for your note-taking during the presentation and can be used in the discussions and activities we have planned for you as well. Over the course of the next 90 minutes, we'll delve into the evidence-based pedagogical shifts that align to the K–6 component of the New South Wales Mathematics K–10 Syllabus. We will examine both the syllabus and the evidence-base as well as practical strategies for the classroom, emphasising the connectionist approach, working mathematically, and the benefits of using explicit teaching practices when teaching mathematics.

We've designed this session to support schools to understand the significance of a connectionist approach to mathematics, reflect on a recent mathematics lesson and pinpoint the working mathematically processes addressed, identify opportunities to integrate working mathematically questioned prompts into the planning phase, define what we mean by explicit teaching and recognise 7 explicit teaching practices and to highlight opportunities in lessons and lesson sequences to incorporate explicit teaching practices.

Let's begin our session by exploring the connectionist approach as a framework for the effective teaching of primary school mathematics. The National Council of Teachers of Mathematics or the NCTM defines mathematical connections as the ability to recognise and use connections among mathematical ideas, understand how mathematical ideas interconnect and build on one another to produce a coherent whole and to recognise and apply mathematics in context outside of the Mathematics Syllabus.

So let's focus on this quote on the screen from Linda Gojak.

'Taking a mathematical concept and considering how it originates, extends, and connects with other concepts across grades will help us as teachers develop a deeper understanding of the content. It is then that we can plan instruction that ensures that our students regularly make connections to help them make sense of the mathematics they're learning.'

That's one of the big goals in our mathematics teaching, to make sure that mathematics makes sense for our students. We also know from Mike Askew's research on effective teaching of mathematics that teachers who teach mathematics through this connectionist lens are highly effective in progressing students' learning. Siloed, isolated pockets of mathematical knowledge is no longer enough to be a successful mathematics learner.

Today's students need to be able to problem solve and work mathematically in an interconnected and interrelated way, building on their prior knowledge and making connections between mathematical ideas to increase both conceptual and procedural understanding.

We view the connectionist approach not as an instructional model. It is more of an organisational design or a view on what mathematics is. It highlights learning that is connected.

Mathematics really is a complex, intertwined network of ideas with many interrelated concepts. Imagine the mathematical content in the syllabus as an intricate underground train network of ideas with each topic and concept interconnected by tunnels or platforms representing the relationships and connections between them.

Like a well-designed train system, the content network identifies connections across focus areas and transports learners from one understanding to another, allowing them to navigate through as they develop increasingly sophisticated mathematical ideas and understandings.

Navigating this train system requires a map, which in education is provided through our curriculum and effective teaching methods. So just as passengers rely on a map to comprehend where they're going, learners need well-structured lessons to help them see the mathematical connections and apply these connections in different areas of the syllabus.

On the screen, you can see a conceptual model of some of the Stage 3 connected focus areas. Looking at this web of intertwined content, we can see the intersection with Multiplicative relations and Two-Dimensional spatial structure as students calculate the areas of rectangles using familial metric units and find the area of composite shapes. We can also see a connection between Multiplicative relations, Represents numbers and Three-Dimensional spatial structure.

Students use multiplicative and place value understanding when they're exploring volume and connecting decimal representations to the metric system. I'm sure you can see many more of these interconnections on our network model.

In the words of Jo Boaler, 'Maths is a subject full of beautiful connections. It is not a long list of disconnected topics.'

The new Mathematics Syllabus really identifies these beautiful connections for us.

Making connections when planning learning sequences helps us as teachers to uncover the mathematics curriculum as opposed to covering the curriculum, which is a phrase coined by Marilyn Burns in 2022. By making these connections, we can shift our practise to a more student-centred sense-making approach that builds students conceptual understanding of mathematics as a whole. Catherine Attard, who you may know, a prominent mathematics researcher, has highlighted some questions we can consider when planning to surface connections in our classrooms.

She encourages us to ask ourselves the following questions, so do we make connections to previous lessons and prior assessment of student learning? Make connections to a topic covered in other areas of the curriculum? Make connections to students' current interests? Make connections to a previous topic where they may have discussed a similar idea in another form? Cover one idea in a variety of ways? And do we aim to make connections with mathematics and the world for our students?

And I love this quote from Anthony Walshaw saying 'Tasks that require students to make multiple connections within and across topics help them appreciate the interconnectedness of different mathematical ideas and the relationships that exist between mathematics and real life.'

We're going to take a moment now to pause here as your facilitator will lead a short discussion on the connectionist approach.

Carolyn Matthews

Now let's take a look at working mathematically as a key component of effective teaching of primary school mathematics. 'As an essential part of the learning process, Working mathematically provides students with the opportunity to engage in genuine mathematical activity and to develop the skills to become flexible and creative users of mathematics.' Working mathematically is the thinking and the doing of mathematics, and students learn to work mathematically by applying the processes of communicating, understanding of fluency, reasoning, and problem solving to the content within our syllabus. It is important to note here that there is one working mathematically outcome across the K–10 Mathematics Syllabus.

The working mathematically outcome does not have specific content groups or content points as it's not a focus area. As such, the working mathematically outcome should not be assessed, taught, or reported on in isolation. We must assess students' proficiency with the processes in conjunction with stage-appropriate content. Our students learn to work mathematically by using the processes in an interconnected way. The coordinated development of these processes results in students becoming mathematically proficient.

Our students are communicating when they describe, represent, explain, and reason about mathematical situations, concepts, methods and solutions. It's important to ensure that we provide opportunities for our students to communicate their thinking in different ways. This might include written, oral, graphical or symbolic forms or through advanced gestures or signing.

Our students are demonstrating understanding of fluency when they connect related ideas and represent concepts in different ways to solve both new and unfamiliar problems. This conceptual understanding then supports retention and fluency.

Facts and methods learnt with understanding are easier to remember, which in turn builds our students' fluency to recall knowledge and concepts as well as use known facts with automaticity and efficiency.

Fluency and understanding are interconnected. When our students explain their thinking, deduce and justify their strategies, when they prove that something is true or false, and when they compare or contrast related ideas and explain their choices, they're demonstrating reasoning.

Our students are problem solvers when they form mental representations of problems by applying mathematical relationships to familiar and unfamiliar problems, and by devising novel solution methods when needed. The syllabus emphasises the crucial relationship between the 'what' – the content of the syllabus, and the 'how' – the working mathematically processes.

Within the mathematics team we often use the analogy that the content are like passengers in a car and the working mathematically processes are the car. This interdependence lies at the core of our new syllabus as we aim for our students to exhibit progressively sophisticated development of the working mathematically processes right across the stages.

This sophistication is embedded deeply in their application and understanding of the mathematical content.

Given that the overarching working mathematically outcome is interwoven throughout the new syllabus content, it is essential to assess working mathematically in tandem with the mathematics content outcomes. The complexity of the working mathematically processes evolves through each stage of learning and correlates with the increasing intricacy of the mathematics content outcomes.

Monitoring a student's proficiency in working mathematically can occur over time. For example, in Additive relations, it is evident in the choice of a strategy appropriate to the task and the use of an efficient strategy for the student's current stage of learning. So whilst the outcome is the same, it looks and it sounds different at every stage of learning.

Students develop their ability to work mathematically by applying the processes to increasingly complex Stage-based content. As a student progresses, they learn more sophisticated and targeted ways to do and think about more complex mathematical content. The significant shift in embedding working mathematically within the content in the new syllabus really offers us the opportunity to breathe life into these processes through purposeful and authentic mathematical tasks, integrating the working mathematically outcome into each content outcome.

Along with the examples and the teaching advice, it represents a major change in the syllabus and will profoundly influence our approach to the teaching of mathematics. The 3–6 microlearning module 'Elaborating on working mathematically K–6' is an additional source of professional learning that provides examples and a resource set on how the working mathematically outcome can be assessed within stage-based content.

The working mathematically processes – they're really brought to life in the verbs of our syllabus, the verbs connect the 'what' and the 'how'. One of the great features of the new syllabus is that the learner action verbs have been included in the syllabus within the content points. So let's have a closer look at some of the learner action verbs in closer detail.

This visual display on the screen has been adapted from the work of Burrows, Clarke and Raymond and it links the working mathematically processes to verbs as purposeful learner actions that could be used to guide effective mathematical classroom practice.

As teachers, we can use these verbs in our planning and in our conversations with students, while encouraging students to both demonstrate them, but also verbalise them in their own communications with their peers. These verbs offer us clues as to what it might mean to be a mathematician, to think mathematically and to articulate mathematical thinking in ways that demonstrate deep understanding. These verbs could be used to support teacher planning when designing learning sequences that connect syllabus content to the working mathematically processes. They could be used as question stems to support, prompt and challenge students' thinking and encourage students to think deeply about the mathematical content.

For example, you may ask students to describe how they manipulated numbers to solve an additive relations task, providing them with the opportunity to draw on the processes of understanding and fluency and communicating. In another lesson, students might investigate and solve a multiplicative relations problem and persuade or convince their peers or you as the teacher that they have found all possible solutions. Here you would be able to see students problem-solving, communicating and reasoning skills in action. In the upcoming activity, you will have the opportunity to discuss some working mathematically question prompts that might guide your mathematics planning.

We're going to take a minute here to pause, as your facilitator will lead you through a working mathematically activity.

Nadia Walker

Explicit teaching has a strong evidence-base and is acknowledged as being important and effective. When teachers adopt explicit teaching practices, they clearly show students what to do and how to do it. Explicit teaching is about making the learning clear. This allows teachers to manage the cognitive load of students throughout a learning sequence and provides the right balance of challenge and support for every learner. When explicit teaching is done well, it is very student centric, interactive and flexible.

Explicit teaching is when teachers clearly explain to students why they are learning something, how it connects to what they already know, what they're expected to do, how to do it and what it looks like when they have succeeded.

Students are given opportunities and time to check their understanding, ask questions and receive clear, effective feedback about aspects of performance. I've bolded some of the words in this What works best CESE quote that really stand out for me. 'Clearly explain the why', 'how it connects', and I also love the emphasis on ‘students checking for understanding, asking questions, and receiving that clear, purposeful feedback' on their learning.

'In the complexity of a diverse classroom, effective teachers use knowledge of their students, a deep understanding of the syllabus and of the mathematics, alongside a variety of assessment strategies, to determine which particular explicit teaching practice or combination of practices will best support their students' learning.'

From this statement, we can see that explicit teaching is not a recipe or a prescribed set model. Using explicit teaching practices is not limited to a particular type of task, activity or lesson plan either. Explicit teaching practices are elements within an effective lesson. Teaching is a complex thing, and as the slide says, 'Teachers make pedagogical choices and significant decisions continually throughout a lesson.'

From determining the learning intentions and success criteria for the lesson, to checking for understanding, to modelling and questioning, to facilitating class discussions, explicit teaching practices all aim for clarity. Clarity for the teacher – 'We have to know what our students are learning and know the syllabus expectations'. Clarity for the students – 'What is the mathematics that they're learning and why?' And clarity of the mathematics itself – 'Do we know the progression of the content, the misconceptions that hinder learning the models and representations to support that concept and the connections that we know are so important?'

Explicit teaching practices can help us to make sure there is clarity in all of those things.

Carolyn Matthews

So, what do we mean by the word ‘explicit’? The dictionary definition defines explicitness as 'The quality of stating something clearly so that the meaning is easy to understand.' One interpretation of an explicit teaching model is the ‘modelled guided, independent’ or the ‘I do, we do, you do’ approach. This model is discussed in the NESA ‘Evidence-based practices for planning and programming’ publication, which states that the stages of explicit teaching should not be viewed as linear or even cyclical. Rather, they are interdependent and decisions about when to move to the next phase of teaching are always determined by student responses. One lesson may have an ‘I do, we do, we do again and then you do’ approach and another lesson may have a ‘you do first, then I’ll do, then you do again’ approach, depending on what the students have shown us they need.

At times, it may be necessary to move backwards and forwards between providing instruction, guiding practice and providing opportunities for student independent practice. So, when we see that the explicit teaching can come at different times within the lesson, depending on student responses and their learning progress on a task, mathematics teachers do not need to be locked into an order that is repeated over and over again in the exact same way in every lesson.

The advantage of this is that the explicit teaching occurs when the student requires it, rather than when the lesson plan or the model dictates it.

Part of the confusion and the debate around explicit teaching practices is the misuse or interchangeable use of terminology. We are talking about a set of practices that help to make complex concepts clear and understandable, and while it can include teacher-led components, the focus is on the students’ learning. These explicit practices aim to actively engage students to promote questioning and encourage exploration and ensure mathematical understanding. They include a variety of strategies and are fluid elements of an effective lesson. As you learn more about explicit teaching practices in this workshop, you’ll understand that there are multiple ways explicit teaching can be put into practice.

We're aiming for learning experiences that get our students in task and not just on task, and tasks that promote connections and student engagement in mathematics include activities such as open-ended tasks, rich tasks, problem solving and investigations, tasks that spark our students' curiosity.

When we use explicit teaching practices within open-ended tasks and rich tasks, in problem solving and investigations, we bring the aim and the rationale of our new syllabus to life. We can make learning mathematics engaging and creative, while at the same time ensuring that students receive just the right amount of teacher guidance for their learning needs and to make the mathematics clear.

So what is one way of conceptualising explicit teaching practices in mathematics? This diagram comes from the Mathematics Hub, and this resource is supported by the Australian Government Department of Education, the Australian Curriculum Assessment and Reporting Authority, the Victorian Department of Education and Training and the NSW Department of Education. It highlights 7 features of explicit teaching in mathematics. These seven practices can be enacted in a variety of ways, throughout various phases of the lesson, to make the mathematics learning explicit and effective. There isn’t any single ‘best practice’. Instead, effective teachers choose the right practice for the right time to create an optimal learning experience for their students and to make the mathematics clear. These 7 practices include purposeful planning before the lesson even starts – making connections, feedback, questioning, modelling, knowing the learners needs, and dialogue.

A quick word about dialogue and class discussions. During mathematical discussions, students develop language, they build mathematical thinking skills and create mathematical meaning through collaborative conversations. Mathematics is a social subject, and we learn it through talking, explaining, debating, exchanging ideas and asking questions. Do you use number talks in your lessons? Do you incorporate talk moves? There is a lot of evidence to show that these are powerful tools to incorporate dialogue in your maths lessons.

This is one of my most favourite quotes from Van de Walle. ‘The value of student talk in mathematics lessons cannot be overemphasised. As students describe and evaluate solutions to tasks, share approaches and make conjectures, learning will occur in ways that are otherwise unlikely to take place.’

Nadia Walker

Making informed instructional decisions focused on our students' learning needs is a really important part of knowing when to be explicit, when to use modelling and guided demonstration, when to give feedback, when to check for understanding. I really liked this clever analogy that I heard recently.

A professional golfer wouldn't play a whole game of golf only using a 9 iron.

Now, I'm not a golfer, but I can understand how using just one club could get me through that 9 or 18 holes, but it's not going to produce my best shots or my best score. I need to know how to use all of the clubs, including the drivers and the wedges because they offer my game different benefits, and it's the same with teaching mathematics.

Explicit teaching practices are all incredibly effective, but I don't want to just rely on one. Modelling or demonstration isn't the only explicit practice I want to use. In fact, if I only used modelling to be explicit, then I'm limiting my effectiveness and the experience for my students in learning mathematics.

The McKinsey report on how the world's most improved school systems keep getting better categorises actions and interventions from a whole range of schools. Schools working from the poor to fair performance category were focusing on supporting students in achieving the literacy and maths basics, providing set routines and scaffolding for teachers, and fulfilling basic student needs. But when schools are working in the top performance category, the great to excellent, their focus is on innovation, experimentation and flexibility, and we want all NSW schools to be in the great and excellent category.

So let's have a look at a task now where explicit teaching practices are used with innovation and flexibility. I'd like to share a classroom story with you to illustrate how these explicit teaching practices can be used in different lessons with different tasks for different student needs. I recently watched this task as it was taught to a class of Stage 1 students.

The teacher drew this open number line on the board and said, 'I'm starting at 17, and want to get to 39 in exactly 3 jumps. What would your jumps be and can you explain your strategy and your reasoning?'

So this is an open-ended task. Because of the design of the task and its multiple solution pathways, the teacher didn't want to model how to complete this task before the students had time to think and have a go for themselves. We can say this lesson started with a 'you do' moment. That doesn't mean that the students were discovering the learning for themselves or exploring these important concepts on their own. These students were definitely not left in the learning wilderness.

On the contrary, the classroom teacher knew the students’ learning needs very well and was clear that they had enough prerequisite knowledge and mathematical experience with both addition concepts and the open number line to engage with this type of task.

Because the teacher had planned a lesson with an awareness of learner needs and had previously used modelling, questioning, feedback and dialogue to prepare the students, this type of task was appropriate for the students. The teacher was also going to use it as a formative assessment moment to gain further information about what the students could demonstrate independently.

Due to the openness of the task, the teacher expected and hoped there'd be more than one successful strategy. This task sparked interest and curiosity in the little Stage 1 students – engagement levels and sweaty brains were high. So this lesson didn't start with modelling or demonstration, but the teacher did explain the task and facilitated a brief dialogue about the concept of addition. As a class, they reviewed the open number line as a tool to support their thinking. So they were set up for success before they dived into the 'you do' moment.

Now I'm going to describe to you what I saw the teacher do during the lesson and as I go, I'd like you to refer to the model of the 7 practices on the slide to see how many of those explicit teaching practices the teacher used.

While students were engaging in the task, the teacher identified some students who required more support and teacher guidance. The teacher used modelling to demonstrate one way of approaching the task and questioning while she created a worked example. The students discussed what the teacher did and were guided to transfer the learning to their own strategy to complete the task.

These students were in a novice phase of learning. They were thinking about new ideas and needed explicit guidance to support them. And then after the class completed the task, the teacher brought all the students back together and used dialogue to make the mathematics clear and build connections. The teacher chose students to share and communicate their approaches. The students used modelling to demonstrate their thinking and strategies. The teacher used the students' worked examples to highlight the mathematics and extend the strategies beyond. The teacher gave feedback during this dialogue. The teacher also modelled correct mathematical language while making further mathematical connections between student strategies.

So during the lesson, the teacher used 2 moments for explicit teaching – one that we call ‘just in time’ and a ‘whole class moment’ at the end of the lesson. These explicit teaching moments, the just in time and the whole class, are not mutually exclusive. Engaging in explicit teaching practices at the beginning of the lesson, the middle of the lesson and the end of the lesson comes from what the student needs are. This flexibility calls for us to be agile and adaptive, responding to our students in the moment.

Carolyn Matthews

We'd like to take a moment now to discuss novice and proficient. Students can be both novice and proficient in different areas of mathematics curriculum at the same time, and therefore, they might require different levels of explicitness and guidance. Let's take for example, “James”, who is a Stage 2 student who has shown a really good understanding of place value concepts. He's able to petition and rename multi-digit numbers successfully in standard and non-standard form. He makes connections with these ideas with additive strategies using non-standard partitioning to assist with efficient mentor computation. He really is demonstrating proficiency in this area.

On the other hand, James is a novice learner with concepts related to 2-dimensional spatial structure. He requires additional teacher guidance, support, and more explicit modelling to help him to tesselate designs by reflecting, translating and rotating shapes. James' teacher knows that he requires different combinations of teaching practices and approaches to meet his learning needs across these 2 different areas of the syllabus.

We do want to highlight some further evidence on explicit teaching practices. According to CESE and What works best, the use of formative assessment is critical in explicit teaching. Formative assessment allows teachers to accurately determine students' current level of understanding and decide how much guidance is required. Without formative assessments, teachers may assume that students need much more or much less support and guidance than they actually do.

Coming back to the NESA document that we spoke about earlier, it states that 'Stages of explicit teaching should not be viewed as linear or even cyclical, rather, they're always determined by student responses.' We said before, this may look like an 'I do, we do, you do' in one lesson, but it may look very different in other lessons. If we have a look at Hattie's table of top effective teaching strategies, here are a few of the explicit teaching practices that we've talked about with you today. We can see that classroom discussion, the use of effective dialogue for learning, this has an effect size of 0.82, remembering anything above 0.4 has been determined to have a positive effect on student learning. We can also see planning, feedback and setting appropriately challenging goals as all ranking highly. Problem-solving teaching has an effect size of 0.67, with direct instruction and explicit teaching at 0.59 and 0.57 respectively.

Well, we're going to take a minute now to pause as your facilitator will lead you through an explicit teaching practices activity.

Nadia Walker

Some people think that these statements on the slide sit on 2 opposite sides of the mathematics fence, whereas we see them as complimentary.

Maths is about getting the answer, and maths is about patterns and relationships. Maths is black and white, right or wrong, and maths is about creativity and thinking. Teaching and learning mathematics is a beautiful, rich thing when we see these seemingly opposite philosophies as working together. And Peter Saffin's quote on the slide here captures that beautifully.

We'd love to encourage you to open up some conversations in your schools about these, so-called 'opposite' ideas about mathematics. Hopefully after this presentation, you'll agree with Carolyn and I that using explicit teaching practices is important in mathematics, yet it is just as important for students to develop problem-solving and reasoning skills through a variety of lesson types, for them to see the connections across mathematical ideas and to dive deeply into the working mathematically processes.

We would like to thank you for your participation in today's workshop. We hope you have found moments that have affirmed your current practice, challenged your thinking, and sparked some good conversations.

If you would like to contact the Primary Curriculum Mathematics team, we can be reached at the mathematics K–6 inbox. That is mathematicsK6@det.nsw.edu.au.

We would really appreciate you taking a few moments to complete the evaluation on the screen for us. Thanks again.

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