1. Introduction

The beliefs of teachers have been identified for some time as a challenge for the reform of mathematics instruction (Stipek, Givvin, Salmon, & MacGyvers, 2001). These beliefs are partly subject specific, related to the nature of school maths (Correa, Perry, Sims, Miller, & Fang, 2008). This study contributes to the international body of work by focusing on a small group of teacher researchers working in schools in England that are implementing a mastery maths text book based scheme informed by a Singapore Maths approach. The Singapore Maths approach was developed by the Singaporean Ministry of Education during the 1980s in a scheme that depends on use of text books and includes aspects of a mastery approach, for example by spending more time to investigate each topic in depth. In our study we do not seek to associate raised pupil attainment with a mastery approach, rather we investigate the complexity of change in classroom practice and of teacher beliefs within the context of adopting such an approach to maths. The commercial scheme, including textbooks, pupil work books and teacher guidance materials, is based on a scheme used in Singapore but has been amended for use in England and is entitled Maths - No Problem!™.

Previous research has investigated mathematical subject knowledge and beginning teachers' beliefs (Cooney, 1985; Paolucci, 2015). In this study the focus is on experienced teachers' underpinning beliefs during implementation of a practical mathematics scheme engaging them in new pedagogical knowledge as well as new mathematics subject knowledge. The seven teacher researchers contributing to this study were in years two and three of a wider and sustained curriculum development project that included three initial workshops, classroom experimentation, supportive classroom observation with coaching on two or three occasions, and primarily provision of the textbooks, workbooks and teacher guides that form the Maths - No Problem!™ scheme. Teachers may have different conceptions of mathematics as a discipline and of ‘school maths’. It seems likely that their personal experience of schooling, as a pupil and then as a teacher, will be a strong influence alongside any formal higher education experience of mathematics they may have gained during an undergraduate degree or teacher education programme (Beswick, 2012). Teacher beliefs are a significant influence on their classroom practice and are relatively difficult to change despite the efforts of teacher educators and policy makers (Meirink, Verloop, & Bergen, 2009). It is important to consider individual teacher beliefs, for example in the way that might involve resistance to reform approaches to teaching, but also to consider the influence of teacher discourses and professional relationships on their beliefs (Priestley, Biesta, & Robinson, 2015). This study will consider beliefs to be on the same dimension as knowledge: ‘Those things we ‘‘more than believe’’ we refer to as knowledge and those things we ‘‘just believe’’ we refer to as beliefs' (Leatham, 2006, p. 92). The study adopts a collaborative practitioner research approach to ask the question: What are the self-reported pedagogical beliefs of teachers during the implementation of a mastery maths curriculum development project?

2. School maths

In relation to this study, it is important to consider how mathematics as a subject discipline is transformed to become ‘school maths’ (Bernstein, 2000). Bernstein identified three areas of rules by which knowledge is transformed from its site of production, for example by researchers in the university, to the classroom, to become the content of lessons in schools. These three areas consist of: distributive rules, related to knowledge production; recontextualising rules, related to official curriculum policy and the local pedagogical influence of teachers; and evaluative rules, related to reproduction of knowledge by pupils in classrooms, tests and examinations (Bernstein, 2000; Puttick, 2015). With regards to mathematics, this process appears to be influenced by beliefs held about the contested nature of the subject itself.

Lakatos (1976) identifies two contrasting perspectives. On one hand, influenced by Euclidean methodology with its deductivist style, mathematics may be conceived as a set of eternal, immutable truths. This mathematics may be transformed (Bernstein, 2000) into a form of school mathematics that is associated with a pedagogy dominated by teacher demonstration followed by individual practice and high stakes testing with strict rules and right or wrong answers (Hudson, Henderson, & Hudson, 2015). On the other hand, mathematics may be seen as a human activity, and it is this human mathematical activity that produces mathematics – referred to as ‘mathematics in the making’ by Polya (1957, p. xxxii). This mathematics may be transformed into a school mathematics that presents the subject as ‘fallible, refutable, and uncertain and which promotes critical thinking, creative reasoning, the generation of multiple solutions and of learning from errors and mistakes’ (Hudson et al., 2015, p. 377).

Within British and American culture this transformation of mathematics into a school subject can be seen to be significantly influenced by two inter-related and damaging myths concerning mathematics: first that only some people, due to natural talent, can do mathematics; and second that being good at mathematics is a clear sign that you are one of the most intelligent people (Boaler, 2016). The influence of these myths on children is revealed in studies of their disposition towards maths based on the language they use (Mazzocco, Hanich, & Noeder, 2012) and their level of anxiety around maths (Maloney, & Beilock, 2012; Ramirez, Chang, Maloney, Levine, & Beilock, 2016).

Perhaps as a result of these cultural myths, maths teaching in schools in Western cultures tends to consist of teacher demonstration and telling, followed by individual practice. Attempts to move towards a more constructivist, active learning approach in mathematics, one that might contribute somewhat to mathematics being more like Polya (1957, p. xxxii) ‘mathematics in the making’, have struggled to become embedded in schools (Hudson et al., 2015).

There are a number of important factors to consider as to why this may be the case. In Primary schools in England, for children aged 4–11 years old, the teachers generally teach a class across the curriculum, they are not mathematics specialists. Another significant contextual factor is that there is a high accountability policy framework in England (Ball, 2013) including a detailed National Curriculum with high stakes school inspections. The individual educational and workplace histories of teachers also influence their beliefs about maths and how to teach it as a school subject (Rogers, Cross, Gresalfi, Trauth-Nare, & Buck, 2011). Previous major reform influencing the teaching of mathematics in Primary schools in England has highlighted the significance of teacher identity and the need for professional learning to explicitly engage with this element of becoming a mathematics teacher (Brown & McNamara, 2011). In particular, as they develop their identity as teachers of mathematics during engagement in a curriculum development project, teachers are likely to change their criteria for successful learning in important ways, especially if they have been helped to focus on the development of children's mathematical thinking (Gabriele & Joram, 2007). Teachers may feel most effective when they teach mathematics by ‘telling’ and this is connected to the belief that the answers to mathematical problems are in books with the teacher acting as interpreter (Smith, 1996, p. 391). Alternatively, Cognitively Guided Instruction (CGI) emphasises teachers gaining an increased understanding of children's development of mathematical thinking during problem-solving (Fennema et al., 1996). Working with CGI, Moscardini (2014) argues for a view of pedagogy that includes underpinning values and beliefs and he highlights in particular values related to inclusion.

3. Singapore mathematics

Many schools in England are adopting mastery approaches to maths teaching influenced by South Asian approaches. The approaches vary somewhat but share some common characteristics. The whole class, of mixed prior attainment, moves at broadly the same pace through a maths curriculum that is not too content heavy. Lessons usually begin with whole class engagement with a contextualised problem. Collaborative work includes considerable use of concrete materials, manipulatives, and dialogue around possible solutions. The tasks are carefully selected and there is an emphasis on mathematical variation and connections. In line with early thinking on mastery learning there is an underpinning assumption that, under appropriate instructional conditions, virtually all students can and will learn most of what they are taught (Block & Anderson, 1975). It is important to note that Singapore Maths is strongly influenced by international theory and research evidence. For example, a key principle of the Singapore approach is known as the Concrete-Pictorial-Abstract heuristic (Yew Hoong et al., 2015) which is based on Bruner's enactive, iconic and symbolic modes of representation (Bruner, 1966). The Maths – No Problem!™ text book scheme is based on a scheme approved in Singapore and adapted for use in England. Compared to widespread practice in England the Singapore approach places less emphasis on differentiation by task or content and largely avoids in-class grouping by prior attainment (Micklewright et al., 2014). It is both a strength but also a limitation of this study, that it focuses only on teachers using Maths - No Problem!™

In a large-scale study of primary schools in England using a mastery approach influenced by Singapore Maths, a modest impact on learning was identified (EEF, 2015; Jerrim and Vignoles, 2016). However, importing strategies from other cultures is not straightforward. A useful study investigated the maths achievement of western born children of East Asian descent living in Australia and showed that they performed more strongly in maths no matter what teaching strategies were used in their schools (Jerrim, 2015). Alongside this, Wong et al. (2009) suggest that, due to the unique cultural, political, religious and racial composition of Singapore, their approach to mathematics education is unable to be emulated in other countries. Despite this, it can be seen that many aspects of the approach are based upon seminal research and theory from around the world as well as being inspired by the British Cockroft report (Cockroft, 1982; Wong et al., 2009). In this study we investigate teachers' beliefs as they implement the Maths - No Problem!™ textbook scheme within a sustained curriculum development project. The textbook scheme includes student workbooks and web-based teacher guidance materials to accompany the textbooks.

In understanding classroom practice in school maths in England it is important to take account of these strong and interwoven cultural, educational, social, commercial and even political influences as they help to shape the classroom practice of a teacher and their interactions with individual children.

4. Mindset and mathematics

In social-cognitive theory, an individual's sense of self-efficacy, their belief that they can have some control over aspects of their life to achieve particular performances, is a major influence on their personal agency (Bandura, 1986) and has an impact on motivation and academic performance (Schunk, 1991). Developing manageable goals and experiencing mastery by achieving them is motivating and helps to develop self-efficacy (Bandura, 1997). There has been a considerable focus on self-efficacy in maths showing that performance, for example in problem-solving, is strongly influenced by self-efficacy because the individual's beliefs affect the way they use their existing knowledge and skills (Michaelides, 2008; Pajares & Kranzler, 1995). Self-efficacy in maths is generally measured by asking research participants, using a Likert type scale, to judge their confidence in solving particular maths problems. There are key principles around specificity in measuring self-efficacy by relating it closely to particular tasks (Michaelides, 2008).

Related to the broader concept and body of work on self-efficacy, Carol Dweck and colleagues have focused on beliefs around fixed and malleable intelligence and developed mindset theory (Dweck, 1999; 2006). Mindset varies along a dimension characterised by fixed or growth mindset and is measured using responses to a Likert type scale focused on beliefs about intelligence (Dweck, 1999). A growth mindset is a belief that the harder you work, the smarter you get. It may be explained in an accessible way to learners by considering the brain to be a muscle, the more you exercise it, the stronger it becomes. The reason for explaining mindset to learners is that it is possible to nurture the development of a growth mindset (Dweck, 1999, 2006). One of the weaknesses of much mindset theory research is that even when it does focus on beliefs around intelligence within the context of schooling it is often generic, meaning it is across the curriculum. There is a need from more research on domain specific mindset, for example in particular curriculum subjects such as maths. Even Mindset Theory studies that are set within the specific domain of maths will often evaluate the impact of a generic Mindset training intervention, rather than the impact of a reform maths approach that may influence Mindset (Blackwell, Trzesniewski, & Dweck, 2007). In this study we are interested in the possibility of ‘situated mindset’, meaning teacher or pupil belief about intelligence within the context of school maths (Boaler, 2016) rather than a more general mindset (Dweck, 1999, 2006; Hymer & Gershon, 2014).

In England, projects such as ‘Learning Without Limits’ have demonstrated the need to work initially with teachers and other adults working in a school in order to challenge underlying beliefs and language before considering the impact of labelling children (Swan et al, 2012). Mindset is related to differences in cultural beliefs, for example in the way individuals in Japan and North America attribute reasons for failure and respond to it (Heine, Kitayama, Lehman, Takata, Ide, & Matsumoto, 2001). Mindset beliefs seem likely to be tangled up with children's experiences of social class and disadvantage and a study in Chile has shown how growth mindset beliefs may help to mitigate the effects of poverty on attainment (Claro, Paunesku, & Dweck, 2016).

A considerable challenge arises in considering self-efficacy and mindset, which is that the teacher's self-efficacy and mindset affects their classroom behaviours and has considerable influence on the beliefs of their students (Tschannen-Moran and Woolfolk Hoy, 2001). In the specific context of school maths, studies have shown how teacher beliefs influence their design and facilitation choices in the classroom (Cross, 2009). Levels of teacher enthusiasm and self-efficacy are related to classrooms in which children are more likely to adopt mastery-oriented goals (Lazarides, Buchholz, & Rubach, 2018) and these are the kinds of classrooms that are likely to nurture children's development of growth mindset. The significance of teacher beliefs is further complicated by the cultural dimension of self-efficacy and this is relevant to approaches such as Singapore Maths because studies have shown differences in the beliefs, for example, of Asian origin students compared to non-immigrants living in western countries (Jerrim, 2015; Klassen, 2004).

5. Grouping by prior attainment

Research review indicates that grouping students based on prior attainment involves a wide range of practices and that the impact of grouping is complex (Kutnick, Sebba, Blatchford, Galton, & Thorp, 2005). This research review highlighted that approaches to in-class grouping may need to vary depending on age and curricular area. In-class grouping by prior attainment, particularly in mathematics, has been widespread across Primary schools in England. A more recent review clarifies the position that overall, setting or streaming by prior attainment does not have a significant impact on levels of attainment except for a negative impact on lower sets and streams (EEF, 2015).

Unfortunately, this generally agreed position on the impact of grouping has not been taken on board by policy makers in England. This has arguably influenced how schools, school inspectors, teachers and even the general public, consider grouping by prior attainment (which in schools in England is generally and misleadingly referred to as grouping by ‘ability’). This situation has implications for social justice and it is possible to identify at least seven key problems that may cause low attainment by students allocated to low sets or streams (Francis et al., 2017):