Critical mathematical thinking process in definite integral applications: an action, process, object and scheme based on cognitive styles

Abstract

Higher-order thinking skills are essential in calculus learning. However, research examining how students with different cognitive styles construct mathematical critical thinking processes within the action, process, object and scheme (APOS) framework when solving integral application problems remains limited. Furthermore, the integration of cognitive styles with the APOS framework has not been widely explored. This study investigates differences in students’ mathematical critical thinking processes based on cognitive styles using the APOS framework. This study used a qualitative case study design involving 20 students, and four subjects were purposively selected for in-depth analysis, representing field-independent (FI) and field-dependent (FD). Data were collected through critical thinking tests, the group embedded figures test (GEFT), and semi-structured interviews, and analyzed using data reduction, data presentation, and data verification aligned with the APOS stages. The results showed two contrasting reasoning patterns. FI students reached the schema stage with a complete thinking trajectory (understanding the problem, formulating a strategy, performing calculations, and partial evaluation) supported by object-process actions. In contrast, FD students were limited to the action stage with problem-understanding thinking processes. These findings contribute by showing that the development of critical thinking is non-linear and influenced by cognitive style, and emphasize the importance of adaptive learning strategies in calculus.