Infinity, Paradoxes and Contexts

This dissertation explores conceptions of infinity of undergraduate students using paradoxes of infinity as a research tool. My particular attention is on the role of context in which each paradox is presented on students’ attempts to address seemingly paradoxical situations. This dissertation is structured around four stand-alone papers with an introductory chapter and a concluding chapter. The first paper is a study on the struggles of a group of undergraduate liberal arts students in engaging with the super tasks in the Thomson’s Lamp paradox, as well as the paradox of Green Alien, which is an isomorphic version of Thomson’s Lamp presented in a fictitious setting. This study shows that the participants have a robust process view of the infinite iterative processes in the super tasks and that even the new totality stage in the APOS perspective is difficult for them to reach. Painter’s Paradox and the struggles of a group of undergraduate Calculus students in engaging with this paradox is the focus of the second paper. Findings of this study indicate that the participants’ difficulty in reconciling the finite volume of the infinitely long Gabriel’s horn could be an epistemological obstacle or a didactic obstacle. The third paper explores how the thinking of a group of mathematics honours undergraduate students is affected by the context in Polygons Task. Data in this study suggests an extension of the ‘Same A-same B’ intuitive rule of Tirosh and Stavy (1999) to an infinite sequence of objects. The last paper develops variants of the well-known Hilbert’s Grand Hotel paradox by introducing room fees and discusses how these variants can be used in Calculus instruction. Participants found the cognitive conflicts elicited by the paradoxes difficult to resolve but they seem to improve their problem solving heuristics and metacognitive skills by engaging with the paradoxes. In Painter’s Paradox, Thomson’s Lamp and Green Alien many participants reduced the level of abstraction by contextualizing further and avoided resolution. Decontextualization seemed to be very difficult for the participants. Findings in this dissertation suggest that the context of a paradox could be a hindrance that diverts students’ attention from addressing the situation mathematically.