Abstract

Despite its centrality to deductive reasoning, hierarchical inclusion remains one of the most persistently misunderstood foundations of school geometry. This study examined preservice mathematics teachers’ understanding of quadrilateral properties and their hierarchical relationships, drawing on Van Hiele’s theory of geometric thinking and Shulman’s pedagogical content knowledge framework. An explanatory mixed-methods design was employed in which 190 undergraduates completed a structured test, and 20 participated in focus group discussions. The results revealed that participants’ geometric reasoning was largely confined to the lower Van Hiele levels. Focus group responses indicated reasoning characteristic of Level 0 (Visualisation) and Level 1 (Analysis), with participants reducing definitions of quadrilaterals to side count and demonstrating little awareness of inclusion relationships. The findings also revealed that while some participants could identify isolated properties of shapes such as squares and rectangles, their explanations were frequently incomplete, contradictory, and grounded in prototypical visual features rather than necessary and sufficient conditions. Taken together, the findings highlight structural fragility in preservice teachers’ geometric knowledge and emphasise the need for hierarchy-focused, deductive reasoning in teacher education programmes.

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