Understanding error patterns in students' solutions to linear function problems to design learning interventions

Reasoning with and about linear functions (LF) is considered essential knowledge for college readiness, but evidence shows that students experience difficulty in this topic. There were two overarching aims of the reported studies. One was to assess students' understanding of LF and discern the cognitive underpinnings of common errors they make in these types of problems. The second was to explore designs of a learning intervention that can ameliorate these misunderstandings. In Study 1, analyses of performance on Verbal Description, Table, and Graph LF problem types showed substantial and interpretable errors on the latter two. Errors reflected systematic conceptual and procedural misunderstandings associated with interpretation of structural features of Table and Graph LF problems. The results for LF Table and Graph problems suggested that students could benefit from error focused interventions. In Study 2, information about common errors from Study 1 was used to inform the design of instructional interventions using three worked example conditions: (1) error detection and correction, (2) error correction only, and (3) no error (control). Results of quantitative and qualitative analyses showed enhanced performance in all intervention conditions and across both problem types and were particularly impactful for both problem types when error correction was required in worked examples. Findings from both studies are discussed in terms of implications for learning environments, including initial instruction and diagnostic assessment.