An investigation of preservice teachers' content knowledge of transformation geometry and an analysis of their conceptual definitions based on the van hiele model

The aim of this study is to examine the content knowledge of pre-service elementary mathematics teachers on transformation geometry and to analyze their conceptual definitions based on the Van Hiele model. The study adopted a case study design, one of the qualitative research methods. The study group consisted of 56 pre-service teachers studying at the faculty of education of a medium-sized state university located in the western part of Türkiye during the 2024–2025 academic year. Participants were selected using the criterion sampling method, one of the purposive sampling techniques. To determine the participants' content knowledge, the Transformation Geometry Content Knowledge Scale was developed. In the analysis of the data, content analysis, descriptive analysis, thematic content analysis, and frequency–percentage values were employed. The research findings revealed that the pre-service teachers' content knowledge regarding transformation geometry concepts was generally insufficient or superficial. The definitions made for symmetry, translation, rotation, and other types of transformations were mostly at a visual and intuitive level, with limited use of mathematical language, representations, and conceptual explanations. According to the Van Hiele model, most of the explanations were concentrated at the second level, and no definitions were found at the fourth or fifth levels. In addition, common conceptual errors identified included confusion about direction, incorrect use of reference points/lines, ignoring the principle of equal distance, failure to recognize vectorial movement, mixing up different transformations, and inability to express composite transformations as single transformations.