En este artículo se reportan los resultados de una investigación que tuvo como objetivo identificar el papel que juega el contraejemplo en la elaboración de definiciones matemáticas. En particular se analizó el caso de la función convexa en un grupo de estudiantes de nivel universitario. Los elementos teóricos y metodológicos primarios que sustentaron el trabajo descansan en los aportes sobre pruebas y refutaciones, elementos de heurística, lenguaje y axiomática y en el debate científico en cursos de matemáticas. Como resultado fundamental se destaca que el debate científico favoreció mediante la discusión en torno al concepto de función convexa y a la formulación de contraejemplos. Estos contribuyeron en la comprobación de casos particulares, la formulación y refutación de conjeturas, hasta aproximarse a la presentación clásica de la definición. Establecemos también que el contraejemplo es un recurso didáctico y una herramienta mediadora que contribuye en los procesos de elaboración y comprensión de la definición matemática.
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 ' with the sense of variation of. 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.
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 (Taylorʼs 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.
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