Mathematical identities and trajectories of secondary school learners Aarifah Gardee 472195 Supervisor: Professor Karin Brodie A thesis submitted to the School of Education, Faculty of the Humanities, University of the Witwatersrand, Johannesburg, in fulfillment of the requirements for the degree of Doctor of Philosophy Johannesburg, January 2022 i Declaration I, Aarifah Gardee declare that this thesis is my own, unaided work. It is being submitted for the degree of Doctor of Philosophy at the University of the Witwatersrand, Johannesburg. It has not been submitted before for any other degree or examination at any other university. Aarifah Gardee 24th day of January 2022 ii Dedication For my loving parents: Yoosuf and Soraya Gardee. May the Almighty show mercy upon you, as you have been merciful to me. And in memory of my beloved grandfathers: Moulana Abdul Rahman Gardee and Mr Ahmed Essop. May the Almighty be please with you and elevate your status among those who are guided. Aameen. iii Acknowledgment I begin in the name of the Almighty, the most gracious, kind and merciful. All praises and thanks are due to Him, the Lord of the worlds, for blessing me and giving me the strength and guidance to conduct and complete my thesis. I would like to sincerely thank my wonderful supervisor, Professor Karin Brodie for her patience, kindness, time, effort, encouragement and wisdom. Thank you for shaping me in becoming a better writer and researcher. Thank you for your interest in my development, not only as a researcher, but as a person. You have been both my role model and teacher and there are no words that can fully express how grateful I am to you for everything. To my beloved parents, Yoosuf and Soraya, no words can adequality describe my gratitude for all the love, care, encouragement and support over all the years. You have made my journey in search of knowledge so easy and I will forever be indebted to you. Thank you for always being there for me and for being proud of me. Thank you for always motivating me to work harder, try my best and never give up. Thank you for everything. To my dearest siblings, Muhammed, Abdur Rahman, Ibrahim and Maseeha, I would like to thank you for all your support and for always being there for me. Thank you for celebrating every PhD milestone with me and for cheering me up when I had my ‘PhD blues’. To my lovely grandmothers, Maryam Gardee and Fareeda Essop, thank you for being so proud of me. A special thank you goes out to both the pilot and the research schools who participated in this research. I would like to thank the principals, teachers and learners for their invaluable assistance and participation. Without their help, this research would have not been possible. My gratitude is extended to my colleagues and every editor and reviewer who read my manuscripts. Thank you for your careful reading of the manuscripts, thought provoking feedback, detailed critical engagement with the texts and helpful suggestions. Your feedback assisted me in deepening my thinking and becoming a better writer and thinker. Finally, I would like to thank the NRF for their funding. The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the NRF. iv Abstract Research on learners’ mathematical identities has become increasingly prominent over the last two decades. Despite this interest, various reviews of research on learners’ mathematical identities argue that there is a need for the concept of identity to be more clearly defined and conceptualised in theoretically coherent ways that contribute to the concept being empirically accessible. I propose the use of critical realism as a theoretical framework, which has not been used before in this field, to define and research learners’ mathematical identities as they are constructed in classrooms, in a non-reductionist manner. Mathematical identity is defined as a social phenomenon, existing in the real world, which emerges from relationships between learner agency, personal identity and social identity, both offered and constructed, but are not reduced to any of these three mechanisms. Building on the theoretical framework, two explanatory frameworks are developed to support the application of the theoretical framework to data. The first explanatory framework relates teachers’ practices with learners’ mathematical identities, and the second explanatory framework relates peer interactions with learners’ mathematical identities. Most of this study focuses on the first explanatory framework to further our understandings of how learners and teachers contribute to the development of learners’ mathematical identities. A small part of the study shows how the second explanatory framework can be used to understand how and why learners construct certain identities in peer interactions. The methodological approach of this study is a qualitative research approach. Data for this study were collected over two years in a technical high school in Johannesburg over two years in the third school term in both years (August - September), for two weeks in 2015 (Grade 9) and four weeks in 2016 (Grade 10). In total, nineteen learners participated in the study over both years, eight in the first year, with an additional eleven in the second year, along with their five mathematics teachers, two in the first year and three in the second year. Data were collected in the form of videotaped lessons, field notes, photographed learner notebooks and audiotaped interviews with participants. Using the first explanatory framework, my analysis of the data points to several important findings, which are: the teachers’ narratives and actual practices, which incorporate teachers’ pedagogical approaches and social relationships with learners are related, and jointly influence the social identities offered to learners; the social identities offered by the teachers, which were informed by how the teachers spoke about errors, the learners and learner v thinking, strongly informed, but did not determine the mathematical identities constructed by learners as learners also played a role in the construction of their mathematical identities; learners were more willing to construct social identities of affiliation with their classroom communities when they felt that their teachers cared about them and others in the classroom; and that learners’ mathematical identities changed, some negligibly and other substantially, over the two-year period in response to changes in teachers. It is recommended that teachers remain conscious of the kinds of opportunities they present to learners to learn mathematics, the kinds of narratives they tell about learners and the types of pedagogical approaches they utilise in the classroom, as these narratives and teacher practices inform the kind of opportunities that teachers provide learners to learn mathematics. vi Table of contents Content Page number Declaration I Dedication ii Acknowledgement iii Abstract iv Chapter 1: Introduction 1 Chapter 2: A framework for learners’ mathematical identities: A critical realist perspective 24 Chapter 3: The influence of teachers on learners’ mathematical identities 51 Chapter 4: Relationships between teachers’ practices and learners’ mathematical identities 60 Chapter 5: Social Relationships between Teachers and Learners, Learners’ Mathematical Identities and Equity 90 Chapter 6: Relationships between teachers’ interactions with learner errors and learners’ mathematical identities 104 Chapter 7: A framework for analysing learners' mathematical identities in peer interactions: accounting for personal and social identities 128 Chapter 8: Conclusion 156 List of Appendices Appendices Page number Appendix A 175 Appendix B 179 Appendix C 183 1 Chapter 1: Introduction 1.1. Background to this study In South Africa, as in other parts of the world, many learners give up studying mathematics when presented with the opportunity to do so, which is unfortunate for various reasons (Andersson, Valero & Meaney, 2015; Darragh, 2013; Umugiraneza, Bansilal, & North, 2017). In our country, mathematics is an important subject to gain access to tertiary education, especially in fields of engineering, accountancy, health sciences and other mathematically orientated disciplines. So mathematics becomes necessary for access to further education and employment opportunities. Dropping mathematics also has epistemological consequences since being proficient in mathematics enables learners to understand the usefulness of mathematics in their daily lives, by developing their reasoning and critical thinking capacities (Brodie, 2010). Treating the study of mathematics as a ‘gate-keeper’ to further education and offering epistemological benefits to learners are somewhat instrumental, viewing the study of mathematics in schools as a means to an end. Perceiving mathematics as instrumental denies learners the opportunity to be “motivated by aspirations of understanding mathematics” (Skovsmose, 2012, p. 5), which involves appreciating mathematics as a body of knowledge, influenced by its own culture and history. In this sense, mathematics is associated with its own ways of working and thinking and can be perceived as being intellectual, sensible and reasonable (Brodie, 2010). The aesthetic appeal of mathematics can influence who we are as learners and doers of mathematics, as well as thinkers more generally. Our current Curriculum and Assessment Policy (CAPS) for Grades 7-9 reflects each of the above-mentioned uses of mathematics. In terms of mathematics being a gate-keeper to tertiary education, CAPS states that a specific aim of mathematics is to “prepare the learners for future education and training as well as the world of work” (DBE, 2011, p. 8). In terms of the epistemological benefit of learning mathematics, CAPS states that mathematics is essential to “enhance logical and critical thinking skills, accuracy and problem solving, that will contribute in decision making” (DBE, 2011, p.8). CAPS also highlights the aesthetic appeal of learning mathematics by stating that an aim of teaching and learning mathematics is to develop “an appreciation for the beauty and elegance of mathematics” and “a spirit of curiosity and a love for mathematics” (DBE, 2011, p.8). 2 Despite the many ways in which mathematics is important as stressed in our curriculum policy, the results of our Grade 9 learners’ performance in the Annual National Assessments (ANA) and the Trends in International Mathematics and Science Study (TIMMS) in mathematics are disheartening. In 2014, the average national percentage for Grade 9 learners who wrote the ANA mathematics assessment was 11% (DBE, 2014). The 2014 results for Grade 9 were worse than those in 2012 and 2013, 13% and 14% respectively. Worse still, the percentage of learners who achieved more than 50% in these assessments in 2014 was 3% (DBE, 2014). An analysis of these results suggested that learners did not possess adequate knowledge regarding the concepts and vocabulary of algebra and geometry (DBE, 2014). The Annual National Assessments have been discontinued since 2015. A more recent TIMMS study in 2019 has shown some improvement in the results of Grade 9 learners in comparison to previous years, yet South Africa is still one of the five poorest performing countries in the world (DBE, 2019). The report states that 59% of all Grade 9 learners who participated in the study have not acquired basic mathematics knowledge (DBE, 2019). Aside from the difficulties experienced with learning mathematics in Grade 9, the 2021-2022 Annual Performance Plan report showed that the highest repetition rate with learners repeating the same grade during the 2018 academic year involved Grade 10 learners (approximately 22% of boys and 16,8% of girls in Grade 10) (DBE, 2021). Research in South Africa illustrates that learners’ socio-economic conditions and a lack of resources influence poor performance in schools (Bayaga et al., 2010; Sailors, Hoffman & Matthee, 2007). However, Graven and Heyd-Metzuyanim (2014) argue that research does not explain why South Africa performs worse in mathematics in international tests such as the TIMSS in comparison to its neighbouring countries which are less wealthy. Success in learning mathematics is not solely dependent on learners’ abilities and socio-economic factors. Learners’ emotions, interest, agency, relationships with their teachers and peers, and how they position themselves as members of their mathematics classroom communities also contribute to how they learn mathematics. To understand learning in its complexity, Darragh (2013) argues that it is necessary to focus on the “whole person and how they become a learner of mathematics” (p.216). To obtain a comprehensive understanding of learning, it is necessary to consider learners’ mathematical identities. 3 1.2. My journey My journey into understanding learners’ mathematical identities began with my Honours degree in 2014, where I investigated how a teacher dealt with errors in her mathematics classroom (Gardee & Brodie, 2015). The theoretical framework informing my study was constructivism, which enabled me to understand how individuals actively construct knowledge, and thereby, construct misconceptions and make errors. When analysing my findings, I realised that merely attributing the cause of learner errors to the individual construction of knowledge was very narrow. The lack of complexity in my research made me question how other mechanisms, such as learners’ social environments, pedagogical experiences, agency and future projections influence how learners make sense of mathematics. For these reasons, I thought that the study of learners’ mathematical identities was most suitable to enable a researcher to understand the learner as a ‘whole’ human being (Darragh, 2013). After all, complex problems, like understanding learning and identity, require complex solutions (Goldacre, 2009). I then embarked upon researching learners’ mathematical identities for my Masters in Education degree in 2015. Since most of the articles that I read were based on situated theory, I decided that a sociological perspective was most suitable to study identities. My Masters study was focused on understanding the identities and learning trajectories of eight Grade 9 learners in two mathematics teachers’ classrooms. Data were collected as videotaped lessons and audiotaped interviews with participants, and I found that there were six dominant mechanisms which influenced learners’ learning and identities: learners’ past and current mathematics teachers, their experiences of marks in mathematics, agency, family support, the transition between primary and secondary school, and learners’ future projections. Among these six mechanisms, the most important mechanism for learners was their current mathematics teachers, involving their pedagogical approaches and the kinds of social relationships shared with learners. When reflecting on the Masters study after the write up of my dissertation, I realised that I was unsure of what ‘identity’ meant. In the Masters study, I used Wenger’s (1998) and Sfard and Prusak’s (2005) definitions of identity as participation and narrative, yet something seemed to be missing in their definitions. Could identity be something so simple as our participation in communities? Or our socially shaped narratives? Or both? It seemed too simplistic to reduce identity to either and I struggled to find an alternative definition in the 4 literature which made sense to me. I also struggled to understand the gravity of my findings, and I even questioned my theoretical framework. I was unsure how all the mechanisms that I found worked together to produce identity. I struggled to understand learner agency, since my current sociological theoretical framework limited my understanding of how learners could exercise agency as motivated by individual forces, rather than social forces. I felt that I was sitting with a huge jigsaw puzzle, staring at a bunch of pieces of the puzzle, but I could not create the picture based on my current understanding of the literature on identity in mathematics education. The process was frustrating! I made multiple attempts to represent the relationships between mechanisms, all unsuccessfully. What was missing, I later realised, was a clear and perhaps realistic (to me at least) philosophical paradigm. During the interval between completing the write up of my Masters dissertation and beginning my doctorate degree, I embarked on what my supervisor would call a ‘me’ search, rather than ‘research’. I asked myself: Am I merely a product of my social environment? Are others able to influence me to learn or not learn so easily? Where is the ‘me’ in all the social discourse? Where is my agency? What about my emotions? My feelings? My motivations? Surely then, learners as individuals must have their own motivations, emotions and aspirations, which would inform how they agentively participate and construct their identities? Despite all my reservations regarding my Masters research, I was pleased when I found out that I passed my Masters dissertation and was provided with the option to upgrade the Masters to a PhD, which I took. Once I began working on the PhD, all of my past concerns regarding my work motivated me to answer all the questions that I had in mind. I decided that the first step was to define identity. 1.3. The problem called “identity” The last two decades have seen what Chronaki (2013) calls the “identity turn” in mathematics educational research, with the construct of identity being increasingly researched and recognized as being influential to understand learners’ learning of mathematics (p. 2). The concept of identity offers valuable insights into how learners engage with mathematics in different classroom contexts (Andersson, et al., 2015; Boaler & Greeno, 2000; Cobb, Gresalfi & Hodge, 2009), their relationships with other significant people in relation to learning mathematics (Heyd-Metzuyanim, 2013, 2015; Sfard & Prusak, 2005), the role of discourse in 5 relation to learners’ learning of mathematics (Bishop, 2012; Heyd-Metzuaynim, 2015; Heyd- Metzuyanim & Sfard, 2012; Wood & Kalinec, 2012; Wood, 2013) and the role of certain social categories, such as race, gender and class, in relation to learners’ learning of mathematics (Nasir & Cobb, 2007; Gholson & Martin, 2019). Yet, the question of what identity is and how the concept can be used in research remains a topic of concern (Darragh, 2016; Graven & Heyd-Metzuyanim, 2019; Radovic, Black, William & Salas, 2018). Over the past few years, there have been several reviews of the concept of mathematical identity aiming to develop suitable definitions of identity (Darragh, 2016; Graven & Heyd- Metzuyanim, 2019; Radovic et al., 2018). Darragh (2016) analysed 188 articles on identity within mathematics education with a focus on their theoretical underpinnings, research methods and definitions of identity. She found that socio-cultural frames underpinned much identity research, with most studies employing qualitative methods and that there were very few theoretical articles on identity in mathematics education. She identified two distinct frames informing identity research: psychological and sociological frames. For psychological frames, identity is regarded as an acquisition, or something inherent to people. Psychological frames consider the area of affect, involving beliefs, goals motivations and attitudes. Darragh (2016) argues that even though the area of affect makes a valuable contribution to knowledge, a strong research tradition concerning the affective domain already exists in mathematics education. She argues that the affective “domain does not need to be re-branded as identity. Doing so muddies waters already filled with a variety of definitions” (p. 10). Instead, she argues that researchers should employ sociological frames, which consider identity as an action, enabling the understanding of social context, positioning and people’s experiences of mathematics teaching and learning. Polarising the psychological and sociological frames from each other may provide an incoherent definition of identity, as suggested by Radovic et al. (2018) in their review of 69 articles on learners’ mathematical identities. Radovic et al. (2018) identified three dimensions for researchers to consider when defining identity, which are the social/subjective dimension, the enacted/representational dimension and the change/stability dimension. For the first dimension, they argue that researchers should consider both the social aspect of identity, understood best using sociological frames, and the subjective/individual aspect of identity, understood best using psychological frames. They argue that ignoring the subjective aspect, by only focusing on the social aspect, is limiting, by ignoring the formation of the self and the important role of individual agency in relation to identity. Conversely, focusing exclusively 6 on the subjective aspect, by only considering the role of the subjective/individual on the construction of her/his identity, leads to decontextualized understandings of identity, without accounting for the important role of social influences (Radovic et al., 2018). Instead, Radovic et al. (2018) argue that researchers should consider both social and subjective aspects since they both contribute to the emergence of learners’ mathematical identities. For the enacted/ representational dimension, they argue that some researchers view identity primarily as actions, while others view identity as primarily representational. They argue that there is an “overemphasis” on the representational aspect, as research on identity predominantly considers identity as narratives, and collects data in the form of narratives (Radovic et al., 2018, p. 34). Ignoring the enacted aspect of identity is limiting, by neglecting the mechanisms related to the emergence of identity in real contextual practices. However, ignoring the representational aspect of identity is also limiting, by neglecting the role of social discourses on the production of identity. Again, Radovic et al (2018) suggest that researchers consider both enacted and representational aspects of identity to provide operational definitions of identity. For the change/stability dimension, Radovic et al. (2018) showed that most research suggests that identity is malleable, rather than fixed, which is currently a widely accepted view of identity in mathematics educational research (see Bishop, 2012, Graven & Heyd-Metzuyanim, 2019). Graven and Heyd-Metzuyanim (2019) conducted a more recent review of 47 articles on identity in mathematics education. Like Darragh (2016), they found that there was a very small number of theoretical articles on identity in mathematics education and that most articles utilised qualitative methods and some form of socio-cultural framework. Like Radovic et al. (2018), Graven and Heyd-Metzuyanim (2019) also found that “an overwhelming majority” of studies consider identity as narrative (p. 368). Some of their other important findings are summarised below: • Only a small number of studies followed learners through their transition from one phase of schooling to another. • Researchers did not often differentiate between the identity they authored about a participant and the identity authored by the participant him/herself. • There seems to be a widening gap between studies which consider the area of affect and studies on identity. 7 • Most importantly, a weakness in mathematics education lies in stating what precisely identity is and how to empirically access and study identity. 1.4. Purpose of this study and research questions Based on the reviews discussed, there is a need for the concept of identity to be clearly defined, theorised and conceptualised in a way that contributes to the concept being empirically accessible (Darragh, 2016; Graven & Heyd-Metzuyanim, 2019; Langer-Osuna & Esmonde, 2017; Radovic et al., 2018). Further, we do not know much about South African secondary school learners’ mathematical identities, and how these identities are related to learners’ learning of mathematics in their mathematics classrooms (Brodie, 2017). The purpose of this thesis is to define, theorise and empirically apply the concept of mathematical identity to understand learners’ mathematical identities and trajectories in a secondary school in South Africa. Therefore, the main research questions that guided the study are: 1. What is mathematical identity and how can the concept be used in researching secondary school learners’ mathematical identities? 2. How do learners and their mathematics teachers, contribute to the development of learners’ mathematical identities? After much discussion with my supervisor, we agreed that I would complete my PhD that includes publications. Like pieces of a puzzle, five interrelating papers were written to answer the two main research questions, to form the big picture. Since this PhD is an upgrade of my Masters study, I had already analysed eight Grade 9 learners’ mathematical identities in a school in Johannesburg. My supervisor and I agreed that it would be in the best interest of my study to return to the same school the following year to follow the trajectory of these learners, and to analyse their mathematical identities in their Grade 10 year. In doing so, I aimed to address the gap in empirical research identified by Graven and Heyd-Metzuyanim (2019), by following learners as they transitioned from Grades 9 to 10. For most secondary school learners, learning mathematics and developing mathematical identities occur in the classroom (Anderson, 2007). Based on the most important finding in the Masters study, i.e., learners’ mathematical identities were strongly informed by their mathematics teachers’ practices, involving their pedagogical approaches and the kinds of social relationships shared with learners, I decided to focus my research on how the 8 relationships between teachers and learners contribute to the development of learners’ mathematical identities. To do this, I wrote four papers that primarily focused on the relationships between teachers, learners and learners’ mathematical identities. 1.4.1. The papers The first paper is a theoretical paper and was written as an extension of a conference paper presented at the International Congress on Mathematical Education (ICME) conference, which focused on answering the first main research question. Paper 1: 1. What is mathematical identity? 2. How can secondary school learners’ mathematical identities be researched? The next three papers focused on applying the framework developed in the first paper to empirical data, with a specific focus on answering the second main research question. These papers focused on how learners and their teachers contributed to the development of learners’ mathematical identities (papers 2-4): Paper 2: a. How are teachers’ actual practices and their narratives of their practices related? b. How do different teacher practices, both actual and narrated, influence how learners construct their mathematical identities? Paper 3: a. What influences teachers who teach using similar traditional pedagogies to develop social relationships equitably or inequitably with learners? b. How do social relationships influence how learners construct their mathematical identities? Paper 4: a. How do teachers speak about and interact with learner errors in their mathematics classrooms? b. What influences how teachers speak about and interact with learner errors? 9 c. How do teachers’ approaches to errors influence learners’ mathematical identities? As I was analysing the data collected in 2016, I realised that learners’ interactions with their peers were also important for the development of their mathematical identities, which led me to revisit the data from the Masters study. Initially, I struggled to find an analytical framework to analyse the data on learner interactions with their peers, and after publishing the first three papers (Gardee, 2019, 2021; Gardee & Brodie, 2022), I came across the work of Wood (2013). According to Wood (2013), learners’ mathematical identities can be examined on a micro-scale as micro-identity, or a macro-scale as macro-identity. Micro- identities refer to the moments when identity is constructed and captured during interactions and these identities can shift as learners interact with others in a lesson (Wood, 2013). Macro- identities are the identities that emerge over long periods and these identities are enduring, in the sense that they are persistent, stable and long-term constructions of who a person is (Wood, 2013). I realised that for all of my published papers, I analysed learners’ mathematical identities on a macro-scale, as macro-identity. So for the final paper, I modified the existing framework developed in paper 1 to understand how learners’ mathematical identities emerge during peer interactions. Paper 5: 1. How do learners construct and offer micro- and macro-identities during peer interactions in mathematics classrooms? 2. How do learners’ personal and social identities inform the micro- and macro-identities that they construct and offer during peer interactions? A conference paper presented at the International Group for the Psychology of Mathematics Education conference (IGPME) was written as an overview of the study and is included in the thesis to support the reading of the empirical papers. The order of the papers presented above is not the order of publication. I worked on multiple papers at the same time and some of them got published before others. For the full list of publications, please see section 1.7, and for more about my experiences of publishing papers, please see chapter 8. I will now describe the theoretical orientation of the thesis, followed by a discussion of the research design and methodology. 10 1.5. An outline of the theoretical orientation of the thesis Critical realism is the theoretical underpinning of this thesis and is used to define, theorise, and conceptualise identity, without reducing identity to psychological frames, sociological frames, representational or enacted aspects of identity. To date, critical realism has not been used previously to define and conceptualise learners’ mathematical identities, meaning that the use of this theory offers a novel contribution to conceptualising identity in mathematics education. Critical realism was initially developed by Roy Bhaskar (1978), and he describes critical realism as a philosophy with an under-labouring agenda: the philosophy seeks to clear the philosophical path for the clear expression of science. Bhaskar (1978) developed critical realism as a response to two existing and popular theoretical paradigms: empiricism, which postulates the existence of an external reality corresponding to observations of reality; and interpretivism, which postulates that the world does not exist independently of thoughts, subjective meanings and the language describing it. Researchers usually consider critical realism to be the “middle way” between empiricism and interpretivism, as critical realism maintains the advantages of empiricism, focusing on the existence of an external reality, and the advantages of interpretivism, such as the theory-laden nature of observations, while avoiding potential drawbacks such as relativism, and reducing individuals to discourse or observation (Tikly, 2015, p. 243). There are several phases to Bhaskar’s work, but his initial work, also called basic critical realism, has been most influential in the social sciences (Elder-Vass, 2021). This thesis draws on Bhaskar’s (1978) work on basic critical realism. There are three initial claims made by critical realists: 1. there is a difference between the transitive world of knowing and the intransitive world of being; 2. the social world is systematically open; and 3. that researchers should be focusing on the ontological depth of social reality (Scott, 2010). I provide a brief account of these three claims here, and a more detailed account in paper 1, which is presented in chapter 2. For the first claim Bhaskar (1978) distinguishes between the intransitive ontology dimension and transitive epistemology dimension. The intransitive dimension, which refers to objects in the world, involving both natural and social structures and mechanisms, is distinct from and cannot be conflated with the transitive dimension, which refers to knowledge about the world, which is subject to change. 11 For the second claim, Bhaskar (1978) distinguishes between open and closed systems. He claims that the world is an open system where “no constant conjunctions of events prevail” (p. 2), while closed systems, like laboratories, are those “systems where a constant conjunction of events occurs” (p. 3). In laboratories, scientists exclude certain mechanisms that could interfere with the regularities they wish to investigate. In open systems, there are multiple causal mechanisms that operate and give rise to events, while in closed systems, one or two mechanisms are isolated to understand the effects of the isolated mechanisms (Tikly, 2015). For the third claim, Bhaskar (1978) argues for depth realism, meaning that the purpose of science is to discover the underlying structures and mechanisms that are responsible for occurrences. Bhaskar (1978) defines generative mechanisms as “nothing other than a way of acting of a thing” (p.42), which endures and is exercised under the appropriate circumstances. In the real world, certain combinations of generative mechanisms give rise to a phenomenon, which is known as the process of emergence (Tikly, 2015). The role of science is to identify and describe these mechanisms, by means of the transitive dimension. In the context of this thesis, identity is defined as a social phenomenon existing in the real world, emerging from relationships between three generative mechanisms: personal identity, social identity and agency. In paper 1, which is presented in chapter 2, I show how I used the theory of critical realism to define and develop a framework to research learners’ mathematical identities. 1.6. Research design and methodology While each paper briefly describes the research design and methodology used to collect and analyse data, details of the research design, participants, collection methods, methods to enhance rigour and ethical considerations, which are not discussed in the papers, are explained here. The methods of data analysis are discussed separately in each paper, since these methods differ based on the focus of each paper. 1.6.1. Research design From a critical realist perspective, qualitative approaches, also called intensive approaches, are important to provide explanations of why certain objects or events occur (Danermark, Ekström & Karlsson, 2019). According to Bhaskar (2014), critical realism is primarily 12 interested in developing rich explanations of mechanisms that are responsible for the emergence of objects and events. Qualitative/intensive approaches serve this purpose, by providing deep causal explanations of why certain objects or events occur. In other words, qualitative/intensive approaches involve discovering generative mechanisms and investigating how these mechanisms manifest themselves and interact with other mechanisms in contexts (Danermark et al., 2019). The process of discovering mechanisms responsible for the emergence of learners’ mathematical identities and understanding the interactions between these mechanisms assisted me in answering the first main research question, which involved defining identity and understanding how the concept of identity can be used to research secondary school learners’ mathematical identities. Qualitative/ intensive approaches were also necessary to assist me in answering my second research question, which involved understanding how learners and their mathematics teachers contributed to the development of learners’ mathematical identities, by considering teachers’ and learners’ experiences and interpretations of the learners’ mathematical identities. The importance of understanding how people experience and interpret events requires the qualitative/intensive approach (Danermark et al., 2019). Further, qualitative/intensive research may provide diverse perspectives and a clear understanding of the complexities involved in a particular situation (Creswell, 2012). From a critical realist perspective, it is important to seek a number of different sources of information to develop a context-specific analysis of identity (Marks & O’mahoney, 2014). 1.6.2. Research site and participants The school This study took place in a selective, technical high school in Johannesburg over two years, in the third school term in both years (August - September), for two weeks in 2015 (Grade 9) and four weeks in 2016 (Grade 10). The school focuses on mathematics and physical science and offers courses aimed at enabling learners to gain entrance into university. Being a technical school, the school does not offer mathematical literacy. The school is selective, and learners write an aptitude test to ensure that they have sufficient mathematical and physical science background prior to accepting them. The learners in the school come from both working and middle-class families. The end-of-year mathematics results in this school was low. For example, at the end of the 2016 academic year, the Grade 10 grade average for mathematics was 22,78%. So even though this school may not be representative of all schools 13 in South Africa, in terms of this school being a technical school, having learners from both working and middle-class families and being selective of the learners who accessed the school, the school faced similar challenges as other schools in South Africa, in terms of learners’ poor performances in mathematics (see Spaull, 2019). There were two reasons for choosing to conduct my research in this school. The first reason was that the school is accessible from the University of the Witwatersrand. The second reason was that the school was part of a project run by my supervisor, Professor Brodie, which involved the creation of a ‘mathematics club’, aimed at developing learners’ proficiency in mathematics, supporting learners’ mathematical agency and developing learners’ mathematical identities (see Lampen & Brodie, 2020). The teachers Five mathematics teachers participated in the study: two in the first year, Mr Moyo and Mr Molefe (Grade 9), and three in the second year, Mr Sithole, Mr Ndlovu and Mr Ncube (Grade 10) (all pseudonyms). All of the teachers were qualified to teach mathematics. Mr Moyo was newly qualified and had joined the school at the end of the second term (May), Mr Molefe had five years of experience teaching mathematics, Mr Sithole had thirty-two years, Mr Ndlovu, eight years and Mr Ncube, five years. The teachers’ choice of pedagogy and style of interacting with the learners were different, which made them suitable participants for this study. Working with a range of teachers provided me with a stronger understanding of how different pedagogies and teacher-learner social relationships contributed to learners’ mathematical identities. The learners I initially selected Grade 9 learners as research participants for this study because according to the TIMMS and ANA results, Grade 9 learners performed extremely poorly, indicating that Grade 9 learners face serious challenges in secondary school when learning mathematics. In selecting the learner participants, a “purposeful sampling” method was used (McMillan & Schumacher, 2010, p. 138), which requires the researcher to purposefully select individuals who will provide information about the matter being researched (Creswell, 2012). Since I was not familiar with any of the learners in the classroom, I requested each of the two Grade 9 teachers to choose two pairs of learners in their mathematics classroom. Asking the teachers to select learners also gave me an indication of how the teachers positioned these 14 learners in the classroom. The Grade 9 learners were selected using the following criteria: one learner who is a relatively low achiever in mathematics and someone who sits next to her/him and one learner who is a relatively high achiever in mathematics and someone who sits next to her/him. Studying the identities of learners who displayed a range of achievements in assessments provided me with possibilities to understand how different relationships between generative mechanisms contributed to the development of learners’ mathematical identities. In Grade 10, two of the initial Grade 9 participants transferred to another school and I continued to observe the remaining six learners from Grade 9 to follow their trajectories. I also selected an additional eleven learners, who were seated next to the initial participants. I asked the Grade 10 teachers whether each learner was a low, average or high achieving learner, which enabled me to understand how the teachers’ ideas about learner performance influenced teacher and learner interactions. In Grade 10, learners sat in rows of twos, threes or fours and all learners in the same row as an initial participant were invited to participate in the study. Table 1 provides detail of the sample. There are three exceptions to the selection method (symbolised with an * in the Table), and the rationale is explained after the Table. Grade Pseudonym of teacher Pseudonym of learner Mathematics performance 9 Mr Moyo Senzo Low achiever Lane (sat next to Senzo) Average achiever Shane High achiever Jack (sat next to Shane) High achiever 9 Mr Molefe Tess Low achiever Sunny (sat next to Tess) Inconsistent Sindiswa High achiever Jennifer (sat next to Sindiswa) Low achiever 10 Mr Ncube Tess (participant from Grade 9) Low achiever Thandi (sat on the right of Tess) High achiever Amy (sat on the left of Tess) Average achiever Thabo (sat on the left of Amy) Average achiever Mike* (sat behind Tess) Low achiever Jimmy (sat next to Mike) Low achiever 15 10 Mr Sithole Senzo (participant from Grade 9) Low achiever Lebo (sat between Senzo and Shane) High achiever Shane (participant from Grade 9) High achiever Lane (participant from Grade 9) Inconsistent Tommy (sat next to Lane) Low achiever Jack (participant from Grade 9) Inconsistent Karabo (sat next to Jack) High achiever Phillip* High achiever 10 Mr Ndlovu Sunny (participant from Grade 9) Low achiever Fred* (sat in front of Sunny) High achiever Edward (sat next to Fred) High achiever Table 1.1: Selected learners Prior to data collection in 2016, Tess (in Grade 10) was unsure as to whether she would like to continue participating in the study. When she initially agreed to participate, I asked the learners who sat in the same row as her to participate, and they all agreed. She then decided not to participate. Her teacher, Mr Ncube, suggested that I ask Mike and Jimmy instead, who like Tess, were both low achievers, since the other learners who agreed were high and average performers. Both learners agreed to participate. Tess changed her mind again and decided to participate, so all six learners were included in the study. Like Tess, Karabo, who sat next to Jack in Mr Sithole’s classroom, was initially hesitant to participate. Mr Sithole suggested that I ask Phillip, who sometimes worked with Jack, to participate in the study. Karabo then agreed to participate, so I included both him and Philip in the study. In Mr Ndlovu’s classroom, the learner who sat next to Sunny did not agree to participate in the study. Mr Ndlovu suggested that I ask Fred and Edward, who sat in front of Sunny, since both learners were high achieving learners, which would enable me to study the identities of learners with different achievements. 1.6.3. Methods of data collection In both years, data were collected in the form of videotaped and audiotaped lessons, audiotaped semi-structured interviews with learner and teacher participants, field notes and photographed learner notebooks (in Grade 10). 16 I videotaped approximately 8 hours of lessons for each teacher. The video recorder was placed in the front of the classroom, which enabled me to focus it at times on the learners who were of interest to my research, and at other times on the teacher and the board. Videotapes are important to capture nonverbal forms of communication like facial expressions and gestures. McMillian and Schumacher (2010) argue that researchers “can trust participants’ responses more if their body language is congruent with their verbal statements”, enhancing the validity of my findings (p. 363). Observing the interactions in the classroom using the videotapes enabled me to “see things that might have been unconsciously missed, to discover things that participants might not freely talk about in interview situations” (Cohen, Manion, & Morrison, 2000, p. 305). In a previous research study, I noted the difficulty of capturing learner interactions with a video recorder. To eliminate the issues of poor sound that may pose a threat to capturing data using videotapes, I placed audiotapes on the desks of the learner participants to ensure that all interactions were captured as clearly as possible. All teacher and learner participants were interviewed individually after the videotaped lessons were collected, and these interviews were captured on an audiotape. According to Silverman (1993), interviews are useful in research as they provide the justification, motives, emotions and reflections of the interviewee. These interviews were semi-structured, which allows for flexibility of discussion and for the interviewer to probe deeper and pose other questions to understand the interviewee’s perspective (Freebody, 2003). It was necessary to capture all this detail on an audiotape as it would have been very difficult to document all the details in notes while participating in the discussion. In both years of data collection, each teacher was interviewed once. In the first year of data collection, learners were interviewed once, and in the second year, twice. The reason that the learners were interviewed twice in the second year was because the interviews in the first year took a lot of time, so I decided to split the interview into two parts, the first focusing on their current experiences of learning mathematics in Grade 10 and the second on their past experiences of learning mathematics in Grade 9. The second interview in the second year provided additional validity to the data collected in the first year. See Appendix A and B for the learner interviews and Appendix C for the teacher interviews. I took field notes in both years to supplement the data from the videotapes, by noting down my thoughts on the lessons, learner participation and important events that occurred in and 17 even out of the classroom, which may have not been captured on the video camera. The notes informed both the interviews and the data analysis. I also took photographs of the Grade 10 learners’ notebooks to supplement the data from the videotaped lessons. I did not take photographs of the Grade 9 learners’ notebooks, as I did not have permission to take the pictures, since they were supposed to write the ANAs immediately after data collection. 1.6.4. Rigour Ensuring the accuracy of one’s findings is crucial for results from research to be useful and meaningful. I piloted the study, which enabled me to improve the “reliability, validity and practicality” of my coding system and research tools (Cohen et al., 2000, p.260), and is discussed further below. Other strategies used to enhance the validity of the analysis, such as peer debriefing and triangulation, are discussed in the articles. Piloting the study To enhance the reliability and validity of my study, I piloted the study over a one-week period in 2015 with Grade 9 learners in a different school, which is similar demographically and is located near the school in which I carried out my final study. The reason I piloted the data collection instruments of my study was so that I could establish whether the instruments provided me with data which were useful for my study, whether the questions were ambiguous, how long it would take me to interview the teachers and learners and whether observations would yield useful data (Cohen et al., 2000). Piloting my study also provided me with the opportunity to test my coding system and to refine it. I observed two lessons of two different Grade 9 mathematics teachers in the pilot school. In conjunction with observing the lessons, I audiotaped two pairs of learner interactions, one pair in each teacher’s class, who were selected by each of their teachers. I practised coding the data using an open coding system and checked my interpretations with my supervisor. Prior to piloting my study, I had selected a mixed method research approach for this study. I intended to use a sequential explanatory mixed method design which involved analysing quantitative data first followed by analysing qualitative data (McMillan & Schumacher, 2010). Applying the quantitative approach first, I intended to collect data from many learners through questionnaires to enable me to identify how the different mechanisms contributed to the development of learners’ mathematical identities. Thereafter, through observing 18 classrooms and conducting semi-structured interviews, I would seek to obtain explanations about some of the trends found. Unfortunately, the results from the questionnaires in the pilot study were invalid because learners responded extremely positively to almost all of the statements (such as they enjoy mathematics, the way their teacher teaches, perform well in assessment and think they are good in assessment). The response bias could be a result of the learners assuming that they were providing me with answers that I wanted. Graven, Hewana & Stott (2013) encountered a similar problem in their study whereby learners answered questions based on what they thought was the correct expected response. Therefore, I changed my research design to qualitative/intensive research. Interestingly, the data from the interviews did not reflect the same situation as the questionnaire. It could have been that these learners who were interviewed were comfortable with me because I spoke to them individually prior to collecting data. Piloting the interviews gave me feedback as to whether the types of questions I used were valid. Based on the findings from the pilot interviews, I modified the interviews. 1.6.5. Ethical considerations I obtained ethical clearance from the University of Witwatersrand School of Education Ethics Committee and my protocol numbers are 2015ECE023M and 2016ECE015D. Permission was also obtained from the Gauteng Department of Education to do research in the school for both years, and in the pilot school. Informed consent was requested from the principal, the teachers, and the parents of the learners as well as the learners themselves. The teachers and the learners were told that participation is voluntary, and they could withdraw their participation at any time. 1.7. An overview of the thesis This thesis is divided into eight chapters. In this chapter (chapter one), I provided the background to the study, my personal rationale for conducting the study, the problem statement, purpose, research questions, theoretical orientation, research design and methodology for the study. The next seven chapters consist of interrelated manuscripts that form the central body of the thesis. For the completion of the thesis, it is required that I publish a minimum of four papers in reputable, peer-reviewed journals. Of the four papers required by the University, one of the 19 manuscripts is required to be published, one in press, and two accepted for publication. I have four published papers (chapters 2 and 4-6). In addition, I have included a conference paper and a fifth paper that is under review. While these are not required for the completion of the PhD, the inclusion of both papers are purposeful. The IGPME conference paper provides an overview of all the participating learners’ mathematical identities, which can support the reading of the remaining empirical papers. The fifth paper uses the framework developed in the first paper and applies it to understanding learners’ mathematical identities in peer interaction. Below is a list of the papers, their statuses, and references (Table 2). Chapter Paper Name of paper Status Reference 2 Journal paper 1 A framework for learners’ mathematical identities: A critical realist perspective Published Gardee, A., & Brodie, K. (2021). A framework for learners’ mathematical identities: A critical realist perspective. Didactica Mathematica, 43, 5-29. 3 Conference paper The influence of teachers on learners’ mathematical identities Published Gardee, A., & Brodie, K. (2019). The influence of teachers on learners’ mathematical identities. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education, Vol 2 (pp. 248-255). Pretoria, South Africa: PME. 4 Journal paper 2 Relationships between teachers’ practices and learners’ mathematical identities Published Gardee, A. (2021). Relationships between teachers’ practices and learners’ mathematical identities. International Journal of Mathematical Education in Science and Technology, 52(3), 1-27. 5 Journal paper 3 Social Relationships between Teachers and Learners, Learners’ Mathematical Identities and Equity Published Gardee, A. (2019). Social relationships between teachers and learners, learners’ mathematical identities, and equity. African Journal of Research in Mathematics, Science and Technology Education, 23(2), 233–243. 6 Journal paper 4 Relationships between teachers’ interactions with learner errors and learners’ Published Gardee, A., & Brodie, K. (2022). Relationships Between Teachers’ Interactions with Learner Errors and Learners’ Mathematical Identities. International Journal of Science and 20 mathematical identities Mathematics Education, 20(1), 193- 214. 7 Journal paper 5 A framework for analysing learners' mathematical identities in peer interactions: accounting for personal and social identities Under review Educational studies in Mathematics Table 1.2: Details of manuscripts that make up the chapters of this thesis. In chapter eight, I summarise the main findings and reflect on the study and make recommendations for further research. It is my hope that this study can contribute to the research focused on learners’ mathematical identities in our country, and internationally. 1.8. References Anderson, R. (2007). Being a mathematics learner: Four faces of identity. The Mathematics Educator, 17(1), 7-14. Andersson, A., Valero, P., & Meaney, T. (2015). “I am [not always] a maths hater”: Shifting students’ identity narratives in context. Educational Studies in Mathematics, 90, 143– 161. Bayaga , A., Mtose, X., & Quan-Baffour, K. P. (2010). Social influences on the studying of mathematics by black South African learners. Problems of Education in the 21st Century, 23, 30-40. Boaler, J., & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple Perspectives on mathematics teaching and learning (pp. 171- 199). Westport, CT: Ablex Publishing. Bhaskar, R. (1978). A Realist Theory of Science. (2nd ed.) London: Verso. Bishop, J. P. (2012). "She's always been the smart one. I've always been the dumb one": Identities in the mathematics classroom. Journal for Research in Mathematics Education, 43(1), 34-74. Brodie, K. (2010). Teaching mathematical reasoning in secondary school classrooms. New York: Springer. Brodie, K. (2017). Mathematics pedagogy in South Africa: A human and academic project In P. Webb & N.Roberts (Eds.). The pedagogy of mathematics : Is there a unifying logic? (pp. 175-183). Johannesburg: Real African Publishers. Chronaki, A. (2013). Identity work as a political space for change: The case of mathematics teaching through technology use. In K.le Roux, M. Berger, K. Brodie, & V. Frith 21 (Eds.), Proceedings of the seventh international mathematics education and society conference (pp. 1–18). Cape Town: Hansa. Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education, 40(1), 40–68. Cohen, L., Manion, L., & Morrison, K. (2000). Research methods in education (4th ed.). United States of America: Routledge. Creswell, J. W. (2012). Educational research. United States of America: Pearson. Danermark, B., Ekström, M., & Karlsson, J. (2019). Explaining Society: Critical Realism in the Social Sciences (Second). Routledge. Darragh, L. (2013). Constructing confidence and identities of belonging in mathematics at the transition to secondary school. Research in Mathematics Education, 15(3), 215-229. Darragh, L. (2016). Identity research in mathematics education. Educational Studies in Mathematics, 93(1), 19–33. Department of Basic Education (DBE). (2011). Curriculum and Assessment Policy Statement (Grade 7-9 Mathematics). Pretoria: Department of Basic Education. Department of Basic Education (DBE). (2014). Report on the Annual National Assessments in 2014, grades 1 to 6 & 9. Pretoria: Department of Basic Education. Department of Basic Education (DBE). (2019). TIMSS 2019: Highlights of South African results in mathematics and science. Pretoria: Department of Basic Education. Department of Basic Education (DBE). (2019). TIMSS 2019: Highlights of South African results in mathematics and science. Pretoria: Department of Basic Education. Department of Basic Education (DBE). (2021). Annual Performance Plan 2021/22. Pretoria: Department of Basic Education. Elder-Vass.,D. (2021). Critical realism. In G. Delanty & S.P. Turner (eds.), Routledge International Handbook of Contemporary Social and Political Theory (2nd ed.)(pp. 1- 14). Routledge. Freebody, P. (2003). Qualitative research in education. London: Sage Publications Ltd. Gardee, A., & Brodie, K. (2015). A teacher’s engagement with learner errors in her Grade 9 mathematics classroom. Pythagoras, 36(2), 1-9. Gholson, M., & Martin, D. B. (2019). Blackgirl face: Racialized and gendered performativity in mathematical contexts. ZDM, 51, 391–404. Graven, M., Hewana, D. & Stott, D. (2013). The evolution of an instrument for researching young mathematical dispositions. African Journal of Research in Mathematics, Science and Technology Education, 17(1-2), 26–37. Graven, M., & Heyd-Metzuyanim, E. (2014). Primary learner descriptions of a successful maths learner. In P. Webb, M. G. Villanueva, & W. L (Eds.), New avenues to transform Mathematics, Science and Technology Education in Africa: Proceedings of 22 the 22nd Annual Meeting of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 39-50). Port Elizabeth: Nelson Mandela Metropolitan University. Graven, M., & Heyd-Metzuyanim, E. (2019). Mathematics identity research: The state of the art and future directions. ZDM, 51, 361–377. Goldacre, B. (2009). Bad Science. London: Fourth Estate. Heyd-Metzuyanim, E., & Sfard, A. (2012). Identity struggles in the mathematics classroom: On learning mathematics as an interplay of mathematizing and identifying. International Journal of Educational Research, 51-52, 128-145. Heyd-Metzuyanim, E. (2013). The co-construction of learning difficulties in mathematics- teacher-student interactions and their role in the development of a disabled mathematical identity. Educational Studies in Mathematics, 83, 341-368. Heyd-Metzuyanim, E. (2015). Vicious cycles of identifying and mathematizing– a case study of the development of mathematical failure. Journal of the Learning Sciences, 24(4), 504–549. Lampen, E., & Brodie, K. (2020). Becoming mathematical: Designing a curriculum for a mathematics club. Pythagoras, 41(1), 1-15. Langer-Osuna, J. M., & Esmonde, I. (2017). Insights and advances on research on identity in mathematics education. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 637–648). Reston, VA: National Council of Teachers of Mathematics. Marks, A., & O'mahoney, J. (2014). Researching identity: A critical realist approach. In P. K. Edwards, J. O'mahoney, & S. Vincent (Eds.), Studying organizations using critical realism: A practical guide (pp. 66-85). Oxford: Oxford University Press. McMillan, J. H., & Schumacher, S. (2010). Research in Education Evidence Based Enquiry. New Jersey: Pearson Education Inc. Nasir, N. S., & Cobb, P. (2007). Improving access to mathematics: Diversity and equity in the classroom. Multicultural education series. New York: Teachers College Press. Radovic, D., Black, L., Williams, J., & Salas, E. C. (2018). Towards conceptual coherence in the research on mathematics learner identity: A systematic review of the literature. Educational Studies in Mathematics, 99, 21-42. Sailors, M., Hoffman, J. V., & Matthee, B. (2007). South African schools that promote literacy learning with students from low-income communities. Reading Research Quarterly, 42(2), 364-387. Scott, D., 2010. Education, Epistemology and Critical Realism. Routledge, Abingdon. Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14–22. Silverman, D. (1993). Interpreting qualitative data. London: Sage Publications. 23 Skovsmose, O. (2012). Students' foregrounds: Hope, despair, uncertainty. Pythagoras, 33(2), 1-8. Spaull, N. (2019). Equity: A Price Too High to Pay? In N. Spaull & J. Jansen (Eds.), South African Schooling: The Enigma of Inequality. A Study of the Present Situation and Future Possibilities (pp. 1–25). Springer. Tikly, L. (2015). What works, for whom, and in what circumstances? Towards a critical realist understanding of learning in international and comparative education. International Journal of Educational Development, 40, 237-249. Umugiraneza, O., Bansilal, S., & North, D. (2017). Exploring teachers’ practices in teaching mathematics and statistics in KwaZulu-Natal schools. South African Journal of Education, 37(2), 1–13. Wenger, E. (1998). Communities of practice. Learning, meaning and identity. Cambridge, UK: Cambridge University Press. Wood, M. B., & Kalinec, C. A. (2012). Student talk and opportunities for mathematical learning in small group interactions. International Journal of Educational Research, 51-52, 109-127. Wood, M. B. (2013). Mathematical micro-identities: Moment-tomoment positioning and learning in a fourth-grade classroom. Journal for Research in Mathematics Education, 44(5), 775–808. 24 Chapter 2: A framework for learners’ mathematical identities: A critical realist perspective 2.1. Introduction This chapter serves to answer the first main research question: What is mathematical identity and how can the concept be used in researching secondary school learners’ mathematical identities? In this chapter, I draw on critical realism as a theoretical perspective to define learners’ mathematical identities and show how identities can be empirically researched. Using the work of key critical realists, I identified three higher-level mechanisms: personal identity, social identity and agency; and developed a non-reductionist model to understand how the relationships between the mechanisms are responsible for the emergence of learners’ mathematical identities. Using these ideas and the ideas of other theorists in the field of mathematics education, I developed an explanatory framework for the relationships between learners’ mathematical identities and the social identities offered by teachers in classrooms. This explanatory framework relates two kinds of social identities offered: affiliation with, or marginalisation from, classroom communities; with four kinds of mathematical identities constructed: affiliation with, compliance with, resistance to or marginalisation from classroom communities. The explanatory framework is unpacked in more detail in chapter 4. This chapter also briefly addresses the second main research question: How do learners and their mathematics teachers contribute to the development of learners’ mathematical identities? Following the development of the explanatory framework in this chapter, I then applied the framework to understand the mathematical identities constructed by four learners as they interacted with their teachers in their mathematics classrooms. Each of the four learners participated in different ways and constructed different mathematical identities from each other, which assisted me in understanding how the combination of the three higher-level mechanisms contributed to the emergence of different mathematical identities. I show that to research learners’ mathematical identities, the relationships between all three mechanisms are important and should be considered concurrently with each other in a non-reductionist way. 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Chapter 3: The influence of teachers on learners’ mathematical identities 3.1. Introduction In this chapter, I present a conference paper presented at the International Group for the Psychology of Mathematics Education conference in 2019. This chapter shows how the explanatory framework developed in chapter 2 is in line with Radovic et al.’s (2018) three dimensions discussed in chapters 1 and 2. These dimensions are social/subjective dimension, enacted/representational dimension and change/stability dimension. This chapter also focuses on applying the explanatory framework developed in chapter 2 to all of the teacher and learner participants, by analysing the social identity offered to each learner by their mathematics teacher in relation to the mathematical identity constructed by the learner. There is a sense of trajectory in this chapter, as I show how the initial six learner participants’ mathematical identities changed substantially or negligibly over the two years, as informed by their interactions with different mathematics teachers. I show that the social identities offered to learners were informed by whether the teachers thought that ability or effort were important to learn mathematics. This chapter also points to other important findings, that are further discussed in chapters 4, 5 and 6. These findings are: the teachers’ narratives of who could do mathematics influenced how they interacted with learners and the kinds of social identities offered to learners; learners identified teacher practices as the most important factor that influenced their learning of and identification with mathematics; and that none of the learners resisted the offered social identities. This chapter provides some answers to the two main research questions, by showing, although briefly, how the concept of identity can be used in researching secondary school learners’ mathematical identities and how learners and their mathematics teachers contribute to the development of learners’ mathematical identities. 52 53 54 55 56 57 58 59 60 Chapter 4: Relationships between teachers’ practices and learners’ mathematical identities 4.1. Introduction This chapter builds on the explanatory framework developed in chapter 2, which is briefly expanded on in relation to Radovic et al.’s (2018) three dimensions discussed in chapter 3. This paper describes in detail what each of the social identities offered by teachers entail, and how the three higher-level mechanisms work together for the emergence of learners’ mathematical identities as affiliation, compliance, resistance or marginalisation. In this chapter, I discuss two teachers whose pedagogical practices and social relationships with learners were different from each other. This chapter provides answers to the first main research question, in terms of what mathematical identity is and by showing how the concept of identity defined in chapter 2 can be used in researching secondary school learners’ mathematical identities. This chapter further explores the key findings discussed in chapters 2 and 3, relating to the importance of teacher practices in a more detailed manner, thereby providing some answers to the second main research question: how learners and their mathematics teachers contribute to the development of learners’ mathematical identities. First, I explain that there are two important dimensions of teacher practice found in mathematics education research, which are distinct from each other and are usually discussed independently from each other. The first is between teachers’ actual practices and narrated practices, and the second is between teachers’ pedagogical practices and social relationships with learners. This chapter considers both dimensions, the actual and narrated teacher practices and their pedagogical approaches and social relationships with learners. Second, I relate these dimensions of teacher practice with the explanatory framework to analyse the data. I show how teachers’ actual and narrated practices, involving both their pedagogies and social relationships are related to each other. I argue that teachers, through their narrated pedagogical approaches, specify what is needed from learners to become affiliating members of their classroom communities, while through their narratives of their social relationships with learners, teachers specify who can learn mathematics and become full members of their classroom community. These narratives in turn shape teachers’ actual 61 practices, involving both their actual pedagogies and social relationships with learners, and their actual practices further justifies the teachers’ narratives of their practices. I also show how these teacher practices inform the social identities offered to learners and thereby inform learners’ mathematical identities. I provide a detailed analysis of the mathematical identities of seven learners and show that the social identities offered by the teachers informed but did not determine the mathematical identities constructed by learners. Some of the other findings in this paper are: the ways in which the teachers dealt with learner errors influenced learner participation; the teachers predominantly asked type 1 gathering information questions and rhetorical questions; and the learners in this sample did not resist any the social identities offered, possibly because learners are strongly influenced by teacher pedagogies and social relationships. One of the key recommendations made in this chapter is that more research is needed on teacher-learner social relationships, as I have shown that these relationships have important implications for learners to access mathematical knowledge. 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Chapter 5: Social Relationships between Teachers and Learners, Learners’ Mathematical Identities and Equity 5.1. Introduction This chapter builds on the important concept of social relationships in relation to learners’ mathematical identities, as discussed in chapter 4, taking a slightly different approach. In chapter 4, I discussed two teachers whose pedagogical practices and social relationships with learners were different from each other. In this chapter, I focus on two teachers who taught similarly but shared different social relationships with learners, highlighting that while both pedagogical practices and social relationships are key aspects of teacher practices, it is important that we do not conflate these two aspects of teacher practice. Another key difference from the previous chapter is that I discuss the importance of equity, as I noted that research tends to relate pedagogical practices with learners’ mathematical identities and equity, and that less attention is given to the relationship between teacher-learner social relationships, learners’ mathematical identities and equity. This chapter looks closely at the identities and trajectories of four learner participants over the two-year period and is focused on answering the second part of the first main research question: how the concept of identity can be used in researching secondary school learners’ mathematical identities; and the second main research question: how learners and their mathematics teachers contribute to the development of learners’ mathematical identities. This chapter, like the previous two, stresses the importance of teacher narratives of their social relationships with learners, which informed how they offered social identities, either equitably or inequitably, in their classroom communities. Yet this chapter differs from the previous two, by showing that the core characteristic of social relationships that contributed to the kinds of identities constructed by these learners was the notion of care. Care was discussed by these learners, not only in relation to themselves, but also in relation to whether other learners in the classroom were cared for by their mathematics teachers, highlighting the importance of equitable social relationships for the construction of learners’ mathematical identities. This chapter concludes by stressing the importance of researching teachers’ social relationships with learners, and not only their pedagogical practices, when researching learners’ mathematical identities, as I showed that even though these learners experienced similar pedagogical practices over the two years, their identities changed substantially and 91 were related to the differences in social relationships experienced in their Grades 9 and 10 teachers’ classrooms. 92 93 94 95 96 97 98 99 100 101 102 103 104 Chapter 6: Relationships between teachers’ interactions with learner errors and learners’ mathematical identities 6.1. Introduction This chapter picks up on the importance of the teachers’ interactions with learner errors as part of their pedagogical approaches in relation to learners’ mathematical identities, which was briefly discussed in chapter 4. This chapter focuses on the perspectives on and approaches to learner errors of two teachers, and how these perspectives and approaches informed learners’ mathematical identities. Like in the previous chapters, there is a clear link between the teachers’ narratives and their practices, which in the context of this chapter concerns the relations between their narratives of learner errors and their approaches to learner errors. In this chapter, I show that there were two basic mechanisms that informed how the teachers spoke about and interacted with learner errors: 1. the teachers’ views on the role of learner reasoning in mathematics and 2. their narratives about learners. I show that learners’ mathematical identities were informed by their teachers’ approaches to errors, yet as I have argued in chapters 2, 3 and 4, teacher practices do solely not determine the mathematical identities constructed by learners. The learners also played a role in the construction of their mathematical identities, which was informed by how they exercised agency, and their personal and social identities. All of the learners in this sample wanted to develop their identities as members of the classroom community, so even though they may have not described errors in the same ways that their teachers did, they altered their participation to maintain their membership in the community. In this chapter, I also provide another possible reason why the learners in the study did not resist the social identities offered by their teachers, stemming from an understanding of their personal and social identities. Resisting teacher practices could have a negative impact on learners, as all the learners in this sample, and most of the learners in the wider sample, aspired to pursue mathematics in the future and aimed at achieving some sense of belonging to the classroom community. Resisting teacher practices could possibly lead to the marginalisation of learners who were not previously marginalised, or further alienation of learners who were marginalised from the classroom community by teachers. Like chapters 3 and 5, this chapter is focused on answering the second part of the first main research question, and the second main research question: how the concept of identity can be 105 used in researching secondary school learners’ mathematical identities, and how learners and their mathematics teachers contribute to the development of learners’ mathematical identities. 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 Chapter 7: A framework for analysing learners' mathematical identities in peer interactions: accounting for personal and social identities 7.1. Introduction This chapter is different from the previous four chapters, as this chapter focuses specifically on the relationships between learners’ mathematical identities and their interaction with their peers. I discuss how two pairs of secondary school learners, who were treated similarly by their teachers and who performed similarly academically, constructed and offered mathematical identities. The reason I chose learners who were treated similarly by their teachers was that there are studies that show how teachers inform the ways learners interact with others and construct identities. My purpose was different, as I wanted to develop a framework to define and research learners’ mathematical identities constructed in peer interaction, to show the importance of learners’ personal and social identities. This chapter builds on the non-reductionist model developed in chapter 2 to develop an explanatory framework for the relationships between learners’ mathematical identities and the social identities offered by peers during peer interactions in classroom contexts. This chapter provides another perspective to answering the first main research question, in terms of what mathematical identity is and by showing how the concept of identity defined in chapter 2 can be used in researching secondary school learners’ mathematical identities constructed during peer interactions. Using the model developed in chapter 2, an explanatory framework is developed to relate the kinds of social identities offered by peers, as higher status, lower status or equal status, to the kinds of mathematical identities that can be constructed during peer interactions, as affiliation with or resistance to the offered social identities. An analytical framework is developed and applied to data to enable understandings of how learners construct mathematical identities during peer interactions, and why learners construct certain mathematical identities, by drawing on their personal and social identities. 129 A framework for analysing the relationships between peer interactions and learners’ mathematical identities Abstract Learner interactions have a powerful influence on learners’ mathematical identities and their learning of mathematics. Yet the absence of definitions of identity, which consider both subjective/personal and social aspects of identity, is an important concern in mathematics education research. The purpose of this paper is to propose a framework to analyse learners’ mathematical identities, by considering personal and social identities. To exemplify the framework, we discuss how two pairs of secondary school learners, who were treated similarly by their teachers and who performed similarly academically, constructed and offered micro-identities during moments of a lesson, and macro- identities across lessons. The two pairs were selected so that we could analyse how they constructed and offered identities differently from each other, independently from the influence of teacher positioning or differences in capabilities, with a focus on their personal and social identities. We show that while learners were offered identities by a peer as higher, lower or equal status, they constructed their identities by affiliating with or resisting the identities offered by their peer. Their identity construction was informed by their personal identities, in terms of their motivations for and enjoyment of learning mathematics, and their social identities, in terms of their choices to occupy certain roles in the classroom community. Keywords: Learners’ mathematical identities, Peer interactions; Participation; Narrations 1. Introduction In recent years, researchers in mathematics education have turned to the concept of identity to understand how learner interactions influence their learning of mathematics (Andersson & Wagner, 2019; Bishop, 2012; Wood, 2013). These researchers have shown how learner interactions inform their mathematical identities, in terms of who they are, who they might become, and how they see others, as learners of mathematics (Andersson & Wagner, 2019; Bishop, 2012; Gresalfi & Hand, 2019). Concomitantly, learners’ identities inform their learning, in terms of their engagement in and disengagement with mathematics (Graven & Heyd-Metzuyanim, 2019; Wood, 2013). Learners’ mathematical identities are usually examined on one of two scales: a micro- scale as micro-identity, or a macro-scale as macro-identity (Wood, 2013). Wood (2013) defines a micro-identity as “the position of a person in a moment of time” (p. 780), and these identities can shift in different moments of a lesson as learners interact with others. For example, Wood (2013) showed how in a lesson, a learner shifted his micro-identity from being an explainer of mathematical concepts when positioned as an explainer by his teacher, to being dependent on his peer for knowledge when positioned by his peer as being less mathematically capable than herself, and then back to being an explainer of mathematical 130 concepts after his teacher positioned him as capable again. So the learner’s micro-identity shifted from being mathematically capable to less capable and then back to being capable in different moments of the same lesson, influenced by the different social positions which were offered to him by his teacher and peer (see also Holland & Leander, 2004). So micro- identities are responsive to small changes in contexts, like engaging with different people, mathematical concepts and tasks at different moments in the same lesson (Wood, 2013). Over time, micro-identities can ‘thicken’, as learners come to identify themselves as certain people, forming their macro-identities (Wood, 2013). Macro-identities are also constructed through interactions, but unlike micro-identities, these identities emerge over long periods (Wood, 2013). Macro-identities are more persistent, stable and longer-term constructions of who a person is (Wood, 2013). Macro-identities can be accessed through interviews and by observing learner interactions over time. For example, research on macro-identities shows how learners described their identities constructed over a course (e.g., Boaler & Greeno, 2000; Cobb, Gresalfi & Hodge, 2009), and how learners’ interactions with others contributed to them becoming identified as superior, inferior, smart or bossy across time (Bishop, 2012; Langer-Osuna, 2011). These studies do not show how learners’ identities shift from moment to moment, but rather, how learners’ identities ‘thickened’ over time. Many studies show how learners’ interactions with their peers and teachers inform learners’ learning of mathematics and their mathematical identities, either in moments during a lesson as micro-identities, or over long periods as macro-identities. Some studies show how learners who collaborated with others on activities learned mathematics and developed their identities positively (DeJarnette & González, 2015; Esmonde, 2009). DeJarnette & González (2015) showed that when learners shared ideas and challenged themselves and others, they deepened their learning and reasoning skills. The learners shared the mathematical authority, so none of the learners sustained a particular micro-identity throughout the interaction sequence (DeJarnette & González, 2015). Esmonde (2009) made a similar finding, as she showed how a group of learners collaborated and did not position each other as being more or less capable than the others, so all learners in the group could make their thinking visible and engage with each other’s ideas. Other studies show how one learner dominated peer discussions and did not consider the ideas of others, especially when that learner was identified by others or identified themself as more capable than others (Bishop, 2012; DeJarnette & González, 2015; Lerman, 2001; Esmonde, 2009). Heyd-Metzuyanim and Sfard (2012) showed how two successful 131 learners, Dan and Ziv, were offered different micro-identities by other members of their group based on their relationships with the other learners. Dan was well-liked by others and was perceived as being understandable, even though his ideas were incorrect. Ziv was not well-liked by others in the group, as he projected himself as being more capable than the others. His mathematical ideas were correct, but he was excluded from the group and positioned as not being understandable, even though his peers did not listen to or engage with his ideas. Wortham (2004) showed how a learner was initially identified by her teachers as a good student, and later as disruptive. Over time, the learner’s interactions with her teachers and peers changed, which resulted in a ‘thickening’ of her identity to form her macro- identity, as she became consistently identified as an ‘outcast’ (Wortham, 2004). Langer- Osuna (2016) found that teacher interactions with learners reinforced authoritative positioning and shaped how learners authoritatively interacted with others. Several other studies have also shown how teachers’ interactions with learners informed how learners participated in the classroom and identified themselves and/or others as learners of mathematics (Boaler & Staples, 2008; Hodgen & Marks, 2009; Author, 2019 a, b; Authors, 2021; Author, in preparation; Lerman, 2001; Wood, 2013). So, the influence of others, such as peers and teachers, on learners’ interactions and identities is well explored in research, i.e., most research focuses on the social aspects of identity, rather than the subjective/personal aspects. While there have been attempts to define identity in mathematics education research, the absence of clear definitions of identity with a focus on both, the subjective and social aspects of identity, is an important concern for researchers (Graven & Heyd-Metzuyanim, 2019). The subjective aspect of learners’ mathematical identities requires more attention in research, so that the role of the individual in the construction of her/his identity is not neglected (Graven & Heyd-Metzuyanim, 2019; Radovic, Black, Williams & Salas, 2018). To address these difficulties, the purpose of this paper is to develop a framework to define and research learners’ mathematical identities constructed in peer interaction, with a focus on the subjective and social aspects of identity, as personal and social identity. To exemplify the framework, we focus on how two pairs of secondary school learners, who were treated similarly by their teachers and who performed similarly academically, constructed and offered mathematical identities, as micro-identities in moments of a lesson, and macro- identities over time. We selected these two pairs to analyse differences in mathematical identities, as informed by learners’ personal and social identities, rather than teacher 132 positioning or differences in learner capabilities, which is already well-researched (e.g., Bishop, 2012; Hodgen & Marks, 2009; Wood, 2013). Using our framework, we answer the following research questions: 1. How do learners construct and offer micro- and macro-identities during peer interactions in mathematics classrooms? 2. How do learners’ personal and social identities inform the micro- and macro-identities that they construct and offer during peer interactions? Our first research question explores how learners construct and offer micro- and macro- identities, while the second research question explores why learners construct and offer certain identities, with a focus on their personal and social identities. 2. Conceptual and analytic framework In mathematics education research, the definitions of identity proposed by Wenger (1998) and Sfard and Prusak (2005) are commonly used to understand learners’ mathematical identities. For Wenger (1998), identity is enacted, as a negotiated process related to learning, because learning involves becoming a member of a community of practice. As a person learns through participating in the community, s/he defines her/himself and is defined by others as a member of the community. For Sfard and Prusak (2005), identity is discursive, as a collection of socially constructed narratives by and about a person, which shapes their actions. Narratives can be told by people about themselves, or by significant others, such as stories told by teachers or peers about a learner, which can contribute to the learner’s actions and identity (Sfard & Prusak, 2005). Both Wenger (1998), and Sfard and Prusak (2005), emphasise the social nature of identity and underplay the subjective, personal aspect of identity, involving the notion of self and agency (Radovic et al., 2018). There is a need for theory accounting for the social, subjective, enacted and discursive aspects of identity (Radovic et al., 2018). Combining the work of Marks and O’mahoney (2014) and Archer (2002), with Wenger (1998) and Sfard and Prusak (2005) can serve this purpose. Using their ideas, we define learners’ mathematical identities as being constituted by relationships between personal identities, social identities and agency, which emerge as learners use discourse to define themselves and participate in communities of practice. Personal identity focuses on internal processes and emerges from the embodied and reflexive self, involving a person’s interests, motivations, emotions and practices (Archer, 2002; Marks & O’mahoney, 2014). Personal identity involves an understanding of self and is “a matter of what we care about in the world”, which informs what people prioritise (Archer, 133 2002, p. 15). Archer (2002) explains that engaging with people and tasks inform personal identity in terms of self-worth, since success and failure are reflected emotionally, thereby regulating behaviour. While personal identity is often discussed in relation to personal preferences, the role of the social world is intimately related to personal identity (Archer, 2002; Marks & O’mahoney, 2014). However, neglecting the importance of personal identity in research undermines the importance of the person in constructing her/his identity. Social identity focuses on what people “care about in social roles” (Archer, 2002, p. 17). Marks and O’mahoney (2014) explain that social identity is a “navigated position between personal identity and the way in which people believe they should be perceived in a social setting” (p. 72). So social identity is distinct from yet related to personal identity, shaping how learners position themselves in communities by choosing to embody roles they care about, which are generated by classroom communities. Society makes social identities available to people, through the roles generated by social structures (Marks & O’mahoney, 2014). We refer to these social identities as the social identities offered to learners. In mathematics classrooms, teachers and peers offer learners social identities through the opportunities they provide for learners to participate in and become members of the classroom community. While learners may be offered certain social identities, they exercise agency, by appropriating, or not appropriating, the social identities offered to them. Agency involves learners making active choices, which are influenced by social contexts (Etelӓpelto, Vӓhӓsantanen, Hӧkkӓ & Paloniemi, 2013). These choices can be exercised through compliance or resistance (Gresalfi, Martin, Hand & Greeno, 2008). How learners exercise agency, by participating in classrooms and learning mathematics, is influenced by the social identities offered to them by their peers and teachers, and their personal and social identities. Concomitantly, through exercising agency, learners learn more about themselves, supporting their personal identities, and they can evaluate their membership in the community, supporting their social identities. Drawing from the ideas of Marks and O’mahoney (2014), Esmonde (2009) and Gresalfi et al. (2008), we developed the following diagram to illustrate the relationship between the social identities offered to learners by their peers and learners’ mathematical identities, which can be accessed in moments of a lesson as micro-identities, or across lessons as macro-identities (Fig. 1). 134 Fig 1 Constituents of mathematical identity Mathematical identity is a social phenomenon and emerges from relationships between learners’ personal identities, social identities and agency. Learners’ mathematical identities are shaped by the social identities offered to them by the people they interact with, like their peers and teachers 1. The social identities offered to learners by others can be examined on a micro-scale, as the offered micro-identities at a moment, or a macro-scale, as the offered macro-identities over long periods. The micro-identity offered to a learner can shift in moments, as the learner may be offered different positions by her/his peer in different moments. These offered micro-identities can thicken to form the macro-identities offered, as the peer comes to identify and interact with the learner in certain ways across lessons. In peer interactions, learners can offer their peers micro- and macro-identities of higher, lower or equal status. Micro- and macro-identities of higher status are offered when learners position their peer as being more mathematically capable than themselves (Esmonde, 2009). Micro- and macro-identities of lower status are offered when learners position their peer as being less mathematically capable than themselves (Esmonde, 2009). Micro- and macro- identities of equal status are offered when learners position their peer as equally knowledgeable. Depending on learners’ personal and social identities and how they exercise agency, they can construct their micro- and macro-identities by affiliating with or resisting the offered micro- or macro-identities. Learners affiliating with an offered micro- or macro-identity of higher status may participate in ways that demonstrate that they are more capable than their peer. These learners may be personally invested in developing their social identities as knowledgeable, thereby constructing their identity as higher status. Learners affiliating with 1 Elsewhere, we show how learners’ mathematical identities are informed by their interactions with teachers (see Author, 2019 a,b; Authors, 2021). Agency Learners’ choice: affiliation resistance Social identities offered How learners are positioned by others in the mathematics classroom: Higher status Lower status Equal status Personal identity constructed Learners’ understandings of self in relation to mathematics. y Learners’ mathematical identities Social ident