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This chapter examines how interconnections among race, nativity, ethnicity, and cultural influences are critical for understanding mental health status in the United States. We argue that many sociologists of mental health neglect sociocultural variation—differences in lived experiences that are linked to race, nativity, ethnicity, and cultural influences—when assessing mental health status, when examining considering social statuses as sources of stress, and when selecting predictors to explain variation in mental health status. We challenge sociologists of mental health to investigate sociocultural variation with care and, in doing so, to expose the limits of current knowledge and reveal further the significance of race, ethnicity, nativity, and cultural influences. In terms of future research, we provide prescriptions connected to intersectionality, cultural competencies, control groups, and culture-bound syndromes. As diversity within the United States increases, continued neglect of sociocultural variation’s relation to mental health status becomes irresponsible.

In underscoring the affective elements of mathematics experience, we work with contemporary readings of the work of Spinoza on the politics of affect, to understand what is included in the cognitive repertoire of the Subject. We draw on those resources to tell a pedagogical tale about the relation between cognition and affect in settings of mathematical learning. Our interest is first captured in the way in which one teacher’s priority of establishing an inclusive learning community occasionally harboured what appeared to be pedagogically restrictive conceptions of mathematics. Yet, the classroom practices that produced these conceptions promoted the students’ motivation and provided meaningful access to mathematical learning within the classroom collectivity. In a second example, the postponement of scientific encapsulation in bodily imitations of planetary movement kept alive a shared dynamic sense of an elliptical orbit. In both of these cases, we draw on Spinoza’s work to show how the affectivity of classroom practice constituted conceptions of cognition and of mathematical activity crucially linked to the imperatives of participation.

Research discourses create the analytical frames that we use in observing classroom activities, which in turn create the objects that we research, including “students”, “teachers” and “mathematics”. Such notions cannot be seen independently of the analytical lens brought to them by researchers. And in doing this the researchers themselves reveal something of who they are, and how they see themselves fitting in. The research discourses, and the lenses chosen from within them, however, are set against the complex backdrop of motivations, preferences and trends built into our use of language, of which we are not always aware.

We have concluded our account of the 5-year process through which trainees become fully accredited, the 4 years of their training course and their first year as a teacher in school . Yet as we have indicated, we feel uncomfortable about drawing out a dominant story , beyond the sketches that we have drawn, as if a modal experience is more important than individual accounts. However, our attempts to pinpoint transitions for individuals in a graphic way were not very successful. As we have reached this stage of the book perhaps it is pertinent to spend a little time reflecting on our evolution in terms of our own understandings, as the researchers, of how we have sought to theorise our data. In creating an account of transition in the trainees’ understanding of their future professional task we attempted to introduce Ricoeur ’s work on time and narrative, which we felt could provide an interesting and useful theoretical frame.

This book rethinks mathematical teaching and learning with view to changing them to meet or resist emerging demands. Through considering how teachers, students and researchers make sense of their worlds, the book explores how some linguistic and socio-cultural locations link to prevalent conceptions of mathematics education. The locations include classroom mathematics, spatial awareness, media images of mathematics, curriculum development, teacher education and mathematics education research itself. The book introduces cutting edge theories of subjectivity that trouble more familiar psychological theories of “humans” apprehending mathematical “concepts”. Rather, it suggests that our senses of self and of mathematics result from self-reflections within the various localities in which we live. In foregrounding subjectivity the book shows how mathematics can provoke alternative ways of thinking towards enlivening our transformative capacities. Learning itself is depicted as participation in cultural renewal, where the very mathematics encountered is becoming something new. Addressing teachers, teacher educators and researchers, the book invites the reader to contemplate alternative trajectories of change into fresh ways of being.

This chapter outlines some of the theoretical ground that will underpin the arguments to be made after the empirical material has been presented in Chapter 4. The alternative frames to be offered enable an analysis of how mathematics and its teaching manifest themselves differently in individual and social perspectives. Distinctions between the individual and the social bring with them a host of difficulties with regard to teacher agency. The image of the lone teacher reflecting on her practice and building her control is challenged. The teacher cannot easily separate her own aspirations from those that are demanded of her by her employer. The very definition of the individual human

The realities of the previous generation clearly impact on the next generation. These realities can firm up in relation to familiar forms that can result in conservative modes of education centred on delivery of these forms. The making of mathematics in classrooms now can sometimes be seen as subservient to the forms created and then favoured by the previous generation, who had been making mathematics, according to their own particular image, in the circumstances they had faced. In such instances, mathematics is referenced and anchored to the last people’s models, thereby shielding the dynamic and analytical aspects of mathematical capabilities that might be invoked in the new conditions. In school education more generally, adjustments to knowledge are processed through successive curriculum models and wider educational infrastructure shaped according to the preferences of the day, ostensibly on a long march to “improvement” or to “raise standards”. Curriculums can be variously understood: as an outline of mathematical content; a guide to mathematical learning by children; a cynical ploy to make teachers and children more accountable according to a particular institutionalised account of mathematics; or, a reconfiguration of the discipline itself to meet contemporary needs, etc. The evaluation instruments used are often referenced to objects from earlier knowledge bases. The instruments of assessment carry with them the modes of governance to which we are subject. The acquisition of settled knowledge is easier to govern than a learning attitude centred in experimentation and critique. And the demands of governance can suppress new perspectives and temper intellectual responsiveness.

In Chapter 2 we considered mathematics in the spatial environment. In Chapter 3 we have seen some instances of mathematics in cultural media. This began our discussion of how mathematics is a function of its location, how it is part of us, and how it is built into the spaces we occupy. We now turn to the context of schools where curriculum frameworks filter teacher perceptions of their task of teaching mathematics.

The initial education of teachers is the key focus of this book so let us now turn to consider in more detail how the task of student teachers learning to teach mathematics has been researched within teacher education contexts. We commence with an overview of some of the ways in which the process of teacher education is understood and represented within the research literature. We consider how the constituent elements of the process have been defined, and in particular examine some of the mechanisms through which students have been enabled to articulate their own understandings of the process.

The key objective of this book is to present a theoretically oriented account of the processes through which primary teachers develop their understanding of the task of teaching mathematics. The theoretical framework was created as we faced the task of building a picture of the training process in England, where the regulative apparatus of the central government impacts greatly on teacher practices and on conceptions of mathematics. At one level these are colloquial concerns. Nevertheless, we are seeking to address an international audience as we feel that certain issues might activate a broader interest. Key among these is the formation of primary teacher

In Chapters 2 and 3 I sought to question the past primacy of notions of psychology centred on individuals apprehending mathematical objects. Objects, whether mathematical or artistic, develop meaning relationally to other objects, and in relation to the persons apprehending them. Objects do not have meaning in themselves. They are accessed through stories told about them. These stories structure the reality in which the objects appear. The objects are, in a sense, identified with images outside of themselves, built through the linguistic or symbolic apparatus at our disposal. And further, the persons apprehending such objects can be themselves understood through a variety of analytical lenses. Individuals know themselves through the stories of who they are, told by themselves, or by others. My stand-alone self, understood as some sort of cognitive or biological core, is peripheral to my everyday perception of life. In short, the logic of the psyche does not need to be understood in cognitive or individualistic terms. Meanings can be a function of socially defined filters, such as the curriculums described in Chapter 4, that in turn shape individuals and how they understand themselves.

It may be difficult to achieve consensus on an ideologically free conception of mathematics teaching. Mathematics itself is susceptible to being portrayed through discursive apparatus and caught between alternative accounts. Mathematics is generally a function of the social agendas relating to the circumstances of its practice. In primary education it will be mediated through the discourses pertaining to this arena.

As this book draws to a close, we shall summarise its empirical basis and theoretical intent before building some further arguments based on our discussion. We commenced with a brief examination of how the teaching of mathematics in primary schools might be better understood. We looked at how some primary school trainees and new teachers conceptualised their professional challenge and how this understanding was a function of the training process and of the curriculum frameworks they encountered. The interviews that we carried out provided a way of accessing this emergent understanding whilst at the same time gaining some sense of the trainees’ personal aspirations with regard to their chosen career in teaching.

In thinking mathematically, how does mathematics itself become something different? And how do we ourselves change? To unsettle the ground a little for later chapters, I commence with an exploration of these issues through considering extracts from a personal reflective diary work by a former student, Krista Bradford. The diary was created as a part of a research masters’ degree programme. The enquiry attended to the cross-cultural perception of mathematical concepts during some classroom research. Formerly a primary teacher in the United Kingdom, Krista was teaching mathematics to teenagers for Voluntary Services Overseas in Uganda. The research was carried out within a practitioner enquiry frame as part of a distance education course that I had initiated within a charity-funded project. As Krista had not worked on mathematics at this level since the end of her own schooling, she experienced a steep learning curve. This curve was made steeper as she became more aware of how mathematics was constructed in Ugandan schools. Its derivation from Western curricula compounded difficulties for the students she was teaching. Krista’s raised awareness of the cultural issues also brought into question her own agency within this development context. As a white person from the west she faced the challenge of mediating the externally defined demands of the western inspired curriculum and the more immediate educational needs of her students.

The last chapter considered examples of mathematical objects embodied in the spatial environment. This chapter contemplates how the cultural environment confers particular images on mathematics that further shape how it is perceived, and how it evolves. To begin this discussion I want to consider the notion of objects from a different perspective towards setting the scene for a preliminary consideration of how specifically mathematical objects derive from cultural contexts in everyday life. In the first instance I take, as an example, an art object, not so much as thing in itself, but to consider its provenance as an object. Could an analogy with the apprehension of artistic objects help us to think a little differently about the apprehension and provenance of mathematical objects? How do humans relate variously to mathematical or artistic objects? How are objects situated or created in relations? I shall argue that the objectivity of the piece is a function of its social location.

The last chapter sought to introduce a contemporary understanding of subjectivity in support of the theme of this second part of the book: cultural renewal. A key point of the chapter was to resist essentialist or fixed conceptions of the human subject, since they might fix the conceptions of mathematics that we feel able to entertain, or renewal trajectories specified around objects reconfigured in accounts of renewal. A particular target was Piaget with his reliance on an image of a human passing through successive stages of development common to all. Lacan’s conception of the subject on the other hand is endlessly responsive to new conditions. Lacan argued that Piaget’s conception of the child’s formation avoided the centrality of social demand, initially from parents and later from the wider social sphere. In Lacan’s formulation, the changing symbolic environment and the Real to which it uneasily relates unsettle the individual’s already mistaken understanding of who they are.

As I pointed out in the last chapter Saussure saw language as having the dual function of articulating meaning (process) and communication (product). However, Saussure and also his modern day advocates in post-structuralism see both process and product as contingent on each other. As educators we are often concerned not so much with the learning itself as with giving an account of what learning has taken place (cf. Garfinkel, 1967, pp. 24–31, who discusses the difficulties of moving between what is said and what is being spoken about). We need to decide what has been learnt by a student and how this has been demonstrated through tangible product, whether this be the application of a method, a reproduction of a famous result, or some verbal explanation of work completed, etc. Learning by the student, however, continuously evolves, oscillating between understanding and explanation; that is, between an on-going learning process and statements generated within this process, which become as if frozen in time. Further, this learning encompasses concerns beyond the frame which maybe anticipated by the teacher.

This paper describes the mathematics classroom from the perspective of social phenomenology. Here the classroom is seen as an environment of signs, comprising things and people, which impinge on the reality of the individual child. The paper introduces a framework through which mathematical work is seen as taking place in the imagined world through the filter of the world in immediate perception. This provides an approach to structuring evolving mathematical understanding. It is suggested that mathematical ideas are contained and shaped by the child's personal phenomenology, which evolves through time. Further, I argue these ideas are never encountered directly but rather are met through a circular hermeneutic process of reconciling expectation with experience.

Caught between the individual’s performance of mathematics and society’s overview of what the discipline is, we are faced with the task of providing a coherent account of mathematical learning. How do we reconcile the social and individual dimensions of developing mathematical understanding? Habermas’ recent work in social theory has sought to combine two traditions that dominated theoretical thinking during the nineteenth and early twentieth centuries, namely, positivism (Comte, 1864, see Habermas, 1972, pp. 65–90; Williams, 1976, pp. 238–239) and hermeneutics (Dilthey, 1976, see Habermas, 1972, pp. 140–186; Gadamer, 1962, Ricoeur, 1981). This can be seen as an attempt to reconcile scientific overviews of social situations with the experience of the people living within these situations. It displays a growing recognition of a need to integrate a fuller account of the participant’s understanding within analyses of social situations. Until recently, positivist approaches dominated the research horizon. This style of enquiry was concerned with scientific overviews where action is seen in terms of techniques or means-ends strategies that are not personalised in any way.

How far can mathematics be drawn into a linguistic domain and how far does such a move enable us to clarify the way in which students share their mathematical thinking with their peers and with their teacher? Such concerns will be addressed throughout the book. Before proceeding, however, I feel an example might help in further clarifying some of the issues of concern. I offer an account of someone working in a university seminar specifically concerned with exploring how words are introduced in holding on to evolving ideas (first reported by Brown, 1996 a, pp. 62–64). In the seminar there was an invitation to imagine a particular geometrical configuration. In an attempt to give a more graphic example of the issues of concern in this book I offer the three sentences which initiated the exercise and an account of someone’s developing understanding during the exercise, written immediately afterwards.

In the current world wide climate of rapid economic restructuring, choices for teachers are not clear. It might be argued that any historical era makes its claims as to its own specificity in terms of experiencing social change. Nevertheless it does seem clear that the conceptions of knowledge we hold are susceptible to some considerable challenge at present. David Harvey’s book, “The Condition of Postmodernity” offers an excellent broad based account of such changes. Similarly, Antony Giddens (1999) offers a lucid account of advancing globalisation in his book “Runaway World”.

How then can we hold on to our mathematical thinking so that we can talk about it and understand it more clearly? Addressing this has been far from easy since the whole notion of finding a way of holding things still is complex. Can we, for instance, allow ourselves to suggest that something can be seen as being stable in itself, even before we get on to considering the way such a thing evolves in individual or collective minds? So far we have considered how language might function in organising such mental activity. Whilst we have seen that language provides some sort of medium for creating, preserving and communicating mathematical thinking, there are limits to what it can achieve, unless we broaden its scope beyond its traditional frame. Mathematical thinking, it has been suggested, is susceptible to changes in time, space and the varying perspectives of people experiencing this, which makes it difficult to create a stable relationship with language. Both the “seen” and “who sees it” shift continuously, if only because the person “grows older”. Further, language would need to transcend its role as being primarily descriptive for a functional association to be possible with mathematical experience so viewed.

As we have seen, in recent work in the study of language there have been moves away from seeing words within a language as being a labelling device to denote preexisting phenomena. Rather, language and phenomena bring each other into existence through time, a process through which meaning is produced. I have shown, within an hermeneutic analysis, how language comes to the fore in mediating mathematical experience. Indeed, it becomes difficult to locate mathematics outside of a linguistic frame. Mathematics is accessed through accounts humans offer about it. Any attempt to locate the underlying truth of mathematics results in us encountering what Lacan calls the “lack”; the emptiness which emerges after the final layers of description (“stories”) are peeled away (Brown, Hardy and Wilson, 1993). Whilst hermeneuticians might hold onto notions of truth anchoring our thoughts, post- structuralist accounts resist the notion of such centring and instead occupy the “play” of accounts offered through history. Rather, we enter what Foucault calls “regimes of truth”, or discourses governed by their own internal structures, consequential to society’s view of itself.

This chapter is about development in student mathematical performance. Nevertheless we shall not be focusing so much on how to achieve this. Rather the chapter is about how such development is seen and how it functions as a notion in guiding our actions as researchers and teachers focusing on students progressing through successive stages in a developmentally formulated curriculum. As such it moves on from the phenomenology of the last few chapters and focuses more on how the teacher’s understanding of how his or her students learn mathematics guides the way in which he or she formulates tasks and makes sense of the student’s mathematical work.

Throughout this book I have taken the view that mathematical ideas are not things to be distributed bodily from teacher to student. Rather, mathematics is an inseparable aspect of human practice, a form of structuring placed on various forms of activity. It is located in individuals and only has life in the acts of these individuals. The theoretical perspectives assumed have various implications for the teacher’s classroom practice and how this might be developed. If mathematics is a style of activity rather than a list of content on a curriculum, the task of teaching is not one of delivery but one of initiating students into specific styles of activity. Since the mathematics “owned” by the teacher is part of themselves, his task may be seen in terms of helping students to build the mathematics they already have in themselves. In addressing these concerns, teachers need to work on understanding their own professional practice and how this practice governs the way in which students carry out their work in the classroom. Even if teachers have a clear sense of how children learn mathematics, they need to build an understanding of how their actions as teachers function in guiding students towards achieving specified learning objectives.

In chapter four we observed Chester and his chums engaged in an attempt to count up in twos. Their efforts, however, were not without a little confusion and mischief. Having started with 2, 4, 6, 8, they had somehow reached 27 to be followed by 29, 30, 32 (written as 23), 25, 45, 70, 46, 16, 17, 19. Whilst some seemed to be trying to throw the activity off course, others remained committed to adding a two to the number they were given, although occasional slips were made. I speculated that their actions were governed by the constraints of their immediate situation where they placed the required number next to the number they were presented with but without paying much attention to the overall scheme of things. They seemed to be “going with the flow” and were responding to each other in a lively interaction.