Ensino exploratório e o teorema de Pitágoras: uma análise das manifestações do pensamento geométrico

Abstract

This research aims to identify and characterize manifestations of geometric thinking in students in the 9th grade (students aged 14-15) when working with mathematical tasks that aim to introduce the Pythagorean Theorem. van Hiele's theory was used to characterize students' geometric thinking, and the Exploratory Teaching approach was used to solve mathematical tasks throughout classes. Six classes were conducted, each lasting 50 minutes, with a 9th grade class, in a public educational institution in the State of Paraná, in which the researcher was the leading teacher. The data analyzed were produced from audio recordings of discussions held in the classroom. In the first task, students, when manipulating the sticks, often demonstrated level 0 of visualization or recognition. Evidence of progression to level 1 of analysis emerged, above all, in interactions with the teacher, who encouraged discussion. However, the students in the group analyzed did not independently state the condition for the existence of a triangle, leaving the teacher to formalize this idea, after contributions from other groups. In Task 2, students showed significant progress in geometric thinking, moving from level 0 of visualization or recognition to levels 1 of analysis and 2 of informal deduction or ordering, with some signs of level 3 of formal deduction, especially in the discussion phases and systematization led by the teacher. Task 3, although reduced and developed in a hurried manner, also revealed evidence of student's advancement through the levels of van Hiele's theory, with the majority consolidating their understanding at level 2 of informal deduction or ordering and only a few demonstrating evidence of level 3 of formal deduction, as they deepen their understanding of the Pythagorean Theorem. Finally, it can be said that the teacher, in an Exploratory Teaching environment, created conditions for students to gradually advance through the levels of geometric thinking. However, reflections and considerations regarding Exploratory Teaching can be made, such as the care that the teacher must take when making comments during the students' autonomous work, so as not to indicate to them the strategy to follow, thus reducing the challenge of intellectual tasks; and time management, avoiding starting a task on one day and finishing it only on another class day. As this is the teacher's first experience with the Exploratory Teaching approach, reflections on the challenges of incorporating such a teaching approach into her own practice have the potential to bring professional learning that will possibly have an impact on her practice in the future.