Encapsulation of the defined integral mathematical object of university students: a view from APOS theory

Understanding the concept of definite integrals is a persistent challenge in university mathematics education, largely attributable to traditional teaching practices that prioritise mechanical procedures over conceptual depth. Many students experience a disconnect between the algebraic, geometric, and conceptual representations of the definite integral, leading to a fragmented understanding of the concept. Therefore, this study seeks an alternative to the difficulties students face in understanding the definite integral from a conceptual perspective. To address this issue, the article aims to strengthen students’ understanding of the definite integral in the Bachelor of Mathematics programme at the University of Quindío, using APOS theory. This theory conceives mathematical understanding as a progressive construction of mental schemas, facilitated by cognitive mechanisms such as internalisation, coordination, and encapsulation. The study is framed within a qualitative methodology and a hermeneutic-interpretive approach. Didactic sequences, problem-solving analysis, and focus groups were used as research instruments. The findings indicate that an approach focused on students’ conceptual development, supported by appropriate teacher intervention, promotes deeper and more lasting learning. Consequently, teaching planning that incorporates genetic decomposition, contextualised activities, classroom reflection, and guided exercises improves understanding of the definite integral and supports connections among different representations.