Exploring Undergraduate Students’ Difficulties in Solving Non-routine Algebra Problems Using APOS Theory

Non-routine problem solving is increasingly recognized as a vehicle for developing undergraduate students' mathematical reasoning, yet little research has systematically identified and classified the specific difficulties students encounter when confronting such problems. This qualitative descriptive case study examined how 32 undergraduate students enrolled in Intermediate Algebra expressed their difficulties when solving three non-routine algebra problems across a semester. Drawing on APOS (Action-Process-Object-Schema) theory and process-object duality, a modified analytical framework was developed to identify and categorize the levels, stages, and types of difficulties students expressed in written form. Analysis revealed two developmental stages, pre-object and post-object, and four distinct difficulty types: definition-object, definition-operation, combined-with-other-object, and operation-on-object. Findings indicate that students predominantly struggled when multiple mathematical objects appeared together in unfamiliar contexts, even when they demonstrated competence with individual objects in isolation. Students frequently showed uncertainty about which object to attend to first, misapplied familiar operations across incompatible object types, and focused on individual components while neglecting the broader mathematical structure. This pattern of combined-object difficulty is not fully addressed by existing APOS theory or process-object duality frameworks. The study extends APOS theory by characterizing this combination challenge and offers practical implications for instructional design in undergraduate algebra courses.