Exploring students’ covariational reasoning in sine and cosine functions: A comparison of expected and manifested learning trajectories with dynamic tasks

This study investigates how high school students develop covariational reasoning in the context of trigonometric functions by integrating dynamic GeoGebra applets within a design-based research framework. Guided by a Hypothetical Learning Trajectory (HLT), the research compares the expected progression of reasoning—from coordination of values to smooth continuous covariation—with the actual reasoning manifested by students during iterative instructional cycles. Qualitative analyses of students’ task artifacts and verbal explanations reveal distinct learning trajectories among participants, highlighting the importance of task sequencing, explicit scaffolding, and dynamic visualization for fostering continuous reasoning. The findings inform instructional design by identifying key areas for differentiated support and further refinement of digital interventions in mathematics education.