Onto-semiotic complexity of the Definite Integral: Implications for teaching and learning Calculus

espanolLa ensenanza y el aprendizaje de los conceptos y procedimientos del Calculo, en particular del concepto de integral definida, es un reto para profesores y estudiantes en su trayectoria academica. En esta investigacion, complementamos el analisis realizado por diferentes autores, aplicando las herramientas teoricas y metodologicas del Enfoque Onto-Semiotico al conocimiento y la instruccion matematica. El objetivo es comprender los diversos significados del concepto de integral definida y los potenciales conflictos semioticos a partir de los datos aportados. Centramos la atencion en un primer significado intuitivo, que implica principalmente conocimientos aritmeticos, y en el significado formal integral definida como limite de las sumas de Riemann predominantemente en las directrices curriculares. El reconocimiento de la complejidad onto-semiotica de los objetos matematicos se considera un factor clave para explicar las dificultades de aprendizaje de los conceptos, los procedimientos y su aplicacion para la resolucion de problemas, asi como para tomar decisiones fundamentadas sobre la ensenanza. El analisis metodologico de un texto matematico, que ejemplificamos en este trabajo aplicando las herramientas del Enfoque Onto-Semiotico, proporciona un nivel microscopico de analisis que permite identificar algunos hechos semiotico-cognitivos de interes didactico. Tambien permite identificar algunos estratos epistemicos, es decir, conocimientos institucionales que deberian haber sido estudiados previamente y que suelen pasar desapercibidos en el proceso de ensenanza. EnglishTeaching and learning Calculus concepts and procedures, particularly the definite integral concept, is a challenge to teachers and students in their academic careers. In order to develop an informed plan for improving instructional processes, it is necessary to pay attention to the nature and complexity of the mathematical features of the definite integral, that students are expected to understand and apply. In this research, we supplement the analysis made by different authors, applying the theoretical and methodological tools of the Onto-Semiotic Approach to mathematical knowledge and instruction. The goal is to understand the diverse meanings of the concept of the definite integral and potentials semiotic conflicts based on the given data. We focus attention on a first intuitive meaning, which involves mainly arithmetic knowledge, and the definite integral formal meaning as Riemann’s sums limit predominantly in the curricular guidelines. The recognition of the onto-semiotic complexity of mathematics objects is considered as a key factor in explaining the learning difficulties of concepts, procedures and its application for problem-solving, as well as to make grounded decisions on teaching. The methodology analysis of a mathematical text, which we exemplify in this work applying the tools of Onto-Semiotic Approach, provides a microscopic level of analysis that allows us to identify some semiotic-cognitive facts of didactic interest. This also allows for the identification of some epistemic strata, that is, institutional knowledge that should have been previously studied, which usually goes unnoticed in the teaching process.