First generation didactic engineering in Higher Education: generalization and extension of the Fibonacci sequence

Abstract

This paper reflects on the dissertation developed; in the Graduate Program in Science and Mathematics Teaching of the Federal Institute of Education, Science and Technology of the State of Ceará; SEDUC / CE teacher Arlem Atanazio dos Santos. In this sense, this work aims to perform an analysis of the Didactic Engineering stages that Santos (2017) built in Higher Education. Thus, there are two specific objectives: (i) to highlight the methodological potential (of this engineering) of didactic transposition of nontrivial mathematical models; (ii) provide the reader with the opportunity to develop an epistemological conception of the teaching of history of mathematics with emphasis on the historical-evolutionary process of the Fibonacci model. This dissertation assumed Didactic Engineering as a research methodology in complementarity with Didactical Situation Theory. In a panoramic view, definitions and mathematical relations arising from the generalization and extension of the Fibonacci Sequence were approached. However, for the classroom, the Binet Formula was considered as a model of generalization and extension of this sequence. Thus, following the paradigm of this engineering, it is understood that a didactic transposition of the generalized Fibonaccian model was performed, in which the didactic experience was carried out in Higher Education. This can lead to the development of an epistemological conception of the teaching of history of mathematics during the initial formation of mathematics teachers. Moreover, given the historical-evolutionary process that involves the Fibonacci model, it can be concluded that the emergence of new definitions and properties contribute to broaden the repertoire of the history of mathematics.