The meaning of conics: historical and didactical dimensions

In this paper we shall draw on the analysis of the research project Mathematical Machines concerning the teaching - learning of geometry in high school (grades 9th - 13th). Even if the project is actually broader, we have chosen the special topic of conic sections (or conics), that is supposed to be representative of the whole approach. Our main thesis is that the present meaning of conics is the result of the complex relationships between the different processes of studying conics in different ages, each of which has left some sediment in the names, in the problems, in the means of representations, in the rules of actions, in the systems of control. To investigate it, we shall analyse the historical development of the study of conics, by means of temporal sections, each framed in a different age culture. Even if from today's standpoint, all the conics studied in the different temporal sections can be identified as the same objects, inside each temporal sections different meanings have been built by geometers, to the extent that conics are representative of the development of different conceptualisations of space and geometry over the ages. As a corollary, we assert that it is not possible to build the meaning of conics through only a one-sided approach, as, for instance, through the most widespread algebraic definition.If history is an unavoidable component of the construction of meaning, a didactic problem immediately arises: how is it possible to introduce students to the historical problematic without undue oversimplification? An exemplary teaching experiment will be described to show how the problem of epistemological complexity on the one side and the problem of historical contextualisation are coped with by means of a selection of tasks. Finally the small group study of a special model for parabola, (the orthotome, inherited from the Greek tradition) will be analysed. It concerns how the meaning is constructed by students, through the introduction of a conscious anachronism, that fosters the intentional recourse to different tools, developed in different ages, and allows them to relate different ways of representation to each other.