Data analysis was carried out to examine (dis)continuities that the pre-service teachers faced while connecting the duos and the feedback received during such process. The data were collected through task-based interviews in which the participants were asked to complete angle bisector construction first with PPE and then in DGS, using together with the PPE to support the development of their construction strategies in DGS. The participants of this case study were two pre-service mathematics teachers, who had limited DGS experiences in solving geometry tasks. This paper focuses on pre-service mathematics teachers’ geometrical construction processes while using a duo of artefacts, namely the paper-and-pencil environment (PPE) and a dynamic geometry system (DGS). Also, the process of reaching the most inclusive cyclic quadrilateral affected both her instrumented techniques and her understanding about the hierarchical relations between the constructed quadrilaterals. The analysis showed that the participant’s initial instrumented techniques enabled her to construct subsets of the cyclic quadrilaterals. Data analysis was carried out to examine the participant’s development of construction strategies related to her instrumented techniques and her reasoning regarding the properties of different cyclic quadrilaterals. The data was collected through task-based interviews in which the participant was asked to construct the cyclic quadrilateral without using the circle tool in DGS. For this aim, the hierarchical classification and instrumental genesis constitute the theoretical framework of our study. Equally, it investigates her understanding of the hierarchical relations between the constructed cyclic quadrilaterals. This case study examines instrumented techniques of a pre-service mathematics teacher when utilising a dynamic geometry system (DGS) for the construction of cyclic quadrilaterals. Possible implications for both teacher education and future research are discussed. This finding raises concerns about teachers’ understanding of draggable geometric objects as being mainly associated with one perspective, generating examples, that is more static in nature than dynamic. We also found that teachers’ descriptions of draggable geometric objects were mostly associated with generating examples. We found that continuous variation is characterized by descriptions of an object: having invariant properties under continuous movement having different instantiations/formations containing strong links to generalization. We found that teachers tend to generate many examples as related to invariance however, they did not tend to use maintaining dragging or to enact continuous movements, as well as only few descriptions included links to generalization. In this article, we investigated such understanding from two perspectives: example spaces and continuous variation of an object. There has been little written about the ways in which teachers understand draggable geometric objects in a dynamic geometry environment with respect to variance and invariance-two important mathematical ideas related to the development of spatial perception and geometric reasoning. Possible future research directions and implications to teacher education are discussed. We also saw that teachers’ backgrounds and past experiences can play an important role in their descriptions of invariant properties. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. Specifically, we investigated if they even attend to invariant properties what invariant properties they discern and discuss and how DGE can support such discernment. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers’ knowledge of variance and invariance.
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