Abstract

Misconceptions about limits in introductory Calculus such as the infamous “a function never reaches its limit” have been thoroughly studied in previous research. However, their resolution is rarely documented. Our objective is to contribute to the understanding of the “vanishing” of common misconceptions about limits as students progress from Calculus to Analysis. In addition, we investigate the possibility that early Calculus misconceptions may influence the learning of Real Analysis in such a way that new, related misconceptions are developed about more advanced concepts. To this end, we created a questionnaire devised to uncover seven of the well-documented Calculus misconceptions, as well as three conjectured misconceptions related to introductory Analysis concepts. The questionnaire was administered to ten students actively enrolled in a first or second Real Analysis course. To analyze participants’ answers, we introduced a model of misconception classification which includes six levels. Using this model, we identified consistent incorrect reasonings indicating the possibility that instruction after elementary Calculus has not contributed to the resolution of some misconceptions. We observed that certain students’ answers exhibited what we refer to as “transitional behavior” from one level to another and discuss what this may mean in terms of overcoming misconceptions. In addition, we identified one instance of a student’s learning of Real Analysis potentially being influenced by their Calculus misconceptions. Finally, we briefly considered the presence of misconceptions about fundamental mathematics, such as logical argumentation and mathematical notation, and new misconceptions that students may develop as they learn more advanced mathematics.