Alejandro S. González-Martín

Didactic Situations and Didactical Engineering in university mathematics: cases from the study of Calculus and proof

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic.

Conceptually driven and visually rich tasks in texts and teaching practice: the case of infinite series

The study we report here examines parts of what Chevallard calls the institutional dimension of the students’ learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duvals Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concepts significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests.

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.

The introduction of real numbers in secondary education: an institutional analysis of textbooks

In this paper we analyse the introduction of irrational and real numbers in secondary textbooks, and specifically the propositions on how these should be taught, in a sample of Brazilian textbooks used in state schools and approved by the Ministry of Education. The analyses discussed in this paper follow an institutional perspective (using Chevallards Anthropological Theory of Didactics). Our results indicate that the notion of irrational number is generally introduced on the basis of the decimal representation of numbers, and that the mathematical need for the construction of the field of real numbers remains unclear in the textbooks. It seems that textbooks used in secondary teaching institutions develop mathematical organisations which focus on the practical block.

Introducing the concept of infinite series: preliminary analyses of curriculum content and pedagogical practice

This article was published in the journal, Research in Mathematics Education [Routledge (Taylor & Francis)

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’