From Informal to Formal Understandings: Analysing the Development of Proportional Reasoning and its Retention

This study is a case study within a three-year-long design research study. The purposes of this study are twofold. The first purpose includes investigating whether there is a significant difference in seventh-grade students' proportional reasoning skills before and after a teaching experiment conducted in line with the perspectives of Hypothetical Learning Trajectories and the theory of Realistic Mathematics Education and assess retention of those skills by means of quantitative analysis. The second purpose includes describing the changes and patterns in students' levels of ratio understandings and related uses of informal, pre-formal, and formal representations through the analysis of students' written work in the achievement tests and task-based interview results. Findings of the study showed that the intervention proved successful in enhancing students' proportional reasoning skills and supporting the shift in students' levels of ratio understanding and representation use from informal ones to more formal ones. Students' levels of ratio understanding and representation use are described in detail with several sample student responses from the pre-, post- and retention assessments, which also delineate the shift in students' proportional reasoning skills from informal to formal ones. Several discussions are made to discuss the findings and their potential implications.