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The balancing act: Maths teaching that is knowledge-rich and child-centred

Written by: Andy Ash
8 min read
ANDY ASH, MATHS LEAD, HOLY FAMILY MULTI-ACADEMY TRUST, UK

The national context: Knowledge-rich curriculum and teaching for mastery

Over the past 10 years in England, there has been a shift in national education policy towards what is termed a ‘knowledge-rich’ curriculum (DfE, 2013; Gibb, 2021). This terminology is used to describe a curriculum centred on knowledge acquisition, and has been strongly associated with so-called ‘teacher-led instruction’, which is contrasted with ‘child-centred’ teaching (Gibb, 2017). Debates about these two views have been ongoing for decades and are well reported elsewhere, but some argue that it is a false dichotomy. In this article, I will report on an aspect of my recently completed doctoral research, where I found myself reflecting on this issue in relation to primary school maths. I take ‘child-centred’ teaching to refer to teaching that amplifies the importance of pupil attributes, such as autonomy and collaboration. In addition to this, I take ‘curriculum’ to refer to formal documentation, teacher planning and the lived experience of students and teachers. Therefore, a ‘knowledge-rich’ curriculum is a much broader concept than simply the content outlined in policy documents. It is often the beliefs, knowledge and practices of teachers that influence the treatment of knowledge more than any other curriculum element. In short, a knowledge-rich curriculum is only knowledge-rich if teachers make it so. Through sharing some of my findings, I aim to avoid the dichotomisation of teaching into an ‘either–or’ debate about teaching approaches, and show how a careful balancing act can lead to a genuinely knowledge-rich experience for students.

In England, recent curriculum reform has included the development of a government-funded ‘Maths Hub’ network, which has been working to reform school maths through the implementation of ‘teaching for mastery’ (NCETM, 2023). One of the key practices within teaching for mastery is the use of representations. In mathematics, defining a ‘representation’ is complex, but for our purposes it can be simply understood as a way of making meaning of an abstract idea – for example, the number ‘three’ may be represented in many ways, such as the common Arabic numeral, three fingers or three dots. Therefore, if a knowledge-rich curriculum is desired, the way in which we represent mathematical knowledge is highly important. There is a significant amount of literature about use of representations that leads to positive learning outcomes, which I used to inform my framework in Table 1 (e.g. Carbonneau et al., 2013; Rau and Matthews, 2017), but there is not very much about the types of beliefs and knowledge that teachers need to make this happen. The aim of this article is to shed light on how a knowledge-rich curriculum can be achieved through a careful balancing act between teacher direction, student autonomy and collaboration, when using representations.

My research study

My research involved conducting a case study where I analysed the beliefs, knowledge and practices of one primary school teacher (given the pseudonym ‘Gillian’). I generated data using lesson observations, video-recorded lessons, teacher interviews, textbook analysis and tasks designed to assess subject knowledge. Focusing on one teacher enabled me to dig deep into understanding and explaining how teacher beliefs and knowledge might influence classroom practice. My intention was therefore not to make any global generalisations but to create a working hypothesis. In doing so, I applied ‘legitimation code theory’ (LCT), which is an emerging sociological approach developed by Karl Maton (2014) that has a rapidly growing body of work in education. Born from a frustration in sociological research that ignores or downplays the place of knowledge in society, LCT posits that underlying any social practices are certain organising principles that influence the way in which knowledge is treated. For example, underpinning a teacher’s daily thoughts and actions are certain belief and knowledge structures that influence the experienced curriculum. These structures lead to certain social ‘rules’ becoming the norm, and these are often hidden or subconscious. By trying to understand these in more detail, we can begin to explain what a knowledge-rich curriculum can look like with more clarity and precision. In my study, I took segments of research data and mapped these against elements of LCT. This enabled me to analyse Gillian’s beliefs, knowledge and practice and to develop a working hypothesis about the underpinning structures that lay behind the way in which she used representations.

Representations and knowledge-rich teaching

The focus of my research was investigating teacher beliefs and knowledge and trying to better understand their influence on representation use. In doing this, I conducted a systematic review of the literature and devised a framework for methods of using representations that seem likely to lead to positive learning outcomes, as seen in Table 1.

Table 1: A framework for using representations in ways that are more likely to lead to positive learning outcomes
Uses of representations for positive learning outcomes
Teachers use multiple representations and help students to make connections between them
Representations are used for the purpose of helping students to develop a deep understanding of the abstract concepts of mathematics
Representations are treated as discussion points and the reasons for using them are made explicit
Teachers are clear when making translations between mathematically different representations
Teachers treat representation as a broad concept, acknowledging students’ own perceptions as part of this
Teachers prompt students to verbally reason about representations
Teachers allow opportunities for students to develop their own representations
Representations used have a clear mathematical purpose

According to the literature, if teachers use representations in these ways, then there is a better chance of students developing deep mathematical understanding. In my research, through analysing Gillian’s classroom practice and conducting several interviews and subject knowledge tasks, I ascertained that Gillian had a strong knowledge and belief system that aligned with this representations framework, showing that her use of representations could be described as ‘teacher-led’. Importantly, this was supported by her use of a textbook scheme, which was designed in such a way that it aligned with her own beliefs and knowledge. The significant finding here was that using representations in this way led to Gillian having a strong knowledge focus within her teaching. Because mathematical ideas are abstract, we need representations to help us to grapple with them, and it was Gillian’s precise, teacher-led use of representations that meant that there was also a definite focus on specialised mathematical knowledge in her lessons. She held knowledge acquisition to be a central part of what was important in order for students to succeed in her classroom. In LCT terms, acquiring specialised mathematical knowledge was part of the basis of success for students in her lessons. In this sense, I describe Gillian’s knowledge, beliefs and practices as knowledge-rich.

Knowledge acquisition: Not the whole story

Despite this firm focus on knowledge from Gillian, the LCT analysis also highlighted another aspect of her teaching. She demonstrated a strong belief in mathematical proficiency as being a way of seeing and acting in the world. For her, being a successful learner of school maths required students to act in certain ways as well as acquire specialised knowledge. Analysing her beliefs and knowledge, it was clear that she placed great importance on students developing certain attributes that were less to do with knowledge and more to do with acting in such a way that would support them as learners. This included a belief in school maths needing to reflect the real-life subject of mathematics, as well as students needing to develop autonomy. For Gillian, teaching was a careful balancing act between the acquisition of specialised mathematical knowledge along with social and personal attributes such as resilience, independence, the communication of thinking and collaboration. In her lessons, both the acquisition of specialised knowledge and social learning and personal attributes were the basis of success. In this way, Gillian’s teaching was both child-centred and teacher-led, and this is what helped her to develop a knowledge-rich curriculum. Importantly, there were times where she might downplay the need for specialised knowledge in favour of focusing students on becoming more resilient and independent – this was often towards the start of lessons. Nevertheless, there were also times where the need for any social attributes was downplayed in favour of demonstrating competency with specialised mathematical knowledge – this was often further into her lessons, where there was an emphasis on practice.

What are the implications?

I believe that there are three important things that teachers can take from these findings. First, it is possible for school maths teaching to be knowledge-rich, teacher-led and child-centred. Teachers may consider what social and personal attributes lead to students becoming effective learners of school maths and ensure that these are fostered alongside the acquisition of knowledge. Second, the extent to which teachers make either knowledge acquisition or learner attributes the basis of success in lessons can be adapted, depending on the needs of the learners at any point within a lesson. For example, at the start of a lesson, downplaying the need for specialised knowledge in favour of getting stuck into doing some collaborative mathematical activity may well enable students who struggle to feel ready to learn. On the other hand, once students have had a chance to grapple with the mathematics, later in the lesson teachers may wish to shift the basis of success towards knowledge acquisition, by encouraging students to practise with their newly acquired understanding. Third, the relationship between use of representations and knowledge-rich teaching seems to be a reciprocal one. In my study, Gillian used representations in an effective way, and this led to a knowledge-rich environment in her lessons. Nevertheless, it is important to highlight that this was not an easy job for Gillian. She did refer to tensions, such as preparing pupils for national tests in Year 6, and how at times this made it hard for her to enact the curriculum in the way that I describe here. This is why I call it a ‘careful balancing act’ for teachers.

At a national level, I believe that there are also implications for policymakers. I argue that the pitching of teacher-led, knowledge-rich lessons against child-centred teaching is unhelpful. Such dichotomising could either lead to a backlash, where the importance of knowledge is rejected and teaching succumbs to what Karl Maton refers to as ‘knowledge blindness’ (Maton, 2014, p. 4), or mean that teachers begin to assume that the development of social learning and personal attributes are not important. Neither of these outcomes seem desirable, given that it is possible, as my working hypothesis suggests, for teachers to avoid the dichotomy altogether and develop both in tandem.

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