Uma investigação sobre os níveis dos conhecimentos geométricos de licenciandos em Matemática por meio da Teoria de van Hiele

Abstract

The little emphasis on teaching of geometry in Basic Education until the end of the 20th century and the prospects for its resumption from the beginning of the 21st century are the subject of studies by several authors, who consider that such abandonment was caused mainly by the teachers' unpreparedness in the face of geometry content. Considering this problem, this research aimed to investigate, based on van Hiele's Theory, the undergraduates’ geometric thinking levels in Mathematics during teaching activities and the experience of a didactic sequence on the Pythagorean Theorem. The participants were 22 students from the Federal University of Sergipe, who were enrolled in the subject of Supervised Internship in Mathematics Teaching I in the second semester of 2019. We developed this study involving the initial training of mathematics teachers with the following guiding questions: What is the level of geometric thinking achieved by students in the second half of a Mathematics Degree course? Do these graduates know other geometric approaches to the Pythagorean theorem besides the one usually presented in textbooks? What are the possible implications of experiencing the didactic sequence for the training of teachers of these graduates? The model proposed in van Hiele's theory classifies the students' geometric thinking levels in five, assigning properties to these levels and, in order to guide the pedagogical work of the teacher, organizes learning phases within each level. Therefore, we consider that such a model, being present in the teaching of mathematics, could contribute to the resumption of geometry teaching in Basic Education. Our investigation, which used a qualitative approach, is characterized as an action research, due to the direct contact of the researcher with the participants, being aware of the research stages. The instruments used in data collection were questionnaire, field diary, interviews, and the development of a didactic sequence. From the analysis of the experience, the geometric knowledge of the undergraduates was classified, in its majority, in the third level of the van Hiele's Theory. Although the students are enrolled in the second half of the Degree, having taken many of the course that require mastery of various properties, axioms, and postulates of geometry, which requires knowledge corresponding to the fourth level, they demonstrated to know few geometric representations of the Pythagorean Theorem, as well as the students revealed conflicts between geometric and algebraic aspects when describing the statement of this theorem. We also observed the lack of knowledge of the undergraduate students regarding the van Hiele theoretical model. The intervention contributed to the process of building the teaching knowledge of these graduates in three main aspects, disciplinary, curricular, and professional knowledge. This highlights the contributions of didactic approaches that articulate theory and practice in the initial training of teachers who teach mathematics.