An analysis of gifted students’ proof and reasoning process based on the Toulmin model

This study aims to examine the mathematical reasoning and proof processes of gifted students within the framework of Toulmin’s Argumentation Model. Grounded in the Rich Teaching Model, the instructional design included seven differentiated lesson plans. The research was conducted with two 9th-grade students in a Science and Art Centre over 20 weeks during the 2022–2023 academic year. Employing a design-based research methodology and a qualitative case study approach, the study focused on a non-routine problem from the product phase of the first lesson plan. This task was examined in depth through micro-level case analysis. Data were analyzed using descriptive methods aligned with Toulmin’s model components. Findings indicate that students used diverse representations, models, and notations to make sense of abstract mathematical concepts and integrated dynamic tools such as GeoGebra into their reasoning. Nonetheless, challenges with formal notation were observed. The study underscores the value of structured, technology-supported instruction in enhancing gifted learners’ mathematical thinking.