Study of conics in some metrics: proposal for the development of spatial thinking

Abstract

Conics are essentially worked from their representation in the Usual metric (Euclidean) leaving aside the different representations that can be given when using other forms of measurement; the main objective of this article was to synthesize the most important results of the research work of Antonio and Garzón (2017) on the geometric and analytical study of conics when they are approached from other metrics (taxi, maximum and discrete) and to think on the way in which this subject is being taught. The Ministry of National Education of Colombia has a concern for the detriment of spatial thinking and there is a need to rescue the intuitive and crtitical analyses types in students. The research was carried out under the theoretical-documentary approach where some theoretical and didactic proposals of the conics from the taxi metrics were taken as the main reference; working with conics with different metrics allows exploring new ways of measuring and promotes the development of geometric-spatial thinking throughout different situations that confrontates the student's knowledge. Furthermore, from the theory of representation registers it turns out to be a favorable strategy because "understanding does not mean making a leap from the content of a representation to the purely mathematical concept represented but in relating diverse contents of representation of the same concept" (Duval, 2006, p. 158). On many occasions, a conic is associated with a specific graphic representation. In this research, new algebraic and geometric representations of conics were achieved when the way of measuring is changed, and with this we better understand the definition of each conic as a geometric place, making it clear that there are several ways to approach conics in a school teaching process that favour the development of spatial thinking.