Learner conceptions of the limit concept

Abstract

The intuitive idea of a limit involves getting as close as one wishes to some goal. The classical Greek mathematicians knew of this idea over two thousand years ago. Through the centuries scientists and engineers used the idea to solve problems by calculating quantities that were not otherwise accessible. This made the limit concept important in calculus and in learning calculus. However, the limit concept has been, and still remains, a troublesome concept for learners to grasp. Two factors stand out: the limit concept is complex, and it poses particular cognitive challenges. Calculus reform and other efforts to improve the teaching and learning of calculus have met with some success, but controversy still surrounds the causes for the difficulties and the efficacy of the various approaches. Much still needs to be discovered about how learners think about the limit in calculus. This study investigated how postsecondary students who have successfully completed an introductory calculus course conceptualised the limit concept. Conceptions are accessed by inference. From a qualitative, cognitive theoretical perspective, data collection was carried out through a combination of task-based interviews with four volunteers, and through a written questionnaire designed to explore particular aspects of limits that was completed by fourteen volunteers. The analysis considered themes of how each participant made sense of the limit concept, and the implications for teaching. The findings reveal that the participants' conceptualisations of limit varies by mathematical context and in relation to the participants' academic backgrounds. The study clearly established that limit is a difficult concept to learn and suggests follow-up investigation of what it means to learn a difficult mathematical concept in general. The study also raises the question of the necessity to include limit in introductory calculus courses and suggests follow-up studies to investigate the impact on learning calculus with and without limit. Finally, further investigation is suggested of the relationship between instruction and students' conceptions as a basis for improving learning of limit and other difficult mathematics concepts