Instructional process for the construction of the derivative function: Modelling and simulation in GeoGebra from the ontosemiotic approach

The conceptual and procedural understanding of the derivative remains a persistent challenge in undergraduate engineering education, particularly in bridging symbolic, graphical, and applied interpretations. Despite advances in digital tools, few instructional designs systematically integrate interactive exploration with formal mathematical reasoning. This study addresses this gap by proposing an instructional framework that combines functional modeling with GeoGebra-based simulations, grounded in the Ontosemiotic Approach to Mathematical Knowledge and Instruction (OSA), specifically targeting first-year engineering students in Chile. A mixed exploratory–descriptive design was implemented, combining quantitative and qualitative analyses. A three-session intervention involved 102 students who engaged in tasks assessing derivative understanding across multiple representations, including graphical slopes, symbolic differentiation, and applied rate-of-change problems. Data were collected via performance questionnaires and written productions, with validity ensured through expert review and reliability confirmed via pilot testing. Students exhibited strong proficiency in graphical interpretation and procedural manipulation of derivatives, with success rates exceeding 85%. Conversely, tasks requiring formal argumentation, rigorous use of limit definitions, and theoretical justification showed reduced performance at 68%, highlighting the challenge of connecting exploratory simulations with formal mathematical reasoning. The findings demonstrate that integrating functional dependency analysis, interactive simulations, and OSA principles can strengthen comprehension of derivatives, particularly in geometric interpretations and formal rate-of-change reasoning. This research provides a replicable instructional design that enhances both conceptual insight and procedural competence, offering evidence-based strategies for technology-enhanced mathematics instruction in engineering curricula and contributing to broader curriculum development.