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

Abstract

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.

Keywords:

• university mathematics education
• Theory of Didactic Situations
• didactical engineering
• didactic contract
• milieu

Notes

1. Throughout we use the term “teacher” to designate the person whose role is to facilitate student learning. When the context concerns university level, the term refers to those who teach at university, acknowledging that this may correspond to different types of status: PhD students, lecturers, research associates, etc.

2. Let b ∊ ℜ or b = ∞, and f: [a, b) → ℜ. f is said to be locally integrable within [a, b) if for every x, a ≤ x < b, f is Riemann-integrable in [a, x].

3. ‘Semantic’ refers here to the classical distinction in logic between syntax and semantics: semantics concerns the relation between signs and objects they refer to; syntax concerns the rules of integration of signs in a given system.

4. To analyse an assertion in a dialogical way means placing it within the dialogue in which it takes place.

5. For the complete a priori analysis, see Barrier (2009, pp. 231–259).