Examining proof schemes of gifted elementary students and their mathematically successful peers

This study investigated how fourth-grade gifted students (N = 100) and their mathematically high-achieving peers (N = 100) approached mathematical reasoning tasks. Students’ written solutions to three problems were examined using Harel and Sowder framework, categorizing reasoning as External, Empirical, or Analytical, revealing notable group differences. Gifted students primarily used analytical proof schemes, demonstrating processes such as generalization, abstraction, and formulation of rules. In contrast, mathematically high-achieving peers relied more on empirical proof schemes, justifying their reasoning through concrete examples, drawings, and visual representations. Interestingly, authoritarian proof schemes were observed only in the high-achieving group, reflecting reliance on external authority rather than independent reasoning. These results indicate that gifted students and mathematically successful peers employ qualitatively different approaches to mathematical thinking. The study highlights the importance of differentiated instruction, encouraging shifts between proof schemes, fostering metacognitive awareness, and promoting autonomy to help students develop flexible, deep, and sophisticated mathematical reasoning skills.