Uses and Functionalities of the Angle in the Development of Trigonometric Thinking

Abstract

The treatment of the angle is recognized in various research studies as a fundamental activity in the development of trigonometric thinking; however, it is complex due to its very nature and its undifferentiated treatment in trigonometry. Therefore, this paper aims to show the uses and the seven functionalities of the angle identified during the resolution of a didactic design focused on the development of trigonometric thinking that was implemented as part of the research in 2020. To this end, the authors examine, from a qualitative approach and from the theoretical framework of the Socioepistemology of Mathematics Education (TSME), the solutions given by 12 secondary school students from Mexico City when responding to the didactic design developed by Scholz in 2020. Moreover, the investigators established the analysis of the information in two phases: in phase 1, a triangulation of the evidence (from the field diary, recordings, and student notes) was carried out and the actions and activities were identified (from the TSME). Meanwhile, during phase 2, the uses of the angle were determined according to its multifaceted nature. Subsequently, the functionalities are determined according to the intentionality of the set of uses, actions, and activities. Among the main findings of the 2020 research are the identification of seven functionalities of the angle, such as: construction tool, referent for applying an arithmetic-algebraic tool, referent for applying an arithmetic-trigonometric tool, referent for studying the relations given in the model, referent for operating arithmetically, referent for employing an empirical-metric tool and referent for classifying triangles. Although these uses and functionalities of the angle are employed indirectly and mainly in geometric contexts, they are necessary when used as referents in the treatment of other tools, operations, and essential mathematical constructs in the study of trigonometry. Finally, the investigators recommend for the development of trigonometric thinking to give special consideration to the conceptualization and treatment of the angle, so that its use is more explicit and visible, as it is a necessary reference for the treatment of other mathematical tools, operations, and constructs.