Carl Winsløw

Components of Mathematics Teacher Training

Initial mathematics teacher education is primarily concerned with knowledge—the acquisition of knowledge required for the teaching of mathematics. Opinions as to what exactly comprises this knowledge and how it is best delivered and best learned varies widely across different contexts. In what follows, we will look more closely at this concept of teacher knowledge and how it plays itself out in the context of initial mathematics teacher education.

An institutional approach to university mathematics education: from dual vector spaces to questioning the world

University mathematics education (UME) is considered, in this paper, as a kind of didactic practice – characterised by institutional settings and by the purpose of inducting students into mathematical practices. We present a research programme – the anthropological theory of the didactic (ATD) – in which this rough definition can be made much more precise; we also outline some cases of ATD-based research on UME. Three cases are presented in more detail. The first is a theoretical and empirical study of the topic of dual vector spaces, as it appears in undergraduate courses on linear algebra. The second case concerns a similar study of the practices and theories on limits of functions which students may develop in calculus courses. Finally, a third study illustrates the use of ATD to design and experiment innovative approaches to mathematical modelling in the setting of a first mathematics course for engineering students.

The Effros-Maréchal topology in the space of von Neumann algebras

New concepts of limes inferior and limes superior in the space of von Neumann algebras on a fixed Hilbert space are defined, and the topology corresponding to these notions is related to earlier works of Effros and Marechal. A main technical result is that the commutant operation is a homeomorphism on the space of von Neumann algebras with this topology. Further, the topological properties of several classes and types of von Neumann factors (regarded as subspaces) are determined, and also continuity-type results for Tomita-Takesaki theory are proved. Some applications to subfactor theory are given.

Institutional, sociocultural and discursive approaches to research in university mathematics education

In this paper we present the work generated during and after a Working Group session at the BSRLM conference of March 1 st 2014 entitled Institutional, sociocultural and discursive approaches to research in (university) mathematics education: (Dis)connectivities, challenges and potentialities. In the session we organised a discussion based on highlights from a Special Issue (SI) for Research in Mathematics Education (entitled Institutional, sociocultural and discursive approaches to research in university mathematics education, 16(2)) which we had just finished writing and editing, together with 20 other colleagues from 11 countries. The approaches covered by the SI papers are: Anthropological Theory of the Didactic; Theory of Didactic Situations; Instrumental and Documentational Approaches; Communities of Practice and Inquiry; and, Theory of Commognition. The papers present recent cutting edge research on several aspects of university mathematics education: institutional practices, analysis of teaching sequences, teacher practices and perspectives, mathematical and pedagogical discourses, resources and communities of practice. In the WG session we invited participants to generate university mathematics education research questions in a small group discussion, and then address these to the whole group in order to discuss how different issues could be dealt with by the different approaches covered by the SI. Our overall aim was to explore how these approaches may offer complementary, overlapping and in some cases diverging or even incommensurable points of departure for dealing with such questions. The participant small groups generated the following list of questions: (1) How can issues of equity and gender be explored by the frameworks presented in the SI? (2) What are the praxis and logos in different courses (e.g. in pure and applied mathematics)? (3) What are the distinct differences of the didactic contract in different courses (including those other disciplines with a strong mathematical component)? (4) What communication practices can we discern in students’ writing? In this paper we present short answers from each framework to a (slightly amended version of) one of the research questions asked by the WG participants, namely (4). To this purpose we first outline how we developed a more detailed question based on (4), which, for the purposes of this paper, will act as a common Research Question. We then use this as a platform on which to illustrate the potentialities of the frameworks presented in the SI. We conclude with a few thoughts on ways forward of this work.

A Comparative Perspective on Teacher Collaboration: The Cases of Lesson Study in Japan and of Multidisciplinary Teaching in Denmark

In this chapter, we present and compare two quite different organisations of teachers’ collaborative work: that of lesson study as a means for professional development of mathematics teachers in Japan, and that of Danish high school teachers’ collaboration in the setting of multidisciplinary modules. Each of these turns out to be crucially affected by certain school level paradidactic infrastructures, defined conditions and constraints of teachers’ collaborative work in preparing, observing and evaluating actual teaching. The systematic and comparative study of paradidactic infrastructures is proposed as a way to interpret and ultimately overcome difficulties which arise, for instance in attempts to realise major educational reforms.

Mathematics at University: The Anthropological Approach

Mathematics is studied in universities by a large number of students. At the same time it is a field of research for a (smaller) number of university teachers. What relations, if any, exist between university research and teaching of mathematics? Can research “support” teaching? What research and what teaching? In this presentation we propose a theoretical framework to study these questions more precisely, based on the anthropological theory of didactics. As a main application, the links between the practices of mathematical research and university mathematics teaching are examined, in particular in the light of the dynamics between “exploring milieus” and “studying media”.

Theorizing Lesson Study: Two Related Frameworks and Two Danish Case Studies

Lesson study refers to certain well-established professional development practices for teachers in Japan. Over the past 30 years, the phenomenon drew the attention of scholars in other countries, and their writings have inspired several ‘movements’ of lesson study implementation. As scholars observe both successes and difficulties in these endeavors, the need arises for finer methods to characterize and monitor the processes and objects which go into what is broadly referred to as lesson study. This brief presents an overall characterization of lesson study in terms of the notion of paradidactic infrastructure, in relation to specific adaptations of two related theoretical frameworks. We argue that the use of these frameworks can help sharpen researchers’ understanding of lesson study as a phenomenon. We exemplify the use of these tools with cases from our own work on pre- and in-service teacher development in Denmark.

Language Diversity in Research on Language Diversity in Mathematics Education

Research on educational phenomena, which always occur in cultural and societal contexts, cannot escape the trappings of language. This applies even to research on the teaching and learning of mathematics, despite the relative similarity of the topics taught around the world and the practicality of communicating internationally in a few common languages. In this chapter, we propose a framework which allows us to identify some intriguing effects of the need for shared natural languages to function in the work of researchers. We use this framework to examine some cases of research on language diversity in mathematics education, including research from the present ICMI study. The framework is essentially based on the anthropological theory of didactics, used as a “meta-theory.”

Challenges and Opportunities for Second Language Learners in Undergraduate Mathematics

In this chapter, we describe the challenges and opportunities (related to mastery of the language of instruction) that second language learners face in undergraduate mathematics programs. The presence of such students is nowadays very common in many undergraduate courses due to migration, student mobility, and other factors. We provide examples of various multilingual contexts at the university level, summarize insights from international research on this topic, and present emergent proposals for helping students to overcome these challenges. This chapter highlights the importance of continuing research on the topic of second language learners in undergraduate mathematics courses so that we can offer research based approaches to improve undergraduate mathematics teaching in multilingual contexts. This chapter will provide the mathematics education research community involved in advanced mathematics, as well as instructors and policy makers, to become aware of the issues involved in undergraduate mathematics learning and teaching for second language learners.

CERME7 Working Group 14: University mathematics education

The WG14 papers presented at CERME7 provide ample evidence of growth in this area of research. The nutshell descriptions of WG14 papers that follow are structured around the main themes of the Group’s Call for Papers (such as the teaching and learning of particular topics, pedagogical and curricular issues at university level, the transition from school to university mathematics) and the observation that, beyond staple references to classic constructs from the AMT era, several works employ approaches such as the Anthropological Theory of Didactics (ATD: Chevallard 1985), and discursive approaches (e.g. Sfard 2008). Xhonneux and Henry employed the ATD framework to distinguish between mathematical and didactic praxeologies in the context of teaching and learning of Lagrange’s Theorem in calculus courses to mathematics and economics students. Gyöngyösi, Solovej and Winslow was another: in it a part of a transitional course in Analysis was taught with a combination of Maple and paper-based techniques, resulting in mixed reception and performance by students. A third was Barquero, Bosch and Gascón: from its analyses ‘applicationism’ emerges as the prevailing epistemology of mathematics in science departments, potentially hindering the teaching of mathematical modelling to science students. Several papers employed a discursive approach. Jaworski and Matthews used this to trace university mathematicians’ pedagogical discourse, and suggested links to their ontological and epistemological perspectives. Biza and Giraldo described how computational inscriptions have potentialities and limitations that can be helpful in students’ exploration of newly introduced mathematical concepts. Three papers made use of Sfard’s commognitive framework. Viirman employed this framework to trace the variation in the pedagogical discourses of mathematics lecturers in the course of their introducing the concept of function. Stadler described students’ experiences of the transition from school to university mathematics as an often perplexing re-visiting of content and ways of working that seem simultaneously familiar and novel (for example in the case of solving equations). Nardi outlined interviewed mathematicians’ perspectives on their newly arriving students’