To exist or not to exist: example generation in Real Analysis

Abstract

This article deals with the activity of example generation as a special case of problem solving. We asked university students in the scientific-technological area to produce examples (which may exist or not) of mathematical objects fulfilling given requirements. For the analysis of students' performances, we have developed a model that attempts to grasp the nature of the different stages in the solving process. The discussion of the findings allows us to outline specificities and educational potentialities of example generation activities.

Argumentos variacionales en la comprensión de la concavidad en gráficas de funciones

“…El uso de la gráfica en el cálculo y el análisis es una gran herramienta para visualizar el cambio y dar significado a objetos matemáticos relacionados con la derivada (Aydın y Ubuz, 2015; Furinghetti et al, 2011;Hoffkamp, 2011; Hong y Thomas, 2015; Jones, 2019;Sánchez-Matamoros et al, 2015). Particularmente Furinghetti et al (2011: 230) reportan que la dialéctica "visual vs. simbólico" tiene alguna relación con la diálectica "informal vs. formal", donde formal para los estudiantes significa uso de fórmulas, definiciones y teorema.…”

Generating examples: focus on processes

“…I now present a case (for an analysis carried out from a different point of view, see Furinghetti et al 2011) where an external control stimulates this transition favoring the production of a solution. We can also see the efficacy of generating examples as an educational tool to give meaning to a mathematical definition and to integrate it with the concept image.…”

Students’ views of example generation tasks

“…They conclude that the 'teaching strategy of example generation is not yet understood well enough to be a viable pedagogical recommendation' (p. 1). Furinghetti et al (2011), while acknowledging that the role of examples is often considered to be crucial in problem solving, caution that a focus on examples may also 'make students stick to the explorative stage and inhibit the need for generalization' (p. 219), with students often considering that checking examples constitutes a means of proof. Furthermore, while Edwards (2011) believes that students with a rich and varied example space are more likely to have concept images that are aligned with formal theory, he advises that the use of certain types of example generation tasks as a pedagogical tool may not benefit students who are struggling with the role of mathematical definitions as well as their content.…”