From resource to document: Scaffolding content and organising student learning in teachers’ documentation work on the teaching of series

We examine teachers’ use of resources as they prepare to teach the topic of numerical series of real numbers in order to identify how their personal relationship with mathematical content—and its teaching—interacts with their use of a commonly used textbook. We describe this interplay between textbook and personal relationship, a term coined in the Anthropological Theory of the Didactic (ATD, Chevallard, 2003), in the terms of documentation work (resources, aims, rules of action, operational invariants), a key construct from the documentational approach (DA, Gueudet & Trouche, 2009). We do so in the case of five post-secondary teachers who use the same textbook as a main resource to teach the topic. Documentational analysis of interviews with the teachers led to the identification of their aims and rules of action (the what and how of their resource use as they organise their teaching of the topic) as well as the operational invariants (the why for this organisation of their teaching). We describe the teachers’ documentation work in two sets of aims/rules of action: scaffolding mathematical content (series as a stepping stone to learning about Taylor polynomials and Maclaurin series) and organising student learning about series through drill exercises, visualisation, examples, and applications. Our bridging (networking) of theoretical constructs originating in one theoretical framework (personal relationship, ATD) with the constructs of a different, yet compatible, framework (documentation work, DA) aims to enrich the latter (teachers’ documentation work) with the individual agency (teachers’ personal relationship with the topic) provided by the former

Communities in university mathematics

This paper concerns communities of learners and teachers that are formed, develop and interact in university mathematics environments through the theoretical lens of Communities of Practice. From this perspective, learning is described as a process of participation and reification in a community in which individuals belong and form their identity through engagement, imagination and alignment. In addition, when inquiry is considered as a fundamental mode of participation, through critical alignment, the community becomes a Community of Inquiry. We discuss these theoretical underpinnings with examples of their application in research in university mathematics education and, in more detail, in two Research Cases which focus on mathematics students' and teachers' perspectives on proof and on engineering students' conceptual understanding of mathematics. The paper concludes with a critical reflection on the theorising of the role of communities in university level teaching and learning and a consideration of ways forward for future research

Transition from School to University Mathematics: Manifestations of Unresolved Commognitive Conflict in First Year Students’ Examination Scripts

We explore the transition from school to university through a commognitive (Sfard 2008) analysis of twenty-two students’ examination scripts from the end of year examination of a first year, year-long module on Sets, Numbers, Proofs and Probability in a UK mathematics department. Our analysis of the scripts relies on a preliminary analysis of the tasks and the lecturers’ (also exam setters’) assessment practices, and focuses on manifestations of unresolved commognitive conflict in students’ engagement with the tasks. Here we note four such manifestations concerning the students’ identification of and consistent work with: the appropriate numerical context of the examination tasks; the visual mediators and the rules of school algebra and Set Theory discourses; the visual mediators of the Probability and Set Theory discourses; and, with the visual mediators and rules of the Probability Theory discourse. Our analysis suggests that, despite lecturers’ attempts to assist students towards a smooth transition to the different discourses of university mathematics, students’ errors at the final examination reveal unresolved commognitive conflicts. A pedagogical implication of our analysis is that a more explicit and systematic presentation of the distinctive differences between these discourses, along with facilitation of the flexible moves between them, is needed

The French Didactic Tradition in Mathematics

This chapter presents the French didactic tradition. It first describes theemergence and development of this tradition according to four key features (role ofmathematics and mathematicians, role of theories, role of design of teaching andlearning environments, and role of empirical research), and illustrates it through two case studies respectively devoted to research carried out within this traditionon algebra and on line symmetry-reflection. It then questions the influence of thistradition through the contributions of four researchers from Germany, Italy, Mexicoand Tunisia, before ending with a short epilogue

Research on Teaching and Learning Mathematics at the Tertiary Level:State-of-the-art and Looking Ahead

This topical survey focuses on research in tertiary mathematics education, a field that has experienced considerable growth over the last 10 years. Drawing on the most recent journal publication as well as the latest advances from recent high quality conference proceedings, our review culls out the following five emergent areas of interest: mathematics teaching at the tertiary level; the role of mathematics in other disciplines; textbooks, assessment and students’ studying practices; transition to the tertiary level; and theoretical-methodological advances. We conclude the survey with a discussion of some potential ways forward for future research in this new and rapidly developing domain of inquiry