Pre-service teachers' classroom stagings of proving-related activities and possible effects of Klein's discontinuity

<div><p>Proof is a core element of mathematics. It therefore plays an essential role in university mathematics studies and thus also in the training of pre-service mathematics teachers. However, it has been shown that in-service teachers face problems in adequately implementing proof in their teaching. We consider this break between university studies and professional practice as part of what Klein called the "second discontinuity". Employing an activity-theoretical framework, we investigate in this paper the question how lesson stagings of preservice teachers at the end of their study program indicate systematic difficulties in staging proving-related activity for the classroom and how discontinuity experiences triggered by differences between proving-related practices in school and university could be considered as a possible cause for that. We present results from an observational study with pre-service teachers that reveal specific patterns in their proof-related behavior when planning and delivering lessons, and relate these back to sources for possible discontinuity experiences of pre-service teachers. As a practical implication, our results provide indications of how the topic of proof could be targeted in mathematics didactic teacher education.</p></div>