Abstract:

The issue of integrating technological tools into teaching came up early in the field of mathematics. Being shaped from the start by the proximity between mathematics and computer science, with activities related to algorithmics, programming and computation, the reflection soon got diversified, thanks to technological evolution and software development opening new possibilities of visualization and of direct experimental work on mathematical objects (Artigue, 2011).

So among others, a significant research trend emerged towards Dynamic Geometry Software packages (DGSs) such as Cabri-géomètre, The Geometer’s Sketchpad or GeoGebra (Laborde et Capponi, 1994 ; Arzarello et al., 2002 ; Restrepo, 2008, etc.). These software packages are now largely utilized in the classroom of the industrialized countries, and have a decisive influence on geometry teaching and learning. But as it is the case with many technological tools, there seems to be a gap between the foreseen benefits and the technological practices really set in the classroom (Lagrange, 2009). Indeed, “Teachers and teacher educators […] are searching for guidelines that foster successful integration of new media into teaching practice. Researchers […] have difficulties in providing evidence of improved learning with technological means, as well as in understanding the influence of technology on learning” (Kieran & Drijvers, 2006, p. 205). We posit that conceiving such ‘guidelines’ requires a deep knowledge of the interweaving between the tools and the structured contents they stem from.

Leaning on the instrumental genesis theoretical framing (Rabardel, 1995 ; Artigue, 2002 ; Lagrange, 2005, etc.), our research program is aimed at analyzing the complex relationship between students, teacher, knowledge and tasks in technological teaching environments. The frame draws a distinction between the artefact, the technological object itself, and the instrument which is constructed from it by an individual using it in a given type of activities. We call instrumental genesis the progressive elaboration of the instrument through a back and forth between two processes :
• instrumentation, by which the individual adapts his or her own action to the artefact ;
• instrumentalization, by which the individual adapts the tool through the development of techniques and usage schemes.

Different tasks exploiting DGSs will be analyzed from an instrumentation perspective, to show that dynamical geometry as it is carried out in the classroom is hardly not ‘ruler and compass geometry with some extra facilities’, but is rather a brand new geometry with its own rules and difficulties, these being likely to divert the teaching from its initial goals. We will also propose teaching situations where a shrewd instrumentalization of the DGS allows going beyond its plain functionalities, by fostering rich and significant students’ reflections and by refocusing students’ activity on more standard geometrical concepts and objects (Tanguay & Geeraerts, 2014 ; Tanguay et al., 2013 & 2014).

On the way, we will try to enforce the necessity of arousing, in the mind and understanding of pre-service and in-service teachers, an increasing consciousness about the issue of technology integration into teaching, and of nourishing an epistemologically better sustained reflection, as regards the foundations and subterranean structures of technological tools.