An Instructional Bottleneck in Learning Limit: An APOS Analysis of Calculus Classroom Practice

Abstract

This study examines how calculus instruction provides opportunities for students to engage with research-based models of learning the concept of limit. Using genetic decomposition from APOS theory as an analytic framework, we analyze four lessons taught by an experienced instructor at a U.S. research university. A total of 36 task discussions were coded to determine whether instruction supported evaluating functions at a point, repeated evaluations, or coordination of domain and range processes. Results show that most task discussions focused on evaluating functions at a single point, with limited opportunities for repeated evaluation or coordination of variation. While students were able to compute limits, instruction provided few opportunities to develop a process-level understanding of limit. Beyond individual instructional choices, we also reported that the instructional bottleneck is embedded within a broader institutional ecology as well. We also reported that This study highlights the importance of aligning instructional practices with cognitive models of learning and suggests that greater attention to process-level reasoning may support students’ conceptual understanding of limit.