Adolescents' understanding of limits and infinity
AIM To investigate mathematically able adolescents' conceptions of the basic notions behind the Calculus: infinity (including the infinitely large, the infinitely small and infinite aggregates); limits (of sequences, series and functions); and real numbers. To observe the effect, if any, on these conceptions, of a one year calculus course. EXPERIMENTS Pilot interviews and questionnaires helped identify areas on which to focus the study. A questionnaire was administered to Lower Sixth Form students with 0-level mathematics passes. The questionnaire was administered twice, once in September and again the following May. The A-level mathematicians had received instruction in most of the techniques of the Calculus by May. Interviews, to clarify ambiguities, elicit reasoning behind the responses and probe typicality and atypicality, were conducted in the month following each administration. A second questionnaire, an amended version of the first, was administered to a larger but similar audience. The responses were analysed in the light of hypotheses formulated in the analysis of data from the first 5ample. PRINCIPAL FINDINGS Subjects have a concept of infinity. It exists mainly as a process, anything that goes on and on. It may exist as an object, as a large number or the cardinality of a set, but in these forms it is a vague and indeterminate form. The concept of infinity is inherently contradictory and labile. Recurring decimals are perceived as dynamic, not static, entities and are not proper numbers. Similar attitudes exist towards infinitesimals when they are seen to exist. Subjects' conception of the continuum do not conform to classical or nonstandard paradigms. Convergence / divergence properties are generally noted with infinite sequences and functions. With infinite series, however, convergence / divergence properties, when observed, are seen as secondary to the fact that any infinite series goes on indefinitely and is thus similar to any other infinite series. The concept that the hut is the saue type of entitiy as the finite tens is strong in subjects' thoughts. We coin the term generic hiuit for this phenomenon. The generic limit of 0.9, 0.99, is 0.9, not 1. Similarly the reasoning scheme that whatever holds for the finite holds for the infinite has widespread application. We coin the term generic law for this scheme. Many of the phrases used in calculus courses (in particular hut, tends to, approaches and converges) have everyday meanings that conflict with their mathematical definitions. Numeric/geometric, counting/measuring and static/dynamic contextual influences were observed in some areas. The first year of a calculus course has a negligible effect on students conceptions of limits, infinity and real numbers. IMPLICATIONS FOR TEACHING On introducing limits teachers should encourage full class discussion to ensure that potential cognitive obstacles are brought out into the open. Teachers should take great care that their use of language is understood. A-level courses should devote more of their time to studying the continuum. Nonstandard analysis is an unsuitable tool for introducing elementary calculus.