Geometrical representations in the learning of two-variable functions

Abstract

This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students' notion of subsets of Cartesian three-dimensional space and the understanding of graphs of twovariable functions. APOS theory and Duval's theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students' understanding can be related to the structure of their schema for R 3 and to their flexibility in the use of different representations.

Covariation between variables in a modelling process: The ACODESA (collaborative learning, scientific debate and self-reflection) method

“…Bloch, 2003;Breidenbach et al, 1992;Carlson, 1998;Harel & Dubinsky, 1992;Hitt, 1994;Mesa, 2004;Trigueros & Martínez-Planell, 2010),…”

Students’ images of two-variable functions and their graphs

“…These relationships often involve multiple quantities, but there is limited work that explores how mathematics and science students reason about relationships between multiple quantities and their representations, such as graphs and tables (Montiel, Vidakovic, & Kabael, 2008;Montiel, Wilhelmi, Vidakovic, & Elstak, 2009;Trigueros & Martinez-Planell, 2010;Weber & Dorko, in press). Given the prevalence of ideas in science and mathematics that require representing multiple relationships using functions and their representations simultaneously, it is important to understand how students think about generating and interpreting representations of functions, particularly graphs of functions of more than one variable.…”

“…As another example, Trigueros and Martinez-Planell (2010) proposed and then refined a genetic decomposition of the understandings a student needs to conceive of two-variable functions and their graphs. Subsequently, Martinez-Planell and Trigueros (2012) proposed a genetic decomposition to postulate about constructions the students might make as they think about two variable functions and their graphs.…”

Estructuras Mentales que Modelan el Aprendizaje de un Teorema del Álgebra Lineal: Un Estudio de Casos en el Contexto Universitario

“…Sus fundamentos se basan en la Abstracción Reflexiva formulada por Piaget. Esta teoría ha sido usada y validada para explicar diferentes nociones y/o conceptos matemáticos avanzados, asociados con el nivel universitario; en áreas como cálculo (Valdivia y Parraguez, 2015;Trigueros y Martínez-Planell, 2010), álgebra lineal Salgado y Trigueros, 2015;Parraguez y Oktaç, 2010), álgebra abstracta (Asiala et al, 1996) ecuaciones diferenciales (Trigueros, 2014), fractales (Villabona y Roa-Fuentes, 2016), probabilidad (Vásquez y Parraguez, 2014), teorema del isomorfismo (Mena et al, 2016), entre otros.…”