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The present work presents some examples of data originated from a set of investigations developed in Brazil, at the Federal Institute of Education, Science and Technology of the State of Ceará - IFCE, in the context of teacher education, and the consideration of elements of a historical, mathematical, and evolutionary nature. Thus, from a perspective of theoretical complementarity involving the research elements derived from the notion of the didactic engineering for development and some assumptions of the French aspect of the professional didactics, two sets of data are discussed. The first set involved research carried out between 2017 and 2020. The second set considered involved studies still under development in Brazil, considering the period of 2020 - 2023. The historical landscape of investigation rests on considering the recurring number sequences of Fibonacci, Lucas, Pell, Jacobsthal, Coordonier or Padovan, Perrin, Mersenne, Oresme, Naraynna, and Leonardo. Furthermore, in a broad sense, the study aims to provide a learning scenario for the teacher, affected by a perspective of mathematical knowledge evolution, including the repercussions and applications with current technology. [ABSTRACT FROM AUTHOR]

The present work presents a proposal for study and investigation, in the context of the teaching of Mathematics, through the history of linear and recurrent 2nd order sequences, indicated by: Fibonacci, Lucas, Pell, Jacobsthal, Leonardo, Oresme, Mersenne, Padovan, Perrin and Narayana. Undoubtedly, starting from the Fibonacci sequence, representing the most popular in the referred scenario of numerical sequences, it was possible to notice the emergence of other sequences. In some cases, being derived from Fibonacci, in others, changing the order of the sequence, we thus have the historical study, with the use of mathematical rigor. Besides, their respective recurrence formulas and characteristic equations are checked, observing their roots and the relationship with the number of gold, silver, bronze and others. The results indicated represent an example of international research cooperation involving researchers from Brazil and Portugal. [ABSTRACT FROM AUTHOR]

In Brazil, we are interested in the study of second-order recurrent numerical sequences. Such mathematical content has an enormous potential to provide a differentiated mathematical culture for the math teachers in initial training. However, it becomes important that the teacher in Brazil knows properties beyond the traditional and classic numerical sequences, such as the Fibonacci sequence, Lucas sequence, Pell sequence, Jacobsthal sequence, Narayana sequence, etc. Thus, in the present work, we present a proposal of CAS Maple as technological resources in the study and discovery of new theorems derived from the Mersenne sequence. Throughout the work, we show that the software allows the discovery of theorems from matrix properties derived from the Mersenne sequences and the sequence of Gaussian numbers of Mersenne. The research presented in the paper can stimulate the formation of Mathematics teachers in Brazil. [ABSTRACT FROM AUTHOR]