| Summary: | Acknowledging the value of understanding defining and classifying special quadrilaterals for prospective mathematics teachers (PMTs), the present study attempts to track this understanding so that the progress of thinking from Van Hiele's level 2 to level 3 could be theorised. Thus, a bounded case study sample of PMTs, who had graduated and joined the mathematics teacher preparation diploma at the Faculty of Education, Tanta University in Egypt, were selected and requested to (a) define trapezoid, parallelogram, rhombus, rectangle, and square, and (b) represent the relationship among these quadrilaterals. The data were collected and analysed in two cycles. During the first cycle, participants' responses were scrutinised upon Prototype 1; it was developed based on the literature review to describe levels of special quadrilaterals understanding as faulty, partitional-uneconomical, partitional-economical, hierarchical-uneconomical, and hierarchical economical. Similarly, the researchers replicated the same analytical process in the second cycle in order to validate the levels suggested in Prototype 1. Also, some clinical interviews were conducted to confirm the participants’ representations of relationships among the defined quadrilaterals. The results enabled advancing the hypothetical Prototype 1 to Prototype 2. Prototype 2 reconceptualised the levels of understanding into faulty, slightly economical, fairly economical, and economical, wherein each level was determined based on (a) the economics of the concept definition and (b) the awareness of relationships among other related definitions to the concept defined (recognising subsets and supersets). These results are prospective for further investigations to sufficiently unpack all sub-levels of geometric thinking embedded in Van Hiele’s fixed levels. It also provides basics on proper pedagogical approaches and corresponding interventions to train PMTs effectively teach geometric thinking.
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