Jill Brown

Is successful maths teaching more than method?

What counts as “evidence-based” teaching in mathematics? Given calls by the Grattan Institute to end the lesson lottery and make a maths guarantee, this question matters. Explicit teaching is always part of a high-quality lesson sequence. But defining it as the ultimate pedagogy sidelines the very practices that engage students in mathematical thinking. Students need more than procedural recall and routines without reasoning if they’re going to thrive in a rapidly changing world.

Why is explicit teaching making a comeback?

In signing the Albanese government’s Better and Fairer Schools Agreement, Australian states and territories have committed to provide all students with highly effective evidence-based teaching and equitable learning opportunities.

This is translated as adopting the Australian Education Research Organisation’s (AERO) advice on explicit instruction as “what works best” when it comes to teaching fundamentals like reading and mathematics.

In its explainer on how to optimise learning, AERO describes explicit instruction as follows:

‘Teachers directly explain to students how to complete a task, why the task is important, and how the task relates to and extends their previous knowledge. Demonstrations of how to perform tasks or solve problems are provided, often using worked examples.’

It is important to note that AERO’s valuing of explicit instruction stems from a narrow interpretation of the purpose of school education and what counts as research evidence. The studies AERO favours are typically randomised controlled trials not set in school classrooms. Seeking to transform teaching practice by generalising research findings made in tightly controlled environments is problematic. Why? Because these settings are often worlds apart from real classrooms. As the OECD notes, the reality for teachers is often unpredictable classrooms, where students have diverse and competing needs, resources are limited and time is constrained.

Policy is being used to deliver instructional fidelity

Yet, for the first time, education policy is being used to deliver instructional fidelity. The NSW Department of Education School Excellence Framework states: “Explicit teaching is the main practice used in the school.” In the Victorian Teaching and Learning Model Version 2.0, explicit teaching is the only pedagogy mentioned. 

In both these states, departments of education have produced detailed guidelines outlining what explicit teaching is and isn’t. They also provide lesson banks to help teachers align their teaching practice with these specifications. Across the Catholic system, instructional resources developed by AERO’s preferred partner, Ochre Education, are now widely used.

As a result, the classroom experience for many young Australians is now the use of universal slide decks that follow the “gradual release of responsibility” model. It borrows from literacy research and is sometimes referred to as the “I do, we do, you do” lesson structure. According to AERO, this involves the teacher modelling how to do mathematics and monitoring for 80% of students to achieve mastery before moving to any form of meaningful independent practice. This enactment tends to focus teachers and students on perfecting procedures and algorithms. This leaves less time for real world problem-solving experiences that more holistically develop mathematical thinking.

To adhere to these directives is to ignore decades of mathematics education research. 

So, what is the evidence for effective mathematics teaching and learning?

The truth is there is no magic bullet.

Studies have shown that when teachers combine student- and teacher-centred pedagogies, students do better. In fact, an OECD analysis identified three broad teaching strategies described as active learning, cognitive activation, and teacher-directed instruction. The OECD inked exposure to these teaching strategies with student performance on its PISA mathematical literacy assessment. It found strategies for active learning and cognitive activation were more effective than explicit or direct instruction. However, teacher-directed instruction was what students mostly experienced, despite this mode of instruction being least impactful for mathematics performance. 

Another OECD PISA analysis found teacher-directed strategies can support student success on easier tasks. But as problems become more difficult, students with more exposure to teacher-directed instruction no longer have a better chance of success. This is because too much teacher talk limits students’ opportunities to take ownership for thinking mathematically without close guidance. This insight is consistent with studies that show student-centred pedagogies are particularly effective in developing student initiative, responsibility and working mathematically.

Of course, it is important to teach explicitly and to make mathematical language and representations clear and visible. But flexibility is key. What works is contingent on the circumstances, including curriculum learning outcomes, learner profiles and the mathematical foci for the lesson.

Strong mathematics and numeracy leadership also matters 

The presence of an expert mathematics teacher who has input into school policy decisions and knows how to develop others’ teaching practice is a key feature of schools that perform highly in mathematics. 

For example, a substantive study commissioned by the Australian Chief Scientist analysed data from 52 case study schools. Each of these schools had an increase of 1 standard deviation or more in their NAPLAN results. Data collected from hundreds of school leaders, teachers and students across these settings revealed organisational factors that underpin success. 

These schools were committed to teaching mathematics for deep understanding. They valued student-centred learning, including student talk for understanding. They also took a consistent (not uniform) approach to local curriculum planning and had high levels of teacher autonomy. This means teachers were trusted to select teaching resources and pedagogies that met their students’ needs and interests.

We need to make maths real

A recent report explained that the telling and testing students typically experience in school mathematics is often at odds with developing positive feelings or a long-term interest in the subject. Parents and teachers want mathematics lessons to be  more engaging and real-world relevant so young people learn to use mathematics to think critically and make decisions. 

Studies of student motivation have shown teaching mathematics through interesting and challenging real world examples motivates students to choose the subject in senior secondary years and pursue mathematics-related careers.

That’s why curriculum writers have tried to position young people as active in the process of developing mathematical knowledge, skills, proficiencies and processes. 

It is through actively doing mathematics – not watching slide decks and memorizing procedures – that young people develop the kind of mathematical thinking they’ll need beyond the school gates.

Explicit teaching may be a solution to some problems in mathematics education, but it is not the only solution to all problems, all of the time.

Carly Sawatzki and Jill Brown are mathematics education researchers at Deakin University’s School of Education and in the Centre for Research for Educational Impact (REDI). Laura Tuohilampi is a mathematics education researcher at UNSW and the founder of Math Hunger and Maths for Humans. The authors’ work helps teachers connect the school curriculum with the real world, making mathematics education lifeworthy for today and tomorrow.

The authors wish to acknowledge the contributions of professional educators who provided insights and feedback that shaped this article.

Explicit teaching mandate – a pushback now is critical

Today, NSW teachers will spend  their professional learning session focused on explicit teaching, also known as explicit instruction. 

The NSW Education Department Secretary Murat Dizdar told the ABC: “On day one, term two, which is a school development day, right across 2,200 schools, we will be undertaking explicit teaching learning, in every single school in New South Wales.”

Excessive focus on explicit methods will have side effects and could lead to students not meeting curriculum expectations.

A pushback is critical – explicit teaching is not a magic bullet, nor should it be the single pedagogy in any classroom. Definitions of explicit teaching vary, as do its implementations. The approach to explicit teaching and its effectiveness will depend on the discipline and specific focus in question.

 Learning is complex: Multiple pedagogical approaches are needed

We all agree that teaching and learning are critically important but complex. Teachers are focused on improving student learning. However, in Australia the 3 yearly PISA results over the last 2 decades show a decline in 15-year-old students’ ability to apply their reading, scientific and mathematical knowledge and skills to solve real-life problems.  PISA focuses on the capacity of students to analyse, reason and communicate ideas effectively, to continue learning throughout life, and become successful in the workplace.  One of the highest ranked countries in PISA has mathematical problem solving at the centre of their curriculum framework. In Singapore teachers are highly valued 

Those pushing explicit instruction,do not recognise that the literature doesn’t support its use in mathematics education. It’s either commentary or uses literature focused on research outside the field of mathematics education (e.g., literacy in the early years) and is not drawing on other mathematics education research literature. Other research is in very specific situations, such as students with some specific disability, or where the ‘thing’ being learned is very narrow.

The language used to describe various pedagogical approaches from general to specific matters. Advocates of explicit instruction or explicit teaching often state this should be the main (or only) approach used by teachers and often incorrectly infer it is the only evidence-based approach. Definitions of explicit teaching vary, as do its implementations. Importantly, the approach to explicit teaching and its effectiveness will depend on the discipline and specific focus in question. 

Comparing the pair

Explicit teaching is typically described as teacher-centred. A lesson based on this ideology begins with the teacher presenting their understanding of the lesson focus, followed by an explanation of important ideas, and a demonstration of how to do  particular examples. Students then work on similar ‘tasks’ with teacher support reducing over time as students demonstrate they are able to achieve success independently. Such lessons conclude with the teacher highlighting the important ideas from the lesson. 

Alternative approaches, where students investigate or inquire into mathematical and real-world problems  are typically described as student-centred. A lesson based on this ideology typically begins by considering a real-world situation or mathematical context that demands exploration and application of prior mathematical and/or real-world knowledge and problem-solving processes. As is often the case in social settings (including workplaces), students are encouraged to work on the task both independently and in small groups. The skilful teacher then draws on their planning and observations of students’ learning to orchestrate discussion whereby key ideas and thinking strategies are shared and evaluated by the class. This too, is explicit teaching… but the enactment allows for greater student agency and voice. This interactive, cyclical process might be repeated several times as students are supported to solve the problem.

Is it simply a matter of “Teachers, choose your pedagogy!”?

No. Australia is a low-equity education system. This means our classrooms are highly diverse. The idea that there is one best way to teach all students is not evidence-based and warrants scrutiny. Making judgments on how to teach students well relies on professional knowledge of the school, the students, the curriculum, and the real-world contexts that are important for students to learn about. Planning for student learning, and teaching effectively in the moment, are skills that teachers develop through their initial teaching qualification(s) and practice over the course of their careers. A skilful teacher will adopt a balance of teacher and student-centred approaches, depending on what the learning focus for the day calls for. 

Teaching and learning is complex. Thus, there is no one way for teachers to act in every classroom irrespective ot school type (e.g., mainstream, special education), Year level (F-12 and beyond), discipline in focus (e.g., mathematics), time of year, and even time in a lesson sequence or unit of work.

Once ‘something’ is learned it can be challenging to consider how to best teach that ‘something’ to others. This is why teachers have discipline knowledge, pedagogical knowledge (both general and specific to each discipline they teach) and curricula knowledge. We should value teachers and their knowledge of teaching, initially developed in their University degrees, and developed further as they teach and engage in professional learning – especially that specific to the specific subject and year levels in focus.

How are education systems responding to the debate?

In 2017, the Victorian Government published the High Impact Teaching Strategies, commonly referred to as the HITS. These are based on the work of Hattie’s (2009) meta-analysis of over 800 studies, his 2012 book and work from Marzano (2017). A meta-analysis is a synthesis of many different studies across levels of schooling (early childhood, primary, secondary and tertiary), types of schooling (e.g., mainstream schools, special education) and discipline areas (e.g., English, Mathematics). Hattie’s approach thus aggregates findings from many studies together. This ‘averaging’ approach can be criticised but the top ten strategies are unsurprisingly part of every teacher’s set of competencies. 

Explicit teaching (following Hattie, 2009) is one of the 10 high impact teaching  strategies or instructional practices presented. An argument is made that all 10 HITS: Setting goals, structuring lessons, explicit teaching, worked examples, collaborative learning, multiple exposures, questioning, feedback, metacognitive strategies and differentiated teaching should be part of a teacher’s practice. Some of these practices are described using different terminology elsewhere. Importantly, the HITS are seen as being used alongside other effective strategies by teachers. 

However, in different jurisdictions explicit teaching is presented as ‘all encompassing’  or all central to other more specific strategies including questioning, feedback, connections.

Questioning

If we think about questioning – an essential pedagogical approach in every discipline and Year level, and which all teachers would aim and plan to be effective – different questions have different purposes. The importance a teacher gives to the students’ response can vary greatly. Most secondary mathematics pre-service students would read an article such as Questioning our patterns of questioning to develop an understanding of different patterns of interactions (initiation-response-feedback, funnelling or focusing). In planning and in-the-moment in the lesson, a teacher selects the interaction type depending on the specific focus for learning at that point in the lesson: mainly providing feedback (IRF), or funnelling students to use a specific strategy, or helping students’ articulate their current thinking. Teachers ask important planned questions and respond to student input in ways related to the learning focus.

Aiming for methods that make sense

Any discussion about teaching must be specific to what is intended to be learned by students. Otherwise too much is open to interpretation.

We should be aiming for methods that are understood and make sense to students – these won’t be forgotten in the longer term. Teaching needs to focus on learning opportunities that persist beyond the short term.

Those who expect learning to be evident immediately do not understand what it means to learn or to understand. Learning  is an ongoing process.

Two examples from within mathematics education are included here. Anthony and Hunter’s (2009) review of the characteristics of effective teaching of mathematics discussed explicit language instruction and explicit strategies for communicating mathematics (explaining and justifying) but did not report evidence for explicit teaching as effective teaching of mathematics. Discussing research-informed strategies for teaching mathematics,  Sullivan notes that if explicit instruction is taken to be “drill-orientated approaches, with the teacher doing most of the talking” and mathematical thinking, then this is not conducive to student engagement nor motivation to learn. 

If we look at the curriculum teachers are implementing, it is very clear in the Australian curriculum, both recent and current, that explicit instruction alone will not provide opportunities for students to meet the expectations of the general capabilities, cross-curriculum priorities, nor of specific disciplines (especially mathematics).

The first aim

According to the Australian Curriculum V9.0, the first aim of Mathematics is to: “ensure that students become confident, proficient and effective users and communicators of mathematics, who can investigate, represent and interpret situations in their personal and work lives, think critically, and make choices as active, engaged, numerate citizens.”

This cannot be achieved without students engaged in decision-making about their own learning. Equally, the proficiencies and processes that underpin the mathematics  curriculum cannot be learned solely via explicit instruction.

The school classroom, the people ‘doing mathematics’ should be the learners, not the teachers, hence the term ‘student-centred’. Teachers do their mathematics in preparation for class. Mathematics teachers need to use varied pedagogies, both planned and in the moment.

Irrespective of definitions, teachers plan for effective teaching and have specific learning goals in mind. As a lesson unfolds, teachers make decisions – based on their planning – and use a variety of pedagogical strategies to maximise learning opportunities for all students. All teachers have the learning at the centre of their planning. In the classroom, the teacher should be empowered to make decisions about pedagogy based on their teachers education, prior classroom experiences, the curriculum, and professional learning (especially that focused on knowing how students learn particular ideas in a discipline.

Complex and nuanced

Teaching and learning is complex and nuanced. Thus, there is no one way for teachers to act in every classroom irrespective ot school type (e.g., mainstream, special education), Year level (F-12 and beyond), discipline in focus (e.g., mathematics), time of year, and even time in a lesson sequence or unit of work.

Once ‘something’ is learned it can be challenging to consider how to best teach that ‘something’ to others. This is why teachers have discipline knowledge, pedagogical knowledge (both general and specific to each discipline they teach) and curricula knowledge. 

We should value teachers and their knowledge of teaching, initially developed in their University degrees, and developed further as they teach and engage in professional learning – especially that specific to the specific subject and year levels in focus.

Dr Jill Brown is an Associate Professor in Mathematics Education at Deakin University. She has been working in teacher education for two decades with preservice and inservice secondary, primary and early childhood teachers of mathematics.Jill is internationally recognised for her research in the field of mathematics education. She has an impressive list of publications that focus on mathematical modelling, the teaching and learning of functions, and the use of digital technologies by teachers and students.

What to do when our schools run out of the teachers they urgently need


When faced with a teacher shortage, often schools need to ‘make do’ and ask teachers to teach away from their area of expertise in order to staff classes. That’s called teaching out-of-field and sometimes teachers, put in that position, can feel unsupported and overwhelmed.

The issue of teaching ‘out-of-field’ persists in Australia and internationally –  and has for some time. Out-of-field teaching refers to when teachers teach subjects they are not ‘qualified’ (or specialised) to teach (Weldon, 2016; Hobbs), that is, they do not have the undergraduate study recognised by the state registration/accreditation body nor the teaching methods. 

The question of suitability of a teacher to teach a subject or group of students can be a tricky one in schools. Quality teaching can occur when teachers have gathered expertise over time to teach a subject even when they do not have the relevant ‘qualification’. However, teachers with qualifications or specialisations and a background in the latest teaching methods for the subject are more likely to provide quality teaching.

Faced with an inadequate teacher supply, how do schools address this problem?

In Victoria, the State Government has committed funds to the Secondary Science and Mathematics Initiative (SMSI). We at Deakin have been contracted to design and deliver graduate certificates in secondary mathematics and science. The courses are being delivered online because of COVID restrictions through intensives and offer a mix of content and discipline-specific pedagogy. Supports are provided for teachers as they are challenged to return to study and complete the course while continuing to teach in their schools.  

Teachers from Government schools who are teaching out-of-field in mathematics or science are funded to undertake the graduate certificates. This ‘upskills’ them, makes them qualified, and therefore no longer ‘out-of-field’ but ‘in-field’. 

Why are upskilling programs like the Victorian SMSI important?

1. Research shows that teaching is a ‘learning profession’, where teachers are constantly undergoing professional development, often want to be challenged to try new things by learning ‘on-the-job’ and want to have some agency as to how their career progresses (Hobbs, 2020). Research also shows that teachers who have a background in a subject often lead to better outcomes for their students (Shah, Richardson & Watt, 2020). Some research has shown quite negative impacts for some students and teachers when teachers are given teaching duties beyond their fields of expertise (Du Plessis, Gillies & Carrol, 2014). 

Teachers can feel as if they are in a holding pattern until they can teach what they are passionate about (Hobbs, 2020) and teacher confidence and expertise can be challenged. Students can feel unsupported, and student achievement can be negatively impacted.    

2. Teachers generally feel valued when they are remunerated and recognised for professional learning (Hobbs & Törner, 2019). It is essential funded Government initiatives, university programs, or subject association initiatives deliver outcomes for teachers and schools that make a difference in the classroom and in the professional lives of teachers. The SMSI will focus on contemporary science or mathematics pedagogy, knowledge and practice as integrated, and will be firmly based in teacher practice. The design of the courses and funding arrangements acknowledge the busy lives of teachers and attempts to support schools as they release their teachers. 

3. Upskilling programs specifically designed for out-of-field teachers are not common, although they are available at some universities (e.g., University of Melbourne [https://handbook.unimelb.edu.au/2021/courses/gc-mthed10], Queensland University of Technology [https://www.qut.edu.au/courses/graduate-certificate-in-education-stem-in-education]) and sometimes through professional associations as professional development programs. 

Often teachers receive no recognition or renumeration for undertaking additional qualifications. Therefore, there is little culture of formally upskilling by teachers in some states and territories. Also, research shows that teachers tend to prefer to undertake professional development in their in-field subject (Hobbs, Campbell, Delaney, Speldewinde and Lai, 2020).

The New South Wales teacher accreditation system is such that teachers gain approval to teach subjects when graduating from initial teacher education. These approvals can be updated as teachers undertake studies and meet the requirements for additional subjects. Victoria, however, has no similar mechanism as teachers are registered as ‘teachers’. As with other states and territories in Australia, teachers are required to undertake professional development to renew their registration, although most teachers will choose their in-field subject as the focus of this development. Thus, the value of the SMSI is that funding is provided for teachers to undertake the Graduate Certificate and schools are renumerated for having their teachers out of the classroom while studying. This incentive is needed for teachers to see that the benefits outweigh the costs.

How will a program like SMSI create change?

Two ways. 

  1. The first relates to the fact that the Victorian State Government (like the Tasmanian State Government in 2015) is funding teachers (especially from rural areas) to gain qualifications in out-of-field subjects. This illustrates that there is formal acknowledgement that out-of-field teaching occurs, that it needs to be attended to, and that schools and teachers need to be supported through funding in order to build the pedagogical and content-related expertise. The Victorian Government has applied this strategy with the STEM Catalyst program and the Primary Mathematics and Science Specialists (PMSS) program, illustrating a commitment to upskilling teachers in the ‘STEM’ areas [https://www.education.vic.gov.au/about/programs/learningdev/vicstem/Pages/schools.aspx].
  1. Secondly, such acknowledgement and commitment will have the effect of generating conversation around the ‘out-of-field’ issue more broadly. Whole of system engagement is required, including those responsible for setting policy, school leaders who enact policy, teacher/discipline/principal associations that inform policy and curriculum.  Additionally, teacher unions that represent and protect the rights of educators and school leaders, and universities and academics who provide teacher-ready candidates and support teachers and schools with professional development and research must be engaged.

Change can be created through a national conversation about:

  • system pressures and mechanisms for responding to the issue of teacher distribution, teacher supply and school leadership practices that lead to out-of-field teaching;
  • our expectations for our teachers and schools in terms to teacher ‘qualifications’ versus ‘experience’;
  • developing school practices that minimise the need for out-of-field teaching, and assesses and reduces the potential risk implicated in teaching out-of-field; 
  • how to present this issue to the public; and
  • the data needed to monitor who is teaching what and under what circumstances.

Ultimately, upskilling teachers is one response. The challenge now is for all relevant stakeholders to work together to develop strategies with coordinated actions that demonstrate how Victoria and Australia can lead the world in responding to this pervasive issue. 

Declaration: Deakin University has been contracted to provide the Graduate Certificates in science [https://www.deakin.edu.au/course/graduate-certificate-secondary-science] and mathematics  [https://www.deakin.edu.au/course/graduate-certificate-secondary-mathematics ] and will do so alongside a rich research program that will evaluate the participating teachers’ experiences throughout their qualification.

From left: Associate Professor Linda Hobbs is a Science and STEM educator and has researched in the area of out-of-field teaching for over 12 years. Professor Russell Tytler is Alfred Deakin Professor of Science Education at Deakin University, and has published widely on student and teacher learning, and interdisciplinarity in STEM. Dr Peta White is a science and environmental education senior lecturer at Deakin University with research interests including science and biology education; sustainability, climate change, and environmental education; and collaborative/activist research. Dr Jill Brown is a mathematics educator and researcher and Course Director of the Graduate Certificate Secondary Mathematics

References

Du Plessis, A. E., Gillies, R. M., & Carroll, A. (2014). Out-of-field teaching and professional development: A transnational investigation across Australia and South Africa. International Journal of Educational Research, 66, 90-102. https://www.sciencedirect.com/science/article/abs/pii/S0883035514000457 

Hobbs, L. (2020). Learning to teach science out-of-field: A spatial-temporal experience. Journal of Science Teacher Education. Published online 29 Jan 2020 https://doi.org/10.1080/1046560X.2020.1718315 

Hobbs, L. & Quinn, F. (2020). Out-of-field teachers as learners: Influences on teacher perceived capacity and enjoyment over time. European Journal of Teacher Education, DOI: 10.1080/02619768.2020.1806230  Published online: 01 Sep 2020. https://www.tandfonline.com/doi/abs/10.1080/02619768.2020.1806230 

Hobbs, L. & Törner, G. (2019b). The out-of-field phenomenon: Synthesis and taking action. In L. Hobbs & G. Törner (Eds.), Examining the Phenomenon of “Teaching Out-of-field”: International Perspectives on Teaching as a Non-specialist (pp. 309-321). Dordrecht: Springer. https://www.springer.com/gp/book/9789811333651 

Shah, C., Richardson, P., Watt, H. (2020). Teaching ‘out of field’ in STEM subjects in Australia: Evidence from PISA 2015, GLO Discussion Paper, No. 511, Global Labor Organization (GLO), Essen. http://hdl.handle.net/10419/215639