O modelo de Van Hiele e a teoria dos campos conceituais : complementaridade na conceitualização de prisma e pirâmide - Santos, Rudinei Alves dos (2023)
Paying attention to the student's actions when facing situations that corroborate the process of building geometric concepts is extremely important. This posture favors the organization and/or reorganization of the teacher's activities and mediations, regarding the promotion of the acquisition of mathematical knowledge gathered by the situations. This study, constructed from a qualitative approach, along the lines of the research participants, seeks to contribute to the teaching of Geometry, through research anchored in the Model of the Development of Geometric Thinking of the van Hiele couple (van Hiele's Model), and in the Theory of Conceptual Fields (CBT) by Gérard Vergnaud. From this search emerges the following research question: in what terms are situations in the conceptual field of polyhedrons: prisms and pyramids, elaborated in line with the levels of Van Hiele's Model of the Development of Geometric Thought, contribute to the evocation and revelation of concepts and theorems- inaction, announced by Gérard Vergnaud, in the Theory of Conceptual Fields? Faced with this research question, the objective is to analyze external representations due to mental operations performed by students during the construction process of their proposed solutions to situations in the conceptual field of polyhedra: prisms and pyramids, which allow the identification and enunciation of operative invariants: concepts-in-action and theorems-in-action. The 12 (twelve) research participants are third-grade high school students at IFPA/Campus Santarém, and the data collection instruments adopted are questionnaires; interviews; notes; interventions based on situations from the conceptual field of polyhedrons (prisms and pyramids); videos, and audio produced during the execution of activities. The collected data were qualitatively analyzed in the light of the theories that constitute the theoretical framework of this research: Van Hiele's Model and the Theory of Conceptual Fields, through the identification, enunciation, and discussion of the operative invariants recorded by the students' actions on situations constructed for the van Hiele's first three levels (N0 - Visualization, N1 - Analysis and N2 - Informal Deduction). The analyses show that, initially, the participant’s level of understanding was visual and, despite the need for more situations that allow for discussion, expansion, deepening, and, consequently, understanding of the conceptual field under study, there was an advance in understanding up to N1. N2 was not fully achieved by any participant, confirming that the conceptualization process of geometric concepts does not happen in a brief period. It is noteworthy that the teaching strategy built based on the complementarity between the van Hiele Model and the TCC, in addition to indicating how geometric concepts were conceived over the time of school training, provided moments of reflection for the students, who became aware of the importance of their prior knowledge for the construction of new concepts, and of the teacher (researcher) who, based on the analyses, promoted (re)structuring of activities, aiming at greater interaction among students. This research proved the thesis: situations in the conceptual field of polyhedrons (prisms and pyramids), consistent with van Hiele's Levels of Development of Geometric Thinking, favor actions that lead to student records, anchored in operative invariants (concepts-in-action and theorems-in-action) and allow the teacher to identify, in the representations (conscious or unconscious) expressed by the students, these operational invariants that can contribute to a better evaluation of the conceptualization process of this conceptual field.