Abstract

A study was done in an attempt to use the APOS theory of learning and teaching mathematics to develop and test a teaching cycle for the improvement of students’ conceptual understanding of related rates problems. “APOS” is an acronym that stands for Action, Process, Object, and Schema, and refers to both a theory of teaching and learning and a research methodology in mathematics education. “Related rates problems” refers to problems in Calculus that require finding the rate of change of one value, given the rate of change of a related value.
Part of APOS research methodology is a “genetic decomposition” of the concepts to be learned by the students in terms of the mental constructions that such learning requires. In the present study, the genetic decomposition focused on the mental constructions required for student success during the initial conceptual stages of related rates problems learning. The decomposition was constructed using the author’s knowledge of the subject. The genetic decomposition was used to construct an Action – Discussion – Exercise (ACE) teaching cycle which was then tested on two groups of students. Finally, students were asked to solve related rates problems during an individual interview with the author. Data from students’ involvement in the ACE cycle as well as their work during the interview process were then used to suggest changes to the genetic decomposition and the ACE cycle. These suggestions constitute the results of the study. Their purpose is to improve the starting point for further iterations of experimentation of teaching related rates problems.