Mathematical Relationships in Education

Identities and Participation

Laura Black, Heather Mendick, Yvette Solomon

• Publisher: Routledge
• Year: 2011
• Print ISBN: 9780415996846
• eBook ISBN: 9781135842185
• Topic: Education
• Subtopic: Education General

This book brings together scholars working in the field of mathematics education to examine the ways in which learners form particular relationships with mathematics in the context of formal schooling. While demand for the mathematically literate citizen increases, many learners continue to reject mathematics and experience it as excluding and exclusive, even when they succeed at it. In exploring this phenomenon, this volume focuses on learners' developing sense of self and their understanding of the part played by mathematics in it. It recognizes the part played by emotional responses, the functioning of classroom communities of practice, and by discourses of mathematics education in this process. It thus blends perspectives from psychoanalysis, socio-cultural theory and discursive approaches in a focus on the classic issues of selection and assessment, pedagogy, curriculum, choice, and teacher development.

1
Introduction

Laura Black, Heather Mendick, and Yvette Solomon

This book is the result of a seminar series entitled Mathematical Relationships: Identities and Participation held in the UK between September 2006 and July 2007. The seminars were funded by the Economic and Social Research Council and the British Educational Research Association. We organised them along with Melissa Rodd and Margaret Brown and spent a year touring the UK, starting in London, then heading to Manchester, Edinburgh, Cardiff, and Sheffield, before returning to London for the final seminar. These six seminars brought together a number of researchers and practitioners working in the areas of mathematics education and in education and identity to discuss how we can better understand patterns of participation in mathematics. This book is an edited collection mostly of the papers presented at those seminars, with additional contributions from other international scholars.

The starting point for the series was the apparent paradox of better results in mathematics examinations juxtaposed with an increasing rejection of the subject at the postcompulsory level. This is an international concern, noted in Europe, North America, Australia, and the UK (Boaler, 2000b; Holton, 2001; Seymour & Hewitt, 1997; A. Smith, 2004). To make sense of this situation, we need to focus on the relationships that learners form with mathematics in the context of formal schooling. Central to these relationships is learners’ developing sense of self and their understanding of the part played by mathematics in it. A focus on identity provides a means of understanding the processes that produce this ‘paradox’ and a way of beginning to address it by unpacking learners’ emotional responses, their self-positioning with respect to others, and the surrounding discourses of mathematics education. Explorations of identity are therefore the central focus of this book.

Identity is a relatively new concept in mathematics education, although it is more widely used in educational and social research. The seminars reflected this fact, and we deliberately set out to include some researchers working in the field of identity who were not mathematics education researchers—this was a fruitful move, and the book correspondingly offers space for dialogue between this emerging exploration of identity in mathematics education and these wider theoretical perspectives.

Many disciplines deal with identity, and, within each of these, there are complex and overlapping ways of understanding it. The three perspectives on identity we selected for the series, and hence the book—sociocultural, discursive, and psychoanalytic—have interesting and contrasting things to say on how people relate to mathematics, focussing, respectively, on practices and participation, language and power, and the role of the unconscious. These theoretical perspectives are very broad, and the authors interpret them very differently. Here we offer only the briefest of orienting introductions. Sociocultural theories derive from psychology and anthropology (Holland, Lachicotte, Skinner, & Cain, 1998; Lave & Wenger, 1991; Wenger, 1998) and have been widely taken up in education (e.g., Bloomer & Hodkinson, 2000), including mathematics education (e.g., Boaler & Greeno, 2000). These approaches view identity as being co-constructed through participation in social practice and seek to understand the ‘cultural models’ agents use, and are positioned by, in their identity work. Discursive approaches view identity as the result of the subject’s interpellation into discourse, systems of knowledge, and practice which construct objects; this process is inseparable from relations of power (Foucault, 1980). These approaches are of growing importance within the field of mathematics education (Lerman, 1998) as are psychoanalytic approaches (Evans, 2000) which define identity in terms of an interaction between conscious and unconscious processes and underline the value of focusing on the role of emotional and relational factors (Britzman, 1998). There are many tensions between these approaches. For example, there is a tension between the discourse insistence that we should not look inside people for explanations and the psychoanalytic concern with unconscious processes such as anxiety, fantasy, and defensive strategies. It is precisely these tensions that we feel make bringing these very different perspectives together so productive for addressing our overarching question: What can a focus on identities and relationships bring to understanding issues of inclusion in and exclusion from mathematics?

As we have said, in organising the series, and hence this book, we turned to the broad policy context of the ‘paradox’ of rising performance sitting alongside falling participation. We want to explore how policy is currently played out in and through mathematics education and the impact of policy on learner identities in relation to mathematics. Research indicates that the role of identity is most visible in formal contexts where learners are subject to policy-driven institutional structures which impose categorisations on them as ‘good at’ or ‘not good at’ mathematics via assessment and selection, pedagogy, and curriculum. In addition, the widespread emphasis in neoliberal policy on ‘choice’ presents a further context for (self-) definition. These themes, together with that of teacher development, provide our organising framework. We introduce them briefly here.

An increasing policy emphasis on assessment and selection has led to high levels of anxiety impacting on how learners construct their identities (Reay & Wiliam, 1999; Shaw, 1995). Within mathematics, discourses of ability and the associated practices of setting correlate with gender, class, and ethnicity and their related identities. For example, Gilborn and Youdell (2000) found that ethnic minority and working-class pupils are more likely to be allocated to lower sets regardless of prior attainment, while high-achieving girls in top sets are more likely to position themselves as having ‘less right’ to be there and to experience anxiety (Mendick, 2005a). The experience of assessment-related ability judgements has a long-term impact: university mathematics students continue to depend on positive test results for their identity confirmation (Rodd & Bartholomew, 2006; Solomon, 2007b). Correspondingly, self-selection or choice of whether and how to approach the study of mathematics is gendered (Mendick, 2005b, 2006), classed (Macrae & Maguire, 2002), and raced (Francis & Archer, 2005) and related to perceptions and self-perceptions of dis/abilities (Rodd, 2005). In the area of curriculum, the specialised language forms and strategies used in mathematics teaching also differentially favour some social groups in terms of class (Zevenbergen, 2000), cultural background (de Abreu & Cline, 2003), and gender (Walkerdine, 1998). Research on pedagogy indicates that ‘traditional’ teaching often results in differential participation (Boaler, 1997; Solomon, 2007a); high achievers and those from social backgrounds with greater cultural capital are most likely to join in positive interactions with teachers (L. Black, 2004a, 2004b; Jones & Myhill, 2004). Lower set pupils are likely to experience a ‘polarised curriculum’ (Boaler, Wiliam, & Brown, 2000) which not only limits exposure to mathematics but also obscures the underlying mathematical principles (Dowling, 1998). However, a wide body of research indicates that mathematics can be made more accessible in classrooms which encourage exploration, negotiation, and ownership of knowledge and their corresponding identity shifts (Boaler & Staples, 2008; Solomon, 2008). This has implications for teacher development, but also requires recognition that teachers’ own mathematical identities affect and are affected by the curriculum and assessment context (T. Brown, 1997), and that their own, often troubled, relationships with mathematics impact on the ways that they interact with learners about the subject (Bibby, 2001).

The book has six parts. The first five reflect these policy and practice foci, each addressing a central question—about selection and assessment, choice, curriculum, pedagogy, and teacher development. These five themes are, as the previous literature shows, significant since they act as ‘essential circuits’ of education policy and practice (S. J. Ball, 1994). The treatment of each theme follows a uniform pattern, opening with a short editorial introduction contextualising our question and the responses which follow. The three subsequent chapters then offer replies from sociocultural, discursive, and psychoanalytic perspectives (although not necessarily in that order). Finally, the sixth part contains two chapters in which the authors, both international scholars, respond to the book as whole. We hope that by interacting with the ideas in these pages you will, like us, have a chance to reflect upon your own mathematical relationships and those of other people.

Part I
Selection and Assessment

In their answers to the question ‘How does the concept of ‘ability’ operating within mechanisms of assessment and selection (by school and within school) impact on learner identities in relation to mathematics?’, the chapters in this part focus on the ways in which assessment practices within mathematics education select students as ‘able’, ‘competent’, and ‘normal’. Given that ‘high-stakes’ assessment plays a prominent role within current educational policy and practice in the UK, as in many other countries, it is perhaps unsurprising that we find it to be central in shaping many learners’ relationships with mathematics and the mathematical identities they develop. In light of this, the chapters in this part all address a common theme: They seek to explore how people are measured and measure themselves in relation to mathematics.

Utilising a discursive approach, Anna Sfard (Chapter 2) presents an overview of the dominant macro discourse of quantification and highlights how quantitative assessments (by which we are all measured) permeate all walks of life. She notes that this may be particularly damaging within education, where numbers are secreted within hidden discourses and then become labels which identify actors rather than actions in a particular context, such that once a label is ascribed to a student via their performance in a given activity (e.g., a test or exam), that label becomes the intrinsic property of the student concerned, to be used by teachers or policymakers to differentiate between students. How the label came into being through the original act of performance then falls from view. A prime example of this can be found in the chapter by Jeremy Hodgen and Rachel Marks (Chapter 4), where primary school pupils are labelled (and label themselves) according to their levels within the English National Curriculum—labels which become reified markers, detached from their origins in the children’s examination performance.

Anna goes on to note the potentially damaging effect of such quantified labelling in restricting the number of ‘designated identities’ (Sfard & Prusak, 2005b) that a given student is able to take up in the future. She quite rightly points out that assessment outcomes (particularly summative assessment) can all too easily become statements about the student’s future, and thus one or two ‘local failures’ can bring about automatic positioning at the lower end of the ‘ability’/‘competence’ spectrum in the category of ‘those who cannot’. Thus, in analysing the macro discourses of quantified assessment, she acknowledges its dominance in ‘shaping the world around us’ and calls for further questioning of what the numbers mean—what competencies underlie the labels we use? And are they the kinds of competencies we wish students to have?

Whilst Anna’s chapter highlights the power of what she calls the discourse of ‘numberese’ in positioning people, the next chapter in this part builds on this by looking at how people are drawn towards the discourse of measurement and the labels it proffers. Adopting the psychoanalytic standpoint of Melanie Klein, Laura Black, Heather Mendick, Melissa Rodd, and Yvette Solomon with Margaret Brown (Chapter 3) present us with three ‘mathematical biographies’ which narrate the authors’ turbulent and sometimes ambiguous relationships with mathematics. For them, discourses associated with assessment and selection are central to defining the pain and pleasure which frame our relationships with mathematics—such discourses offer positions which invoke our psychic defenses (against anxieties) in a myriad of ways. Thus, whereas Anna highlights the power of ‘numberese’, this chapter focuses on the psychic damage of such discourses when revealed through individual stories. This exercise is particularly illuminating since it underlines how discourses of assessment and measurement should not be considered as unambiguously problematic since, for some, they are also enabling—individuals can and frequently do use them to construct positive relationships with mathematics (e.g., through personal comparison or competition) in their ‘identity work’ (i.e., in defining who they are). Throughout the chapters in this part (and in this book), we see repeated examples of people using various assessment outcomes to measure or define their success. But as the accounts of Nikki, Rachel, and Zoë show, this is risky business since the threat of failure is ever present. Rachel’s account in particular suggests a desire to compare her achievements with others, despite her knowledge of the ways in which mathematics is typically associated with processes of selection. Such comparisons carry the threat of failure, of course; this threat is powerful and perhaps even attractive—we are reminded of the ‘frisson of danger’ (Bibby, 2006) which appeals in doing such identity work. Therefore, one might argue that the psychic damage of ‘numberese’ is not simply brought about through the positions which assessment and selection discourses make available; rather, we may actively subscribe to, and work, these discourses as we negotiate our fluctuating relationships with mathematics.

Finally, the chapters in this part also address an important question—how might we change assessment practices within mathematics education in order to both challenge the potentially ‘damaging’ consequences brought about by labelling and measurement and reduce the threat of failure which we are so vulnerable to? This is essentially an issue about how we can avoid using assessment as a selection mechanism—to separate those who ‘can’ from those who ‘cannot’—and how we can avoid the inevitable exclusions which stem from this. Here, Jeremy and Rachel’s chapter is particularly relevant. Using both pupil and teacher interview data, they explore the commonality between pupils’ and teachers’ negative experiences of learning mathematics and note the role that summative assessment plays in positioning teachers so that they reproduce the same patterns of exclusion that they themselves experienced. In light of this, they call for a more relational approach to assessment, one which emphasises the development of relationships and learning rather than labelling. Anna makes a similar argument in her discussion of ‘assessment for learning’, with its emphasis on providing teachers and learners with detailed information on their strengths and weaknesses as a more effective pedagogic tool than the summative assessment which she rejects. However, Jeremy and Rachel go on to argue that we should not abandon summative assessment altogether since it is not quantified discourses per se which are stratifying but a lack of critical perspective amongst the teachers who use them. They suggest that we should instead encourage teachers to develop their assessment literacy and a critical understanding of the contexts in which such assessments...

Table of contents

• Routledge Research in Education
• Contents
• List of Figures
• Acknowledgments
• 1 Introduction
• Part I Selection and Assessment
• Part II Choice
• Part III Curriculum
• Part IV Pedagogy
• Part V Teacher Development
• Part VI Endings
• Index