A look into students’ conceptual understanding of the definite integral via the APOS model

Abstract

The research aims to investigate the students’ levels of conceptual understanding of the definite integral and related topics. It made use of the APOS Theory, as a guide in the exploratory investigation and in the analysis and interpretation of the students’ responses in written exams (mastery tests). The instructional treatment is the ACE Teaching Cycle which is designed to help foster the formation of the mental constructions needed for the understanding of the concepts covered in the study. The study made use of one intact mathematics class of the Bulacan State University in the City of Malolos, Bulacan in the Philippines. Using descriptive statistics such as the mean, standard deviation, and percentages, the researcher primarily described the levels of students’ understanding of the specified mathematics concepts qualitatively and quantitatively by the APOS – based levels of understanding consisting of Action, Process, Object, and Schema. Results showed that for items in the mastery tests requiring Action, Process, and up to Object levels of understanding, the students got mean scores ranging from 2.40 to 5.0. Whereas, in items requiring the schema level of understanding, they posted mean scores ranging from 1.50 to 2.78. A deeper analysis of the results revealed that the subjects of the study, in general, attained only the Action to Process APOS levels of understanding across mathematical tasks catering to the following topics - the Definite Integral, Properties of the Definite Integral, the Fundamental Theorem of Calculus and Areas between Curves. From the results of the mastery tests, the ACE Teaching Cycle, which is the instructional component of the APOS Theory, can be contributory in the development of the conceptual understanding of students from all levels of mathematical performance in the definite integral and related concepts.