In mathematics education research paradoxes of infinity have been used in the investigation of students' conceptions of infinity. We analyze one such paradoxthe Painter's Paradox -and examine the struggles of a group of Calculus students in an attempt to resolve it. The Painter's Paradox is based on the fact that Gabriel's horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel's horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel's horn has a convergent series associated with volume and a divergent series associated with surface area. This study shows that contextual considerations hinder students' ability to resolve the paradox mathematically. We suggest that the conventional approach to introducing area and volume concepts in Calculus presents a didactical obstacle. A possible alternative is considered.
In this study we consider a classic paradox of infinity and its variations and suggest how the sources of misleading intuition can be analysed using the concept of uniform convergence of functions. We then examine how six mathematics honour students engage with a variation of the paradox. Despite their advanced mathematical training, the participants experienced considerable difficulty in addressing the presented paradoxical situation. Drawing on data from written responses and individual interviews, we describe the students’ approaches in an attempt to reconcile their intuitive perceptions with their mathematical computations.
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