Mastering by the teacher of the instrumental genesis in CAS environments: necessity of intrumental orchestrations

In this article, we study didactic phenomena identified in integration experiments within our classes, CAS (implemented in calculators). From this study, we show the interest of an instrumental approach to understand the influence of tools on the mathematical approach and on the building of student's knowledge: through a process—instrumental genesis—a calculator becomes a mathematical work tool; this process depends on the tool's constraints and potentialities, on students' knowledge, and on the class' work situations. To analyze the differentiation of instrumental genesis, we then have taken interest in students' behaviour and we propose a method enabling us to constitute a typology of extreme behaviour in environments of symbolic calculators. To take the variety of these genesis into account, the professor needs a particular organization of space and time of the study in the class. We suggest the notion of instrumental orchestration to name this organization. We demonstrate how this notion gives a better definition of the objectives, the configurations and the exploitation modes of different arrangements which aim at constituting coherent instrument systems for each student and for the class.