INVESTIGAÇÃO DE ASPECTOS DA COMPREENSÃO RELACIONAL E DA INSTRUMENTAL EM GEOMETRIA ESFÉRICA

• Autor(a) principal: Fonseca, Jussara Aparecida da
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Repositório Institucional Universidade Franciscana
• Texto Completo: http://www.tede.universidadefranciscana.edu.br:8080/handle/UFN-BDTD/875
• topic: Triângulos esféricos. Compreensão relacional e instrumental. Investigações matemáticas. Ensino Superior.; Spherical triangles. Relational and instrumental understanding. Mathematical investigations. Higher Education.; Ciências e Matemática

This work presents the results of a research that aimed to analyze how undergraduate mathematics students understand metric relations in spherical triangles from Euclidean geometry perspective. In a first study, we found elements that justify our investigation, verifying the importance of the knowledge on non-Euclidean geometries for the understanding of Euclidean geometry itself. This way, we focused mainly in its development in initial teacher training courses. The qualitative and exploratory research was based on the theoretical and methodological assumptions from the mathematical investigations of João Pedro da Ponte and the categories of understanding by Richard Skemp: relational and instrumental. In the mathematical investigations, we researched the notion of task and the subsidies for the organization of our research and teaching sequence, and the types of understanding, elements that would allow us to verify those presented by the participants. For this purpose, tasks were organized in exploratory activities on the Euclidean plane, on the spherical surface and in spherical triangles. Those carried out in the plane aimed to recall some concepts of Euclidean geometry in order to later verify their existence on the spherical surface and, finally, the tasks involving spherical triangles focused on obtaining some of their relationships and properties. Besides these, evaluation activities were used in order to analyze the participants' understanding of the topics developed. Twelve undergraduate students from different semesters who major in mathematics participated in the investigation at Instituto Federal Farroupilha – Alegrete Campus. The results obtained from the analysis of the written records from the participants, which were based on criteria contributed to the mathematical investigations and types of understanding, showed aspects of the relational one on topics of spherical geometry, especially metric relations in spherical triangles. The topics worked on enable the participants to have another view on the development of both geometric knowledge and mathematics itself, since from them they glimpsed the existence of another different geometric model, but as consistent as that of Euclidean geometry. Moreover, the results also pointed to the need to create intermediate levels of understanding, in relation to those described by Skemp: almost relational understanding and almost instrumental understanding.