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

• Burgos, María ^[1] ; Bueno, Seydel ^[2] ; Godino, Juan D. ^[1] ; Pérez, Olga ^[2]
  1. [1] Granada University
  2. [2] Camaguey University
• Localización: REDIMAT, ISSN-e 2014-3621, Vol. 10, Nº. 1, 2021, págs. 4-40
• Idioma: inglés
• DOI: 10.17583/redimat.2021.6778
• Títulos paralelos:
  • Complejidad onto-semiótica de la integral definida: implicaciones para la enseñanza y el aprendizaje del Cálculo
• Enlaces
  • Texto completo (pdf)
• Dialnet Métricas: 2 Citas
• Resumen
  • español

    La enseñanza y el aprendizaje de los conceptos y procedimientos del Cálculo, en particular del concepto de integral definida, es un reto para profesores y estudiantes en su trayectoria académica. En esta investigación, complementamos el análisis realizado por diferentes autores, aplicando las herramientas teóricas y metodológicas del Enfoque Onto-Semiótico al conocimiento y la instrucción matemática. El objetivo es comprender los diversos significados del concepto de integral definida y los potenciales conflictos semióticos a partir de los datos aportados. Centramos la atención en un primer significado intuitivo, que implica principalmente conocimientos aritméticos, y en el significado formal integral definida como límite de las sumas de Riemann predominantemente en las directrices curriculares. El reconocimiento de la complejidad onto-semiótica de los objetos matemáticos se considera un factor clave para explicar las dificultades de aprendizaje de los conceptos, los procedimientos y su aplicación para la resolución de problemas, así como para tomar decisiones fundamentadas sobre la enseñanza. El análisis metodológico de un texto matemático, que ejemplificamos en este trabajo aplicando las herramientas del Enfoque Onto-Semiótico, proporciona un nivel microscópico de análisis que permite identificar algunos hechos semiótico-cognitivos de interés didáctico. También permite identificar algunos estratos epistémicos, es decir, conocimientos institucionales que deberían haber sido estudiados previamente y que suelen pasar desapercibidos en el proceso de enseñanza.

  • English

    Teaching 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.

• Referencias bibliográficas
  • Attorps, I., Björk, K., Radic, M. & Tossavainen, T. (2013). Varied ways to teach the definite integral concept. International Electronic...
  • Borji, V. & Font, V. (2019). Exploring students’ understanding of integration by parts: a combined use of APOS and OSA. EURASIA Journal...
  • Bressoud, D., Ghedams, I., Martinez-Luances, V., & Törner, G. (2016). Teaching and learning of Calculus. Springer. https://doi.org/10.1007/978-3-319-32975-8_1
  • Brousseau, G. (1997). Theory of didactical situations in Mathematics. Dordrecht: Kluwer.
  • Chevallard, Y. (1992). Concepts fondamentaux de la didactique: perspectives apportées par une approche anthropologique. Recherches en Didactique...
  • Contreras, A., & Ordóñez, L. (2006). Complejidad ontosemiótica de un texto sobre la introducción a la integral definida. Relime, 9(1),...
  • Contreras, A., Ordóñez, L. & Wilhelmi, M. R. (2010). Influencia de las pruebas de acceso a la universidad en la enseñanza de la integral...
  • Crisóstomo, E. (2012). Idoneidad de procesos de estudio del cálculo integral en la formación de profesores de matemáticas: Una aproximación...
  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In Tall, D. (Eds.), Advanced mathematical thinking (pp. 95–123)....
  • Duval, R. (1995). Sémiosis et pensée: registres sémiotiques et apprentissages intellectuels [Semiosis and human thought. Semiotic registers...
  • Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational...
  • Godino, J. D. & Batanero, C. (1998). Clarifying the meaning of mathematical objects as a priority area of research in Mathematics Education....
  • Godino, J. D., Batanero, C. & Font, V. (2019). The onto-semiotic approach: implications for the prescriptive character of didactics. For...
  • Greefrath, G, Oldenburg, R., Siller, H. S., Ulm, V. & Weigand, H. G. (2016). Aspects and “Grundvorstellungen” of the concepts of derivative...
  • Grundmeier, T. A., Hansen, J., & Sousa, E. (2006). An exploration of definition and procedural fluency in integral Calculus. PRIMUS, 16(2),...
  • Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics...
  • Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),...
  • Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. Journal of Mathematical Behavior, 32(2), 122–141. https://doi.org/10.1016/j.jmathb.2012.12.004
  • Jones, S, R. (2015). Areas, anti-derivatives, and adding up pieces: definite integrals in pure mathematics and applied science contexts. Journal...
  • Kouropatov, A., & Dreyfus, T. (2013). Constructing the integral concept on the basis of the idea of accumulation: Suggestion for a high...
  • Kouropatov, A. & Dreyfus, T (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM Mathematics Education,...
  • Marton, F., Runesson, U., & Tsui, A. (2004). The space of learning. In F. Marton and A. Tsui (Eds.), Classroom discourse and the space...
  • Mateus, E. (2016). Análisis didáctico a un proceso de instrucción del método de integración por partes. Bolema, 30(55), 559-585. https://doi.org/10.1590/1980-4415v30n55a13
  • Ordoñez, L. (2011). Restricciones institucionales en las matemáticas de 2º de bachillerato en cuanto al significado del objeto integral definida....
  • Orton, A. (1983). Students’ understanding of integration. Educational Studies in Mathematics. 14, 1-18. https://doi.org/10.1007/bf00704699
  • Peirce, Ch. S. (1958). Collected papers of Charles Sanders Peirce. 1931-1935. Cambridge, MA: Harvard UP.ç
  • Pino-Fan, L., Font, V., Gordillo, W., Larios, V., & Breda, A. (2018). Analysis of the meanings of the antiderivative used by students...
  • Rasslan, S. & Tall, D. (2002). Definitions and images for the definite integral concept. In: A. D. Cockburn & E. Nardi (eds.) Proceedings...
  • Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. Journal of Mathematical Behavior,...
  • Serhan, D. (2015). Students’ understanding of the definite integral concept. International Journal of Research in Education and Science, 1(1),...
  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York: Cambridge University...
  • Starbird, M. (2006). Change and motion: Calculus made clear, 2nd Edition. Chantilly, Virginia. The Teaching Company.
  • Stewart, J. (2016). Calculus. Early transcendentals. Boston: Cengage Learning.
  • Tall, D. O. (2004). Building theories: the three worlds of mathematics. For the learning of mathematics, 24 (1), 29-32. https://doi.org/10.1017/cbo9781139565202.011
  • Tall, D., O. & Vinner, S. (1981). Concept images and concept definition in mathematics with particular reference to limits and continuity....
  • Thompson, P. W., & Silverman, J. (2008). The concept of accumulation in Calculus. In M. P. Carlson & C. Rasmussen (Eds.), Making the...
  • Wittgenstein, L. (1953). Philosophical investigations. New York, NY: The MacMillan Company.