Exploring Students’ Mathematical Understanding Through ‘Trial and Error’

Objectives This study aimed to analyze students' trial-and-error processes when solving proportionality problems and to explore the mathematical understanding and thought processes underlying their actions. Methods Two 7th-grade students participated in individual clinical interviews. Their task performance was analyzed, focusing on proportional problems in which all four quantities were relatively prime. The study adopted Beckmann and Izsák’s (2015) theoretical framework, utilizing the variable parts and multiple batches perspectives to examine students’ units coordination activities. Results One student accurately calculated an unknown value using a proportional algorithm but encountered difficulties in visually representing and quantitatively interpreting the result. These difficulties were attributed to the inconsistent application of units coordination across different measurement spaces. The other student exhibited similar initial struggles but ultimately solved the problem by employing the variable parts perspective and using a visual model to successfully transition between two three-level units. Their trial-and-error processes served as meaningful indicators of their developing understanding of proportional relationships and units coordination. Conclusions The findings suggest that students’ errors are not merely procedural mistakes but provide valuable insights into their conceptual understanding. Units coordination theory emerges as a powerful tool for interpreting students' reasoning and supporting their mathematical development. The study highlights the importance of designing personalized feedback strategies and effective instructional practices grounded in an understanding of students’ unit-based reasoning.