| The purpose of this study was to seek ways of improving introductory calculus instruction by investigating the following two questions: (1) How well do students understand the concepts of calculus?, and (2) What kinds of misconceptions, difficulties, and errors do students have concerning the content of calculus?;To provide structure in addressing these questions, a five-level model describing student performance in calculus, similar to the van Hiele levels in geometry, was employed. The keywords characterizing these levels are computational, intuitive, transitional, rigorous, and abstract. Tests on limits and derivatives, whose items measured understanding at the first four levels of the model, were developed and administered to 76 and 83, respectively, university, four-year college, and community college students in first-semester calculus.;Regarding question (1), Guttman scalogram analysis resulted in acceptance of the following hypothesis: For each of the concept areas of limits and derivatives, a student performs to criterion at a given level of the model, only if he performs to criterion at all lower levels of the model. Performance to criterion at levels III and IV, however, was rare. Regarding question (2), misconceptions, difficulties, and errors on the test items (all common even among the better students) were cataloged according to levels of the model.;Based upon the results relative to questions (1)-(2), suggestions for improving instruction with respect to computational skills, concept development, problem solving, and general teaching approach were presented. Among these, it was suggested that computational skills should occupy no more than one-third of students' time, both in and out of class. This would then allow for a more adequate treatment of concept development and problem solving. Furthermore, it was pointed out that reversing the usual pedagogical practice of proceeding from concept to problem, to that of proceeding from problem to concept, might be one way of making instruction more meaningful and motivating to students. It was also argued that the general instructional approach taken in introductory calculus, both by teacher and textbook, should be more intuitive, rather than rigorous, in nature. Finally, recommendations for further research were discussed. |
