Mathematics Proof Competency: A Mixed-Methods Investigation of Self-Explanation versus Explaining to Fictitious Others

<p dir="ltr">This mixed-methods study investigated the differential effects of self-explanation (SE) versus explanation to fictitious others (EFO) on mathematical proof competency among 148 first-year calculus students. Using a within-subjects crossover design, students created video explanations employing both strategies across proof production, comprehension, and transfer tasks over three months. Quantitative analyses revealed no significant differences between SE and EFO in immediate proof production performance. However, MANOVA results showed that EFO students demonstrated significantly superior gains in proof summary (understanding specific inferential steps) compared to SE students, while no differences emerged for holistic comprehension (overall proof structure). After three months, no significant differences were found in proof transfer performance, though this finding should be interpreted cautiously due to contextual testing factors. Qualitative interviews with 19 students revealed that strategy preferences were influenced by explanation confidence, knowledge domain (procedural vs. conceptual), and three key factors: motivation (achievement satisfaction vs. efficiency concerns), learner characteristics (prior knowledge and language proficiency), and learning material complexity. These findings suggest that while both strategies engage similar cognitive processes, EFO's social presence mechanisms may provide targeted advantages for developing fine-grained proof comprehension through enhanced elaboration and metacognitive monitoring.</p>