maths teaching Archives

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.

Lazy, crazy mathematicians and other myths we need to bust

Creating opportunities for students to develop ‘healthy’ images of mathematicians and mathematics is paramount. The images of mathematics or mathematicians that students hold have a huge impact on their learning outcomes. For example, the perceived negative image of mathematicians by students could result in unhappiness in mathematics classrooms or a loathing of mathematics (Hatisaru & Murphy, 2019).

Why is it important? Maths matters because it impacts life quality, income and national development.

Since 2009, I have aimed to understand school students’ images of mathematics and mathematicians (Hatisaru, 2020). What views do they have about mathematicians and their work? What are the connections between students’ views about mathematicians and their attitudes towards mathematics? What views do they have about the needs for mathematics? How do they perceive their mathematics classroom?

Students’ images of mathematicians and mathematics are developed throughout years and impacted by several different factors. From the investigations of myself and others it is clear that experiences in mathematics classrooms contribute to students’ perceptions. Other factors include representations in media and popular culture, and family or society related factors.

For example, Wilson and Latterell (2001) found that in movies, literature, comics, and music mathematicians are portrayed as insane, socially inept. Darragh (2018) too.

Ucar et al. (2010) examined the image of mathematicians held by a group of 19 elementary school students and observed that the students described mathematicians as ‘unsocial, lonely, angry, quiet who always work with numbers’. (p. 131).

Picker and Berry (2000) introduce a cycle of the perpetuation of stereotypical images of mathematics and mathematicians (for example ‘mathematicians are weird’ or ‘mathematicians are asocial people’). According to them, this cycle begins with exposition of different cultural and societal stereotypes via TV, cartoons, books, other media, also via peers and adults through negative repeating phrases.

Among students there is a dominant male perception of mathematicians (e.g., Aguilar et al. 2016; Picker & Berry 2000). In Picker and Berry’s study, which included participants from 5 different countries, students sometimes associated negative or aggressive behaviours to mathematicians such as being large authority ﬁgures, crazy men, or having some special power.

Darragh (2018) examined 59 young adult fiction books to identify the depiction of school mathematics in them. Mathematics was more commonly portrayedto be “nightmarish; inherently difficult; something to be avoided: …”

“Mathematics teachers in particular bore the brunt of negative portrayals and were depicted as ridiculous, sinister, insane, and even dispensable; in short, they were positioned as villains.”

Students then meet teachers who lack awareness of stereotypes of mathematics and mathematicians, and sometimes they themselves hold certain stereotypes. Through teachers and the media, students are affected by certain attitudes such as ‘they must be quick at mathematics to be good at it’, or ‘mathematicians are a privileged group who have the special ability to do mathematics’. These messages and others, according to Picker and Berry, contribute to the formulation of the perceptions of mathematics and mathematicians in students’ minds. Jo Boaler, too, indicates that in her writings on the (important) role of holding a Growth Mindset in mathematics.

Over time, Picker and Berry continue, students develop attitudes and belief systems towards mathematics and mathematicians that may lead to generalisations or stereotypes. The cycle completes with the exchanging of students’ views with others. As a part of society, each student now contributes to others’ images of mathematicians and mathematics.

Given that some students hold negative images of mathematicians and mathematics, and their images are impacted by school-related factors, it is important that school educators are aware of student images.

For about four years now, in my interactions with schoolteachers in several different conferences, workshops, and professional learning events, I have noticed that some teachers use the phrase ‘Since mathematicians are lazy …’ often when they introduce some ‘short-cut’ methods or procedures to their students. Once, for example, the context was solving the problem: 27 + 28 + 13 = ? The teacher’s language practice was: ‘Since mathematicians are lazy, they add 27 to 13 first, which is 40, and then add 28 which gives 68’.

In fact, the mathematical behaviour behind this solution is ‘efficiency’ (Cirillon & Eisenmann, 2011) rather than ‘laziness’. The mentioned ‘lazy mathematicians’ know that, according to the associative property, 27 + 28 + 13 = (27 + 28) + 13 = (27 + 13) + 28. In this case, adding 27 to 13 first is a lot easier than adding 27 to 28 as 7 and 3 makes 10.

Using this property for solving a problem such as 138 + 44 + 12 + 6 = ? makes the calculations even easier: adding 138 to 12 gives 150, and adding 44 to 6 gives 50. The sum of 150 and 50 is 200. Once again, the reason for mathematicians’ desire to use these approaches is not ‘laziness’ but their desire for ‘efficiency’. They also see mathematics as a connected body of knowledge. That is, they use the same property in solving algebra problems (e.g., 17x + 21y + 43x + 19y = (17x + 43x) + (21y + 19y) = 60x + 40y).

In the short term, ‘mathematicians are lazy’ types of messages may appeal to students, but in the long term, they may contribute to the development of (negative) stereotypical images of mathematicians in students. It may prevent students from ‘seeing’ the reasons behind mathematical procedures, and the beauty and connectedness in mathematical ideas. Furthermore, they are morally wrong: Are mathematicians really ‘lazy’? Have we met all mathematicians? Have we measured their relevant attitudes? Were they found to be ‘lazy’ based on those measurements?

While we cannot control messages in the media or popular culture, as also Cirillon and Eisenmann tell us, we could carry and share best messages with our students. My suggestion to schoolteachers, and all other actors in mathematics education including parents and family members, is that we use alternative phrases. Why not use: ‘Since mathematicians are creative …’, ‘Since mathematicians seek to find alternative approaches …’, or ‘Since mathematicians desire to use more efficient ways …’.

These messages are not only more representative and morally more appropriate, but they also have more value in developing images of mathematicians and mathematics in students that are closer to the reality. Moreover, they could contribute to establishing ‘healthier’ relationships between students and mathematicians and mathematics.

Vesife Hatisaru MEdB, MEdM, PhD, MEdD is a lecturer in Mathematics Education (Secondary) in the School of Education, Edith Cowan University Joondalup, and an adjunct senior reseacher in the School of Education, University of Tasmania. She had a long career as a secondary school mathematics teacher before entering academia.