A Contemporary Theory of Mathematics Education Research

For proposal stage:

About the author

Preface

Chapter One                      Introduction

Chapter Two                      Reason to believe

Chapter Two provides a theoretical discussion of how we understand mathematical knowledge. The theory presents rationality and belief as mutually formative dimensions of school mathematics, where each term is more politically and socially embedded than often depicted in the field of mathematics education research. School mathematics then presents not so much rational mathematical thought distorted by irrational beliefs but rather a specific mode of activity referenced to the performance of certain substitute skills and procedures that have come to represent mathematics in the school context consequential to the demands of social management. The chapter considers alternative modes of apprehending mathematical objects derived as they are from this socially defined space. The chapter's central argument is that rational mathematical thought necessarily rests on beliefs set within a play of ideological framings that within school often partition people in terms of their proxy interface with mathematics. The challenge is then seen as being to loosen this administrative grip to allow both students and teachers to release their own powers to generate diversity in their shared mathematical insights rather than being guided by conformity.

 

Chapter Three                  The social packaging of mathematical learning.

Chapter Three considers some of the arbitrary curriculum or assessment criteria that operate in the social construction of mathematics in educational institutions. The advance of mathematics as an academic field is typically defined by the production of new ideas, or concepts, which adjust progressively to new shared ways of being. That is, mathematical concepts are created or invented to meet the diverse demands of everyday life, and this very diversity can unsettle more standardised accounts of what mathematics is supposed to be according to more official rhetoric. For example, the expansion of mathematics as a field often relies on research grants selected to support economic priorities. In schools, economic factors influence the topics chosen for a curriculum. In some countries, for instance, there is a shortage of specialist mathematics teachers that limit curriculum choices and restrict the choice of viable teaching materials, educational targets or models of practice advocated by research in mathematics education. Our evolving understandings of who we are and of what we do shape our use of mathematical concepts and thus our understandings of what they are. School mathematics has been reduced according to ideological schema to produce its conceptual apparatus, pedagogical forms and supposed practical applications.

 

Chapter Four                     The social administration of mathematics subject knowledge through teacher education

Chapter Four describes some recent empirical research in university teacher education. It considers how practices of teacher education impact on classroom practice by new teachers and thus shape the mathematics that takes place. The theme is explored through an extended discussion of how the conduct of mathematical teaching and learning is restricted by regulative educational policies that set the parameters through which teacher education takes place. Specifically, it considers the example of how mathematics is discursively produced by student teachers within an employment-based model of teacher education in England where there is a relatively low level of university input. It is argued that teacher reflections on mathematical learning and teaching within the course are patterned in line with formal curriculum framings, assessment requirements and the local demands of their placement school. Here, both teachers and students are subject to regulative discourses that shape their actions and, consequentially, this regulation influences the forms of mathematical activity that can take place and be recognised as such, but where this process restricts the presentational options for the mathematics in question. It is shown how university sessions can alternatively provide a critical platform from which to interrogate these restrictions and renegotiate them.

Chapter Five                      Rethinking objectivity and subjectivity

Chapter Five digs deeper in to theory to consider further how the mathematical/human interface depends on the mutual dependency of how we understand mathematical objects and of how we understand human subjects. The apprehension of mathematical objects is examined through sessions with student teachers researching their own spatial awareness from a pedagogical point of view. The chapter is guided by the theoretical work of Alain Badiou whose philosophical model develops a Lacanian conception of human subjectivity and defines a new conception of objectivity. In this model, the conception of subjectivity comprises a refusal to allow humans to settle on certain self-images that have fuelled psychology and set the ways in which humans are seen to apprehend the mathematically defined world. The assertion of an object, meanwhile, is associated with finding a place for it in a supposed world, where the object may reconfigure that world in its very assertion. The composite model understands learning as shared participation in renewal where there is a mutual dependency between the growth of human subjects and of mathematical objects. The chapter provides examples from teacher education activities centred in addressing these concerns.

Chapter Six                         Subjectivity and cultural adjustment: a response to socio-culturalism

Chapter Six documents aspects of the discussion that has taken place as a result of socio-cultural theorists responding to my engagement with their work in my book Mathematics Education and Subjectivity. Specifically, the chapter responds to Wolf-Michael Roth's critical reading of the book. (Roth is one of the top authors in the world). His reading contrasted my Lacanian approach with Roth's own conception of subjectivity as derived from the work of Vygotsky, in which Roth aims to "reunite" psychology and sociology. I argue, however, that my book focused on how discourses in mathematics education shape subjective action within a Lacanian model that circumnavigates both "psychology" and "sociology". From that platform, this chapter responds to Roth through problematising the idea of the individual as a subjective entity in relation to the two theoretical perspectives. In line with the broader remit of this present book, the chapter argues for a Lacanian conception of subjectivity for mathematics education comprising a response to a social demand borne of an ever-changing symbolic order that defines our constitution and our space for action. The chapter concludes by considering an attitude to the production of research objects in mathematics education research that resists the normalisation of assumptions as to how humans encounter mathematics.

Chapter Seven                  The evolution of mathematics

Chapter Seven discusses at a more historical level how our conceptions of mathematics and of ourselves as researchers, teacher educators, teachers and students have been transformed through mathematical activity being viewed through the apparatus of schooling and international comparative filters. This model provides an example of how changing practices impact on the social construction of mathematics itself. The chapter argues that the fields of psychology and mathematics each describe realities that are consequential to past human endeavours or conceptualisations. Neither of these fields depict stable truths.