The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgementTown of the source. The thesis is to be used for private study or non- commercial research purposes only. Cape Published by the University ofof Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University An investigation of the co-constitution of mathematics and learner identification in the pedagogic situations of schooling, with special reference to the teaching and learning of mathematics in a selection of grade 10 mathematics lessons at five schools in the Western Cape Province of South Africa Town Megan Jane ChitsikeCape (DVDMEG003 ) of A minor dissertation submitted in partial fulfilment of the requirements for the award of the degree of Masters in Education School of Education, Faculty of Humanities, University of Cape Town University September 2011 DECLARATION This work has not been previously submitted in whole, or in part, for the award of any degree. It is my own work. Each significant contribution to, and quotation in, this dissertation from the work, or works, of other people has been attributed, and has been cited and referenced. Signature: Date: 7th September 2011 i Town Cape of University ii Abstract This dissertation is an investigation of what is entailed in the co-constitution of mathematics and learner identification in the elaboration of school mathematics in a selection of grade 10 mathematics lessons in five working-class schools in the Western Cape Province of South Africa. The study is located within the broad framework of the sociology of education, specifically drawing on Bernstein’s sociological theory of education and its application in the investigation of the relations between pedagogy and social justice. The specific problematic within which my study is located is the constitution of school mathematics in the pedagogic situations of schooling. My theoretical framework consists of the work of Davis (2009b, 2010a, 2010b, 2011a, 2011b) in conjunction with Lacan’s (2006) psychoanalytic notions of the registers of the Real, the Imaginary and the Symbolic, Eco’s (1979) notion of a model reader, and Bernstein’s (1996) discussion of pedagogicTown discourse and the pedagogic device, which I use to fashion a set of resources for describing the co- constitution of school mathematics and learner identification in pedagogic situations. In my analysis I describe the operational Capeactivity making up fifteen grade 10 mathematics lessons selected for descriptionof and analysis. I use these descriptions of operational activity to discuss the realisation of content and the regulation of the learners in these lessons in order to explore the ways in which the extimate relation of the learner to the teacher, and the learner as obstacle to the reproduction of mathematics, appear in the exposition of mathematical content by teachers, and the implications of this for the co-constitution of mathematics and learner identification. University The results of this study show that (1) in most cases in these lessons the ways in which content is realised differs from the content indexed by a particular mathematical topic from the point of view of the mathematical body of knowledge; and (2) that there are instances where the ways in which content is realised also do not correspond with mathematical propositions, definitions, processes, rules and objects. I also found that (3) in most of these lessons, necessity is situated external to mathematics and the regulation of learners in these lessons is thus predominantly under the aspect of the Imaginary. iii Table of contents Declaration ..................................................................................................................................... ii Abstract ......................................................................................................................................... iii Table of contents ........................................................................................................................... iv Acknowledgement ......................................................................................................................... 2 List of figures ................................................................................................................................. 3 List of tables ................................................................................................................................... 4 Chapter 1 ........................................................................................................................................ 5 Introduction 5 1.1 Pedagogy and social justice 5 1.2 The co-constitution of mathematics and learner identification 6 1.3 The teacher, the learner and knowledge 7 Chapter 2 ......................................................................................................................................Town 11 A Review of the Literature 11 2.1 The procedural-conceptual distinction 11 2.2 The relation of pedagogy to ideas about the learnerCape 14 2.3 Learner and teacher identification of 16 Chapter 3 ...................................................................................................................................... 20 Theoretical Framework 20 3.1 The pedagogic device and pedagogic discourse 20 3.2 Model reader to model learner 21 3.3 The Real, the Imaginary and the Symbolic 22 3.4 Operations and their objects 25 3.4.1 The mathematicsUniversity encyclopaedia ...................................................................................................... 26 3.4.2 The regulation of mathematical activity .......................................................................................... 28 3.5 Summary: Propositions 34 3.5.1 Theoretical propositions .................................................................................................................. 34 3.5.2 Empirico-theoretical propositions ................................................................................................... 37 3.5.3 Research hypotheses ....................................................................................................................... 38 Chapter 4 ...................................................................................................................................... 39 A Framework for the Production and Analysis of Data 39 4.1 Research Design 39 4.1.1 The cases ......................................................................................................................................... 39 4.1.2 Data collection ................................................................................................................................ 40 4.2 Analytical framework 40 iv 4.2.1 Primary data production .................................................................................................................. 41 4.2.1.1 Generation of evaluative events ................................................................................................... 41 4.2.1.2 Descriptions of operational activity .............................................................................................. 42 4.2.1.3 Activation of the mathematics encyclopaedia .............................................................................. 46 4.2.1.4 Summary of primary data production process .............................................................................. 49 4.2.2 Secondary data production .............................................................................................................. 49 4.2.2.1 The realisation of content ............................................................................................................. 49 4.2.2.2 The regulation of the learner ........................................................................................................ 53 4.3 Reliability and validity of my study 58 Chapter 5 ...................................................................................................................................... 59 Presentation of Results 59 5.1 Describing the lessons 59 5.1.1 Mathematical topics and procedures ............................................................................................... 59 5.1.2 Time use in the lessons.................................................................................................................... 60 5.2 The realisation of content 62 5.3 The regulation of the learner 65 Chapter 6 ......................................................................................................................................Town 69 Discussion of results 69 6.1 School P1 69 6.2 School P2 Cape 71 6.3 School P3 73 6.4 School P6 of 74 6.5 School P7 76 Chapter 7 ...................................................................................................................................... 78 Four pedagogic practices and their implied model learner 78 7.1 Four pedagogic practices 78 7.1.1 Privileging one method over another .............................................................................................. 78 7.1.2 ChangesUniversity in the domain .................................................................................................................... 79 7.1.3 Specifying the order in which operations must be carried out ........................................................