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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.