La tasa de variación de una función real de variable real: trabajo matemático de estudiantes de educación secundaria
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2021-01-27
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Pontificia Universidad Católica del Perú
Abstract
En la enseñanza del Cálculo, diferentes investigaciones señalan que conceptos como
pendiente, velocidad y tasa variación son significativos y útiles per se, ya que
constituyen la estructura axial de las funciones y el Análisis. A partir de ello, surge
nuestro interés alrededor del aprendizaje de la tasa de variación, pues es necesario
comprender los procesos a través de los cuales los estudiantes dotan de significado
tal concepto. En ese sentido, la presente investigación cualitativa plantea estudiar el
trabajo matemático de estudiantes de último año de Educación Secundaria al resolver
tareas relacionadas con la tasa de variación de una función real de variable real, en la
que se considera a la tasa de variación (media e instantánea) como velocidad (media
e instantánea). Para ello, se diseña y aplica una situación de aprendizaje a cuatro
estudiantes (16-17 años), agrupados en dos binomios, de último año de Educación
Secundaria de una institución educativa en la ciudad de Valparaíso, Chile. Para los
análisis, utilizamos, como marco teórico, el Espacio de Trabajo Matemático (ETM)
formulado por Kuzniak, el cual permite caracterizar el conocimiento y la producción
matemática del estudiante, así como el valor epistémico y cognitivo de las tareas. A
partir de los resultados, se evidencia la activación de las distintas génesis del ETM y
los planos verticales asociados a ellas, así como la identificación de los paradigmas
del Análisis que fueron privilegiados. Se concluye que las acciones y producciones de
los estudiantes giran en torno a la activación y coordinación de las génesis semiótica,
instrumental y discursiva y con ello los planos verticales [Sem-Ins] y [Ins-Dis], así como
los paradigmas del análisis AG y AC.
In teaching Calculus, some researches highlight that concepts such as slope, velocity, and rate of change are significant and useful per se, since they constitute the axial structure of the functions and the Analysis. Base on that, appear our interest on rate of change learning, because it is necessary to understand the processes that the students made to give a meaning to that concept. In regard, this qualitative research sets out to study the mathematical work of last year students of High School when they are resolving tasks related to rate of change of a real function of real variable in which it is consider rate of change (average and instantaneous) as velocity (average and instantaneous). To that end, it was designed and applied a learning situation to four students (16 - 17 years old) grouped on two binomials, of last year of High School of an educative institution on Valparaíso, Chile. For the analyses, we will use, as theoretical framework Mathematical Working Space (MWS) formulated by Kuzniak, which allows us to characterize the student's mathematical knowledge and production, as well as the epistemic and cognitive value of the tasks. Based on the results, it is evinced the activation of the different genesis of MWS and vertical planes associated to them, as well as, the analysis paradigms identification that were identified. In conclusion, students’ actions and productions concentrate on the activation and coordination of the semiotic, instrumental and discursive genesis and with it the vertical planes [Sem-Ins] and [Ins-Dis], as well as the analysis paradigms GA and CA.
In teaching Calculus, some researches highlight that concepts such as slope, velocity, and rate of change are significant and useful per se, since they constitute the axial structure of the functions and the Analysis. Base on that, appear our interest on rate of change learning, because it is necessary to understand the processes that the students made to give a meaning to that concept. In regard, this qualitative research sets out to study the mathematical work of last year students of High School when they are resolving tasks related to rate of change of a real function of real variable in which it is consider rate of change (average and instantaneous) as velocity (average and instantaneous). To that end, it was designed and applied a learning situation to four students (16 - 17 years old) grouped on two binomials, of last year of High School of an educative institution on Valparaíso, Chile. For the analyses, we will use, as theoretical framework Mathematical Working Space (MWS) formulated by Kuzniak, which allows us to characterize the student's mathematical knowledge and production, as well as the epistemic and cognitive value of the tasks. Based on the results, it is evinced the activation of the different genesis of MWS and vertical planes associated to them, as well as, the analysis paradigms identification that were identified. In conclusion, students’ actions and productions concentrate on the activation and coordination of the semiotic, instrumental and discursive genesis and with it the vertical planes [Sem-Ins] and [Ins-Dis], as well as the analysis paradigms GA and CA.
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Matemáticas--Estudio y enseñanza (Secundaria), Cálculo, Velocidad, Estudiantes (Educación secundaria)
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