Investigação da complexificação, generalização e modelo combinatório dos números de Padovan e Perrin com elementos de uma engenharia didática

Abstract

In the analysis of the History of Mathematics curriculum component, a gap was observed in the teaching of recurring numerical sequences. Thus, this study proposes the exploration of Padovan and Perrin sequences, enabling an investigation into their epistemological, cognitive, and didactic aspects. Thus, this thesis aims to analyze the contributions of Didactical Engineering and the Theory of Didactical Situations to teaching models of these numbers, within an investigation of the processes of generalization, complexification, and combinatorics (epistemic-mathematical field), in the context of initial teacher training courses, using a manipulative material. The methodology employed was based on the principles of Artigue’s Didactical Engineering. The theoretical framework of the research drew on Brousseau’s Theory of Didactical Situations to develop didactical teaching situations involving the respective combinatorial models, fostering students’ intuitive thinking and emphasizing the construction of definitions using manipulative materials. The epistemic-mathematical field was developed by selecting combinatorial models through teaching situations for students in the Mathematics Teacher Education program, specifi- cally in the curricular component of History of Mathematics and Mathematics Teaching Practices. This was conducted at higher education institutions such as IFCE Fortaleza campus, URCA Crajubar campus, and UECE Fortaleza and Quixadá campus, involving a total of 48 student participants. The development of the morphic board provided support for studying combinatorial approaches to these sequences. The data analysis led to the conclusion that combining the use of manipulative materials with the Theory of Didactical Situations enabled the 48 students to employ strategies in formulating their definitions. The results show that constructing definitions was employed during the search for the recurrence of sequences, potentially influencing the tea- ching process positively. Finally, the students were able to develop epistemological relationships concerning the sequences, applicable to the teaching of the History of Mathematics in initial teacher education programs for Mathematics.