The Meaning of Conics: Historical and Didactical Dimensions

Abstract

My main thesis is that the present meaning of conics is the result of the complex relationships between the different processes of studying conics during different historical ages, each of which has left a residue in the names, the problems, the means of representations, the rules of actions, and the systems of control. To investigate the present meaning, we may refer to the historical development of the study of conics by means of time periods, each framed in the culture of a different age. Even if from today's standpoint all the conics studied in the different time periods can be identified as the same objects, inside each time period different meanings have been built by geometers to the extent that conics are representative of the development of different conceptualizations of space and geometry over the ages. As a corollary, I claim 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 (as meant by Hanna & Jahnke, 1994), on the one hand, and the problem of historical contextualization, on the other, are coped with by means of a selection of tasks. Finally, a small-group study of a special model for the parabola (the orthotome, inherited from the Greek tradition) will be analyzed. It concerns how the meaning is constructed by students through the introduction of a conscious anachronism that fosters an intentional recourse to different tools developed in different ages and allows the students to relate different ways of representation to each other.