Local instructional theory STEM: Integrating the context of football into parabola learning to support prospective teachers’ flexibility

Mathematics learning for prospective teachers frequently emphasizes procedural proficiency, while the development of mathematical flexibility, particularly the ability to shift representations and adapt problem-solving strategies, remains underdeveloped. This limitation is often associated with learning designs that insufficiently integrate STEM perspectives and meaningful real-world contexts. To address this issue, this study aims to develop a STEM-based Local Instructional Theory (LIT) using a football context to strengthen the mathematical flexibility of prospective mathematics teachers. The research employed a design research approach, a validation study type, conducted in two main phases: a pilot experiment and a teaching experiment. The learning design was iteratively refined through continuous reflection between the Hypothetical Learning Trajectory (HLT) and the Actual Learning Trajectory (ALT). Data were collected through classroom observations, learning videos, students’ written work, and documentation of instructional activities, and were analyzed qualitatively. The findings indicate that STEM-based learning grounded in a football context effectively enhances students’ representational and strategic flexibility, as evidenced by their ability to move among visual, numerical, and symbolic representations and to reflectively evaluate problem-solving strategies. The integration of digital tools such as Desmos and Kinovea further supported the visualization and validation of mathematical models. This study suggests that the developed LIT provides a contextual and innovative framework for improving mathematics teacher education.