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We report a characterization of the reflections made by high school mathematics teachers when discussing their resolutions of three numerical tasks related to the concept of function understood as a mathematical procept. Thematic analysis was used to code and categorize the reflections. Six teachers with at least 5 years of professional experience and an average age of 30 years participated. The results show that the teachers reflected on their actions in the tasks; their content knowledge; their teaching practice; and their professional training. As a result of these reflections, they recognized the need to improve their knowledge and practice of teaching functions in a more proceptual sense. The results contribute to deepen in how to make teachers reflect and incorporate a much more structural vision of functions in their teaching practices and, at the same time, incorporate the importance and role of their professional training in their reflections.

A theoretical-didactic proposal based on variational ideas and the use of GeoGebra is described as a response to a local problem on the understanding of the concept of function and quadratic equation in pre-university students. The theoretical elements of this work are based on the understanding of concepts and the theory of semiotic registers and representations, while the methodological references were based on hypothetical learning trajectories and the use of dynamic software, considered as a heuristic resource. The proposal was validated by means of expert criteria. This elaboration contributes with a proposal that breaks with the classical schemes of presentation and treatment of the content, which favors through the incorporation of variational ideas generated with the dynamic software GeoGebra the understanding of the articulated concepts of function and quadratic equation in pre-university students.

This paper reports the results of three questionnaires applied to sixty-seven students preparing to become university-level mathematics teachers; the questionnaires were focused on knowing their conceptions and their mastery of the representations of functions in the development of power series. The theoretical and methodological background rests on the Mathematics Teacher's Specialised Knowledge (MTSK), specifically in the domain: Mathematical Knowledge (MK). As a result of the analysis of the responses to the questionnaires, it was identified that the notion of development of power series played an important role as a means of justification and that, from the three notions addressed (Taylor's Formula, limited development, development of power series), this was the most prevalent in the mind of the students. These results will be used as the starting point for the development of proposals that improve the teaching and learning of the subject of study.

This paper reports the results of the experimentation of a didactic engineering for the treatment of the sense of variation of functions with pre-university students. The theoretical references of the investigation are grounded in the theory of didactic situations and the methodological elements in the didactic engineering, the use of counterexample and the didactic strategy known as Scientific Debate in Mathematics Courses. As a result of the experiment, it was identified that the methodological resource allowed the students to develop construction processes of the concepts of increasing and decreasing functions; the dynamic context fostered intuitive ideas of increment and decrement, and intuition about the conditions involved; the debate and counterexample on the graphic and algebraic treatments allowed the establishment of the conditions that structure the formal definition of these concepts.

This paper shows the results of the epistemological and didactical analysis of the sense of variation of functions. Specifically, on the conceptions of growth and decay in a function that underlie the demonstrations of the theorem that links the sign of "f" with the sense of variation of "f". The epistemological approach covered the years 1795 to 1912. It was identified that the conceptions of Fourier, Lagrange and Cauchy about growth and decay differ from the conception in the formal current definition; however, the posed procedures and definitions provide elements that foster reconstruction processes of the definitions and properties of increasing and decreasing functions. It is important to highlight that the current definition of growth and decay has a solid foundation on the definition made by Osgood in 1912. The didactical analysis identified that the current text books inherit some of the limitations and inconsistencies of the definitions found on the epistemological approach. The conflicting issues enhance the starting point for the development of a didactic engineering for the treatment of the sense of variation of a function at a preuniversity level.