Integrating new technologies in teaching and learning mathematics leads educational researchers to deal with some fundamental questions (Jones, 1998, Laborde, 2007). We aim to present some of them, by studying the introduction of a Dynamic Geometry Software (DGS) in a classroom with 10 years old students (Assude, 2005). One difference between Geometry by drawing on a sheet (or on a drawing software like Mspaint) and Geometry with a DGS is that the software gives opportunity to the “dragging test”: when elements of a construction are dragged, all the properties employed in constructing the figure are preserved, and only them. So, the dragging test can be used to provide relevant feedbacks which aim students to test by themselves if their constructions are really geometric ones or not. We proposed to three teachers with fifth grade primary school students some situations using the DGS “Tracenpoche” (website). In this communication we will show through a case study how the geometrical construction done by the teacher with a DGS is useful to the students to understand the relations between the geometric objects. In a first time, students can see the construction done by the teacher with the DGS in a perceptive way, for example they can see the construction as a square. But when they move the construction, they can't see a square anymore if the teacher has defined the properties of a rectangle, a plane geometric figure having four right angles. But seeing something else, they may not necessarily appreciate the significance of what is invariant. They have to recognize the geometrical relations that persist in the variations during the dragging test. All the drawings they obtain have the same properties, for example, during the moving of the points they see a lot of rectangles. So the focus is directed not about one form on the screen, but about a set of forms, which have the same geometrical properties.

To describe students' and teacher's joint action, we use the categories of the Joint Action Theory in Didactics (JATD, Sensevy 2011, Sensevy 2012): the didactic contract as a system of habits between the teacher and the students, the didactic milieu as an antagonist system to the previously taught one. Joint action is modelized as a didactic game. In this specific game, the player A (the teacher) wins if and only if the other player B (the student) wins. In order to win the game, the teacher A has to lead the student B to a certain point, a state of knowledge which allows the student B to make the right move in the game (in our case, to see a figure as a realization of geometric properties), and the student has to make this move on his own (proprio motu clause). So the teacher has to be reticent, in a certain way, involving students in a succession of learning games, the modelprovidingthe description of the teacher's game on the students' game, and linking the didactic contract and the didactic milieu. These theoretical categories enable us to understand what could be the source of the changing view of students.

Assude, T. (2005). Time management in the work economy of a class, a study case: integration of cabri in primary school mathematics teaching. Educational Studies in Mathematics, vol 59 p 183-203 Hall, R. (2007). Strategies for video recording: Fast, cheap and (mostly) in control, in S. J. Derry (dir.), Guidelines for video research in education, Chicago, Data Research and Development Center: p 4-14, disponible en ligne : http://drdc.uchicago.edu/what/video-research-guidelines.pdf Jones, K.(1998). The Mediation of Learning within a Dynamic Geometry Environment, In A.Olivier & K. Newstead (Eds), Proceeding of tfe 22nd Conference of the International Group for the Psychology of mathematics Education. University of Stellenbosch, South Africa,vol 3, p 96-103. Laborde, C. (2007), The role and uses of technology in mathematics classrooms : between challenges and modus vivendi. Canadian Journal of Science. Mathematics and Technology Education, 7(1), p 68-92. Sensevy, G. (2011). Overcoming Fragmentation: Towards a Joint Action Theory in Didactics. In B. Hudson & M. Meyer (Eds.), Beyond Fragmentation : Didactics, Learning and Teaching in Europe (p 60-76). Opladen and Farmington Hills : Barbara Budrich. Sensevy, G. (2012). About the Joint Action Theory in Didactics. Zeitschrift für Erziehungswissenschaft Berlin: Springer.