What first-year students’ prior mathematical knowledge reveals for modelling-oriented university mathematics teaching

Background: Students’ prior mathematical knowledge plays a critical role in their success in university mathematics, yet it is often treated diagnostically rather than instructionally. Aim: This study examines students’ prior mathematical knowledge at the start of a university calculus course, with the purpose of identifying conceptual and procedural strengths and weaknesses that are relevant for subsequent learning and for a mathematical modelling intervention. Setting: The study was conducted in South Africa and involved three cohorts of first-year science and engineering students (1874 participants) across the years 2020, 2021 and 2022. Methods: A mixed-methods sequential explanatory design was employed. Quantitative data were collected using a diagnostic test. Welch’s one-way analyses of variance (ANOVAs) and Games–Howell post hoc tests were used to compare cohort performance across content areas. Qualitative analyses of written solutions from a stratified random sample of 90 students were conducted to examine errors and misconceptions. Results: Although overall test performance showed modest improvement across cohorts, students’ results remained low across most content areas. Algebra emerged as a relative strength, indicating procedural fluency, whilst weaknesses were evident in Analytical geometry, Functions and Modelling. Qualitative analyses revealed persistent conceptual and representational difficulties, in interpreting inequalities, graphing functions and constructing and using linear models. Conclusion: The findings suggest that apparent improvements in preparedness may cover structural weaknesses. The study highlights the importance of using diagnostic evidence not merely to identify at-risk students, but to inform responsive teaching that addresses conceptual gaps. Contribution: By integrating large-scale diagnostic data with qualitative error analysis, students’ prior knowledge can be leveraged as an instructional resource to support modelling-oriented teaching in university mathematics.