The instrumental approach: the institutional dimension Michèle Artigue Université Paris 7

The origin of this approach An institutional problem: the contrast between on the one hand, the number of institutional incentives, the richness of research and innovative work, the amazing development of ICT resources for mathematics education and, on the other hand, an educational integration which, globally, tended to remain rather marginal. The desire to question usual explanations (material problems and teachers’ inertia) and to go beyond these.

The first insights Systems of values, issues of legitimacy Scientific legitimacy Social legitimacy Educational legitimacy Technology seen as an educational tool at the service of « permanent » values

The lack of educational legitimacy and its resulting effects Emphasizing the potential and among its different possible facets, those linked with highest educational value: conceptual and strategic thinking, or with recurrent problems such as time constraints. Minimizing the effects of the computer transposition of knowledge. Developing a naturalistic vision of integration, and a naïve vision of teacher’s role.. Trying to seduce teachers by presenting them impressive situations without taking into account the didactic and technological expertise required for their management.

How to approach these problems? A « natural » tool, given my specific culture: the anthropological approach. But also some feeling that I needed something more, and a particular event: meeting with a book by Rabardel which had just came out. Immediatly, the conviction that some coherent and operational tools for analysis could be built from the conjonction of these two approaches.

Some key points in the anthropological approach by Chevallard Mathematical objects are nothing absolute; they arise from institutional practices : « praxeologies » Praxeologies can be seen as complexes of tasks- techniques-technology- theory The advance of knowledge goes along with the routinisation of tasks and techniques, the naturalization of knowledge Knowing = ideoneity with institutional relationships

The anthropological approach An approach leading to give some priority : to the analysis of institutional norms and values; to the interaction between the personal and institutional relationships to knowledge; to the analysis of the computer transposition of knowledge and its possible effects; to possible conflicts between norms and values, the way there are managed and their resulting effects; to the new praxeologies induced by technological change, and their relationships with the ancient ones (both mathematics and didactic praxeologies).

The ergonomic approach To the instrument Instrumental genesis From the artefact InstrumentalisationInstrumentation ConstraintsNew potential

An implementation of this approach Understanding what knowledge about variation of functions can be approached and learnt in a CAS environment: by building some engineering design; by observing its life both at a personal and institutional levels through the collection of different types of data.

The personal dimension: what results from the interview sessions? First interview : understanding the variations of f(x)=x(x+7)+9/x

The second step: symbolic computations CAS gives you everything you need…

Then, coming back to the graphic application

Further verifications using tables and zooms

The instrumental genesis of variation From a personal point of view: a slow progression from the graphic calculator culture towards the CAS culture. an evident dependence of this progression on the evolution of students’ mathematical knowledge. specific phenomena : zapping, over- verification strategies, explosion-reduction phases. How to explain such results ? Connecting the personal dimension with the institutional dimension

The ordinary life of techniques in their relationship with conceptualisation Solving new problems Exploratory phase: Craft work Selection, improvment, institutionalisation of some techniques Routinisation and investment in more complex situations Development of a « theoretical » discourse Personal techniques Offical techniques

What changed with instrumented techniques? During the first experimentation: no official selection, legitimation but not institutionalisation, a « theoretical discourse » reserved to paper and pencil techniques Instrumented techniques remained private objects which were not officially worked out

Some specific difficulties … The diversity of commands and possible techniques The mixture of computer and mathematics knowledge engaged in explanation and jusitications, including new mathematics knowledge The problematic accessibility of technical knowledge The distance with ordinary norms and values of mathematics teaching

Revisiting the dialectics technical/ conceptual: the epistemic value of instrumented work and techniques Standard environments CAS environments Immediate results Step by step solving Multiplicity of accessible results Surprising results New mathematical needs

Becoming aware of such constraints and difficulties: the second experimentation Some essential changes in the design: Selection of commands and techniques official work of institutionalisation and routinisation management of the didactic contract taking into account its necessary evolution With evident positive effects

Some open questions How to have such results influencing practice? What is offered by this approach with respect to others dealing with the problems? Where are the real complementarities? Up to what point, the language of schemes and that of techniques can be made coherent when used simultaneously?