Persistent difficulties in undergraduate mathematics are often linked to students’ reliance on procedural reasoning rather than structural understanding. Guided explicitly by APOS theory (Action–Process–Object–Schema), this study investigates how assessment task design can support the development of higher-order mathematical reasoning in first-year calculus. Two assessment tasks were designed using an Initial Genetic Decomposition (IGD): one focusing on the domain of composite functions and the other on asymptotes under functional transformations. Each task was deliberately structured to provoke mental constructions across APOS levels, moving students from rule-based actions to coordinated processes, object-level conceptualisation, and schema-based generalisation. Data were generated through students’ written responses to a questionnaire completed by the full cohort and follow‑up semi‑structured interviews with selected participants to confirm emerging inferences. Data analysis was explicitly aligned with the IGD, using APOS indicators to identify students’ dominant modes of reasoning. Findings show that most students operate at Action and Process levels, applying rules mechanically with limited coordination of inner–outer function relationships. Fewer students attained Object-level reasoning, where composition and transformation were treated as coherent mathematical entities, and only a small proportion demonstrated Schema-level reasoning, evidenced by generalisation and invariance awareness. The study demonstrates how APOS-informed assessment design can function not only as an evaluative tool but also as a mechanism for fostering conceptual development. It contributes to theoretical discussions on assessment as a catalyst for cognitive growth in higher education mathematics.

Authors:
Mthobisi Ntuli, Durban University of Technology, South Africa

About the Presenter(s)
Dr Mthobisi Luca Ntuli is a Lecturer in Mathematics at the Durban University of Technology. His interests include mathematics education, HOTS, and AI in teaching. He currently leads a SoTL project on student learning and success.

Connect on Linkedin
https://www.linkedin.com/in/dr-mthobisi-ntuli-42650655/

See this presentation on the full schedule – Sunday Schedule

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