Generating examples: focus on processes

Abstract

Constructing an example can be a rich and complex activity, interesting to investigate mathematical thinking and with many potentialities in mathematics education. In this article, I analyse processes involved in example generation, with particular emphasis on production and transformation of signs representing mathematical objects and on generation of inferences. The richness and complexity of these processes will also be shown through the notions of prototypes, concept image and concept definition. This investigation reveals aspects that are significant both in education and for the reflection on cognitive and cultural aspects of mathematical thinking.

Le modalità logico-argomentative nei testi scolastici di geometria della scuola elementare e media in lingua italiana

“…(Balacheff, 2001, p. 14) Le prove che prevedono un esempio generico rappresentano uno stadio intermedio più vicino alle prove pragmatiche o a quelle intellettuali a seconda del processo di produzione della prova, e quindi dello statuto operatorio dell'esempio utilizzato. L'importanza di questo tipo di prova risulta evidente anche dall'analisi di studenti implicati nelle attività argomentative per costruire e giustificare congetture, come sottolineato anche da alcune recenti ricerche sul ruolo dell'esempio nel fare matematica (tra le altre si vedano Antonini, 2011;Antonini et al, 2011;Pedemonte & Buchbinder, 2011). Infine, l'esperienza mentale, che invoca l'azione interiorizzata e si distanzia dalla realizzazione su un rappresentante particolare: questa prova segna il passaggio dalle prove pragmatiche a quelle di tipo intellettuale.…”

Promoting Metalinguistic and Metamathematical Reasoning in Proof-Oriented Mathematics Courses: a Method and a Framework

“…As students reason about given statements, we should not necessarily expect them to only reason about the concept definition itself. However, to use examples fruitfully, students must be able to intentionally vary relevant aspects of chosen examples to observe invariances that might generalize (Antonini 2011;Lockwood et al 2012;Pinto and Tall 2002). Dimensions of possible variation are often revealed by boundary examples, which are members of a category that differ in some significant way from prototypical examples.…”

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

“…Our study deals with example generation in Real Analysis, a rich and interesting field for investigating examples concerning advanced mathematical thinking (Antonini 2006(Antonini , 2011Benbachir and Zaki 2001). In this context, an important tool for our discussion will be the duality between concept image and concept definition, as discussed by Tall and Vinner (1981).…”

“…In(Antonini 2011) the reader can find an analysis of a solution of this task that takes into account the argumentation of impossibility, the proof and the transition between argumentation and proof.…”