Olov Viirman | The function concept and university mathematics teaching | function concept and university mathematics teaching | The Olov Viirman The function concept and university mathematics teaching The function concept This thesis concerns the teaching of mathematics at university level, with a particular focus on the teaching of the function concept. The main aim of the thesis is describing and analysing the teaching practices of university and university mathematics teachers regarding the function concept, and how this concept is constituted through these practices. To this end, video recordings of lectures by mathematics teaching seven mathematics teachers at three Swedish universities were analysed using a discursive perspective, Sfard’s commognitive framework. The observed teaching was traditional in form, with teachers using “chalk talk” – simultaneously talking and writing on the board. The results show that the teaching practices of the teachers belong to two distinct but intertwined discourses – a mathematical Olov Viirman 2014:12 discourse, and a discourse of mathematics teaching. Classifications of important aspects of these discourses are presented, and it is found that the teachers’ discursive practices, while sharing overall form, still display considerable differences. Other results include an analysis of the levels of objectification displayed by the teachers in their discursive constitution of the function concept. The study contributes to a small but growing body of empirical research on university mathematics teaching practice. ISBN 978-91-7063-543-4 Faculty of Health, Science and Technology ISSN 1403-8099 Mathematics DISSERTATION | Karlstad University Studies | 2014:12 DISSERTATION | Karlstad University Studies | 2014:12 The function concept and university mathematics teaching Olov Viirman DISSERTATION | Karlstad University Studies | 2014:12 The function concept and university mathematics teaching Olov Viirman DISSERTATION Karlstad University Studies | 2014:12 Örebro Studies in Educational Sciences with an Emphasis on Didactics 10 urn:nbn:se:kau:diva-31204 ISSN 1403-8099 ISBN 978-91-7063-543-4 © The author Distribution: Karlstad University Faculty of Health, Science and Technology Department of Mathematics and Computer Science SE-651 88 Karlstad, Sweden +46 54 700 10 00 Print: Universitetstryckeriet, Karlstad 2014 WWW.KAU.SE Table of contents Acknowledgements ................................................................................................................................. 3 List of papers ............................................................................................................................................ 6 1 Introduction .......................................................................................................................................... 7 1.1 Aims of the thesis ....................................................................................................................... 10 1.2 How to read this thesis .............................................................................................................. 11 2 Literature review ................................................................................................................................. 13 2.1 The mathematical concept of function ................................................................................... 13 2.1.1 The historical development of the function concept .................................................... 13 2.1.2 Research on the teaching and learning of the function concept ................................. 17 2.2 Research on mathematics teaching in higher education ....................................................... 21 2.2.1 Self-reports and theoretical analyses of teaching practice ............................................ 23 2.2.2 Empirical research on university mathematics teaching ............................................... 25 3 Theoretical considerations ................................................................................................................ 30 3.1 Basic theoretical assumptions ................................................................................................... 30 3.2 Discursive approaches to mathematics education research ................................................. 32 3.3 Sfard’s commognitive framework ............................................................................................ 36 4 Methodology ....................................................................................................................................... 42 4.1 Research questions ..................................................................................................................... 42 4.2 Research design ........................................................................................................................... 43 4.2.1 Preliminary study ................................................................................................................. 43 4.2.2 Main study ............................................................................................................................ 44 4.3 Further methodological considerations................................................................................... 49 5 Overview of the papers ..................................................................................................................... 53 5.1 Paper I .......................................................................................................................................... 53 5.2 Paper II ......................................................................................................................................... 54 5.3 Paper III ....................................................................................................................................... 56 5.4 Paper IV ....................................................................................................................................... 57 5.5 Paper V ......................................................................................................................................... 59 6 Conclusions ......................................................................................................................................... 61 6.1 Teaching practices regarding the function concept............................................................... 61 6.1.1 Mathematical discourse ...................................................................................................... 61 6.1.2 Discourse of mathematics teaching ................................................................................. 65 6.1.3 General features of the teaching practices ...................................................................... 68 1 6.2 The discursive constitution of the function concept ............................................................ 71 7 Discussion ............................................................................................................................................ 75 7.1 Contributions of the thesis ........................................................................................................ 75 7.1.1 Contributions to research on university mathematics teaching practice .................... 75 7.1.2 Methodological contributions ........................................................................................... 76 7.1.3 Contributions to university mathematics teaching practice ......................................... 77 7.2 Reflections on the quality of the study .................................................................................... 78 7.3 Further research .......................................................................................................................... 80 8 References ............................................................................................................................................ 83 2 Acknowledgements In a sense the first seeds of this thesis were sown almost fifteen years ago, when, as a doctoral student in mathematics, I realized that I wasn’t that interested in doing research in mathematics per se. Instead my interests lay more towards the teaching of mathematics, and what Dörfler (2003) somewhat tentatively calls mathematicology, that is the meta-study of mathematics – the study of mathematics “as a given human phenomenon and activity” (ibid, p. 149). However, at the time I didn’t realize that you could conduct research in mathematics education, at least not in university mathematics education. Having discontinued my doctoral studies in mathematics, and realizing that getting a permanent position as a university mathematics teacher without a doctoral degree was almost impossible, I decided to turn to teaching mathematics in upper secondary school, and enrolled in a teacher education program. To get my degree as a teacher I needed to write a bachelor’s thesis in mathematics education, and I knew that I wanted to study conceptual development in mathematics in the context of teacher education. When planning the bachelor’s thesis I met Iiris Attorps, who was to become my advisor on the present thesis. She was very enthusiastic about my idea for the project, and already in the first meeting she and I had sketched an outline of a research project for a doctoral thesis, leaving me happy but somewhat shell-shocked.