Foster Student Understanding of Parallelism Based on the Toulmin Model and Pirie-Kieren Theory

Students’ understanding of the definition and classification of quadrilaterals serves as a crucial foundation in geometry learning because it allows students to make connections between geometric figures based on their essential properties. Previous studies have primarily examined the structure of arguments or levels of geometric thinking, leaving the dynamics of the development of understanding during the argumentation process inadequately explored. This study aims to how collective argumentation fosters the development of junior high school students’ understanding of the definition and classification of quadrilaterals through collective argumentation. The research framework integrates Toulmin’s argumentation model to analyze the structure of arguments with Pirie and Kieren’s theory of the development of understanding to identify how it fosters trajectories of understanding. This study employs a qualitative approach involving a mathematics teacher and eighth-grade students selected through purposive sampling. Data were collected through classroom recordings, student work samples, observations, field notes, and interviews. The results indicate that the development of understanding progresses through the layers of primitive knowing, image making, image having, property noticing, formalising, and observing, accompanied by the process of folding back, while the characteristics of structuring were not fully and explicitly identified in the data. The integration of these two theories shows that the Toulmin model reveals the construction of arguments, while the Pirie-Kieren theory explains the development of understanding that underlies those arguments. These findings emphasize that collective argumentation fosters a mechanism that drives the reorganization of knowledge and strengthens students’ conceptual understanding of the definition and classification of quadrilaterals.