Windows on the infinite : constructing meanings in a Logo-based microworld
This thesis focuses on how people think about the infinite. A review of both the historical and psychological/educational literature, reveals a complexity which sharpens the research questions and informs the methodology. Furthermore, the areas of mathematics where infinity occurs are those that have traditionally been presented to students mainly from an algebraic/symbolic perspective, which has tended to make it difficult to link formal and intuitive knowledge. The challenge is to create situations in which infinity can become more accessible. My theoretical approach follows the constructionist paradigm, adopting the position that the construction of meanings involves the use of representations; that representations are tools for understanding; and that the learning of a concept is facilitated when there are more opportunities of constructing and interacting with external representations of a concept, which are as diverse as possible. Based on this premise, I built a computational set of open tools — a microworld — which could simultaneously provide its users with insights into a range of infinity-related ideas, and offer the researcher a window into the users' thinking about the infinite. The microworld provided a means for students to construct and explore different types of representations — symbolic, graphical and numerical — of infinite processes via programming activities. The processes studied were infinite sequences and the construction of fractals. The corpus of data is based on case studies of 8 individuals, whose ages ranged from 14 to mid-thirties, interacting as pairs with the microworld. These case studies served as the basis for an analysis of the ways in which the tools of the microworld structured, and were structured by, the activities. The findings indicate that the environment and its tools shaped students' understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes, and permitted the students to deal with the complexity of the infinite by assisting them in coordinating the different epistemological elements present. On a theoretical level, the thesis elaborates and refines the notion of situated abstraction and introduces the idea of "situated proof".