Back and forth between Euclidean geometry and Taxicab geometry To foster students' theoretical thinking in digital contexts

High school students are expected to develop skills to understand definitions of concepts in Euclidean geometry, to handle conjectures and proofs, and to use them later to prove other propositions of the theory. Students do not generally find this shift easy. One of the reasons would seem to be related to the understanding of the definition of a concept (Moore, 1994). In geometry, properties of figures derive from definitions within an axiomatic system: a figure is “controlled by its definition” (Fischbein, 1993). Research shows that by exploring the properties of figures and making conjectures, especially through dynamic geometry systems (Baccaglini-Frank & Mariotti, 2010), students can better develop their understanding. Furthermore, comparing different geometric worlds aids helps the advancement of Euclidean knowledge. In our ongoing experience, both high school and undergraduate students are involved in exploratory cooperative learning activities within taxicab geometry (Krause, 1975) using digital tools, and comparing them with Euclidean geometry. They are required to investigate what concepts can be defined in terms of the taxicab distance by analogy with those existents in the Euclidean one (Berger, 2015) as well as whether a theorem or an open problem in one of the geometries is valid or open within the other. We face the issue of promoting students’ conceptual understanding and theoretical thinking by transferring their knowledge of concepts or the validity of statements back and forth between Euclidean and taxicab geometry, with the goal of providing evidence that productive learning processes are activated when students engage in these activities.