An analysis of learners' mental constructions in learning and solving trigonometric equations

Abstract

Learners' construction of mathematical concepts is an issue of global concern. As a result, scholars advocate that mathematical topics identified as problematic for learners should be analyzed through research. This paper reports on an APOS (action, processes, objects, and schema) analysis of Grade 11 learners' mental constructions of trigonometric equations. The aim is to reveal the nature of the learners' mental constructions rather than provide a statistical analysis of their performance. This qualitative study was conducted with 17 purposively selected Grade 11 learners doing pure mathematics. The data used in this study were generated from learners' written responses and semi-structured interviews. Using the constructs of APOS theory, our analysis of learners' mental constructions revealed that most learners are operating at the action stage, which means that their knowledge construction of trigonometric equations is at the developmental stage. We further observed that learners needed external cues to trigger their thinking of how they should go about solving the trigonometric equation. All the procedural steps were explicitly written down, meaning they did not internalized the process of solving trigonometric equations. Furthermore, our analysis revealed that while learners can carry out procedures, meaning they can perform step-by-step procedures, they struggle to integrate concepts. The struggle to integrate concepts emanates from not constructing the concept image of trigonometric equations but rather constructing the concepts as isolated facts. Based on the findings, we argue that for learners to develop a schema for trigonometric equations, instructional strategies must focus on helping learners construct the concept image of trigonometric equations. Construction of the concept image will help learners start to internalize the procedures, thus leading to the interiorization of action to process and then encapsulate the process to an object stage.