Epistemological characteristics influencing didactic choices in course planning – the cases of Basic Topology and Differential Geometry

International audienceThere are many factors influencing the didactical choices made by university mathematics lecturers in course design. We focus on one of these: the inherent structure of the mathematical domain to be taught. Adopting an institutional perspective, the Anthropological Theory of the Didactic, we conduct an epistemological analysis of two courses aimed at second and third year mathematics students, Basic Topology (BT) and Differential Geometry (DG), showing how their different epistemological characters affect these didactical choices. We argue that the main differences stem from their aims and objects of study: where BT aims to develop the theory of topological spaces, and providing a categorization of them, DG aims at producing tools for analyzing geometrical properties of surfaces. In other words, where BT is theory-driven, DG is praxis-driven. We show how this and other differences relate to the course design, and argue this as a case of domain influencing pedagogy

Mathematics lecturers’ views on the teaching of mathematical modelling

The paper reports on the views and use of mathematical modelling (MM) in university mathematics courses in Norway from the perspective of lecturers. Our analysis includes a characterisation of MM views based on the modelling perspectives developed by Kaiser and Sriraman (2006). Through an online survey we aimed to identify the main perspectives held in higher education by mathematics lecturers and the underlying rationale for integrating (or not) MM in university courses. The results indicated that most respondents displayed a realistic perspective on MM when it came to their professional practice. There was a more varied response when it came to their views on MM in teaching. Regarding conditions influencing the use or non-use of MM in teaching, these mainly concerned the mathematical content and the institutional practices

Introduction to the papers of TWG14: University mathematics education

Research on university level mathematics education is a fast developing field as evident in the growth of the CERME University Mathematics Education (hereafter UME) Thematic Working Group. TWG14 was launched in CERME7 (Nardi, González-Martín, Gueudet, Iannone & Winsløw, 2011). After CERME8 (Nardi, Biza, González-Martín, Gueudet, & Winsløw, 2013), its leader team – in collaboration with TWG14 participants and others – worked towards a Research in Mathematics Education Special Issue on Institutional, sociocultural and discursive approaches to research in university mathematics education (Nardi, Biza, González-Martín, Gueudet & Winsløw, 2014) which focused on research that is conducted in the spirit of the following theoretical frameworks: Anthropological Theory of the Didactic, Theory of Didactic Situations, Instrumental and Documentational Approaches, Communities of Practice and Inquiry and Theory of Commognition

Introduction to the papers of TWG14: University mathematics education

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Negotiating different disciplinary discourses: biology students’ ritualized and exploratory participation in mathematical modeling activities

Non-mathematics specialists’ competence and confidence in mathematics in their disciplines have been highlighted as in need of improvement. We report from a collaborative, developmental research project which explores the conjecture that greater integration of mathematics and biology in biology study programs, for example through engaging students with Mathematical Modeling (MM) activities, is one way to achieve this improvement. We examine the evolution of 12 first-semester biology students’ mathematical discourse as they engage with such activities in four sessions which ran concurrently with their mandatory mathematics course and were taught by a mathematician with extensive experience with MM. The sessions involved brief introductions to different aspects of MM, followed by small-group work on tasks set in biological contexts. Our analyses use the theory of commognition to investigate the tensions between ritualized and exploratory participation in the students’ MM activity. We focus particularly on a quintessential routine in MM, assumption building: we trace attempts which start from ritualized engagement in the shape of “guesswork” and evolve into more productively exploratory formulations. We also identify signs of persistent commognitive conflict in the students’ activity, both intra-mathematical (concerning what is meant by a “math task”) and extra-mathematical (concerning what constitutes a plausible solution to the tasks in a biological sense). Our analyses show evidence of the fluid interplay between ritualized and exploratory engagement in the students’ discursive activity and contribute towards what we see as a much needed distancing from operationalization of the commognitive constructs of ritual and exploration as an unhelpfully dichotomous binary

The function concept and university mathematics teaching

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 mathematics teachers regarding the function concept, and how this concept is constituted through these practices. To this end, video recordings of lectures by 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 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

Klein's double discontinuity around the world - the case of calculus

International audienceAs a first step in an investigation of what is done to prepare prospective upper secondary teachers to teach calculus, an international survey of calculus in upper secondary school, at university and in teacher education was conducted. As a result, 14 national accounts, written by experts in the field, were collected. These were then analyzed, focusing on the institutional context, the role of the content in the institutions and differences in content between institutions. Differences and similarities between the countries, as described in the accounts, are presented, and implications of the results regarding how Klein's double discontinuity plays out in the different countries are discussed