This book addresses the question of how we might better understand the task of teaching mathematics to young children. But rather than starting out with a conception of mathematics derived from mathematics' own evolution, we center ourselves instead within the social practices that surround the teaching of the subject in British primary schools today. That is, we do not commence with an a priori conception of mathematics and see what people are saying about it. Rather, we start from what people are saying and see what this points to. We probe how the desires of society have manifested themselves in a societal decision to teach mathematics and how this decision now shapes what we call "mathematics". This extends and develops a conception of how language intervenes in the task of mathematics education presented elsewhere (Brown, 2001). In this present book however, we have a particular focus on trainee and new teachers, with a view to pinpointing how this conception of mathematics manifests itself in their evolving practices. We question how such teachers with many years of experience as a pupil in school might now re-orient themselves towards the demands of teaching mathematics. We also consider how for those charged with providing training for such individuals might better understand the process and impact of this training. The book further questions the way in which we might conceptualize the balance between nurturing teachers to become autonomous professionals responsible for developing and delivering the mathematics curriculum in schools and, alternatively, setting policies that prescribe practices to be followed. We consider whether we should focus our attention principally on the teachers themselves or on the professional space in which they operate. The book is primarily concerned with examining how trainee teachers conceptualize their own professional development, from the time they enter university on a four-year course as prospective initial teacher training students through to their first year of teaching in primary school. It has a particular focus on how they understand mathematics and how they understand their own teaching of the subject in schools. It offers both empirical and theoretical perspectives. Empirically, the book draws in particular on two studies conducted by the authors, funded by the UK Economic and Social Research Council and spanning a four-year period. Both of these studies were concerned with the professional development of trainee teachers with a particular focus on their phenomenological experience of the training process. Theoretically, the book draws on recent work in the field of psychoanalysis, and in particular the work of Slavoj Zizek, as an approach to examining how individual trainee teachers encounter the social framework in which they operate. In tackling this we consider the "technologies of self" that produce teachers in schools. We also look at how we might theorize our empirical findings that locate the discursive formation of school mathematics. To summarize the key strands: Firstly, we are keen to present an account of how trainee teachers understand their own journey into teaching mathematics in the primary school. Secondly, we wish to understand better the conception of mathematics in the primary school and how it might develop. Thirdly, we are keen to offer some discussion of how official policy as presented in government initiatives impacts on such teachers. Fourthly, we are concerned with better understanding the role that research in mathematics education might have in accounting for the process of trainees becoming teachers in the primary school and in stimulating development in this area. Finally, we try to offer a theoretical frame that accommodates evolving and alternative conceptions of mathematics, how it is taught and the social parameters that guide these conceptions.