(Dis)Continuity and Feedback in Using a Duo of Artefacts for Robust Constructions: The Case of Pre-Service Mathematics Teachers Using Paper-and-Pencil and Dynamic Geometry

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). The participants of this case study were two pre-service mathematics teachers, who had limited DGS experiences in solving geometry tasks. 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. 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. Findings indicated that the pre-service teachers had difficulties in comprehending ( 1) the concept of robustness of the construction and (2) the dependency relationships between the objects in DGS, which created discontinuities for them to differentiate between a static representation on paper and dynamic representation in DGS. In this, the use of duos and the feedback different artefacts generated -- i.e., direct manipulation feedback, evaluation feedback and tool operation feedback from the DGS, and strategy feedback scaffolded by the interviewer --, promoted their conception of angle bisector construction.