Translating Cyclic Mathematical Representations: Effects on Students’ Learning Outcomes and Cognitive Processes in Covariation Problem-Solving

Research on students’ understanding of covariation has primarily focused on static, linear models, overlooking the cognitive challenges associated with translating dynamic and cyclic representations. Despite their relevance to real-world applications, few studies have examined how students process covariational relationships in cyclic contexts, thereby limiting effective strategies to improve their problem-solving abilities. Using a sequential mixed-methods design, the study was conducted with 53 first-semester students at STKIP PGRI Sumenep in Indonesia, during the 2024/2025 academic year. Data were collected through pre-tests and post-tests to measure changes in problem-solving performance, along with a closed-ended questionnaire administered to all students and structured interviews with six selected participants, which provided deeper insights into the cognitive processes involved. The findings revealed significant improvements in students’ ability to interpret and solve covariation problems after engaging in tasks that required translating between cyclic graphical and tabular representations. Specifically, students exhibited enhanced learning outcomes in recognizing periodicity, understanding variable interdependence, and applying functional reasoning in cyclic contexts. The cognitive analysis indicated that tasks involving representation translation helped students engage more deeply with the cyclical nature of mathematical relationships. These results suggest that focused instruction on representation translation can foster stronger cognitive engagement with complex mathematical concepts. The study emphasizes the importance of incorporating cyclic models into early mathematics instruction and recommends that curriculum developers prioritize representational fluency. Further research is needed to explore the broader implications of such instructional strategies in diverse mathematical contexts.