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Abstract Background Functional thinking is characterized as a specific way of thinking in relationships, dependencies, and changes. Hence, beyond mathematics, it is also crucial for other (STEM) disciplines as well as for everyday situations. In particular, dealing with different representations of functions and changing between them are core function-related competencies, which are correspondingly needed for the formation of appropriate concepts and flexible problem-solving in various situations. Therefore, this study investigated students’ (N = 856) competencies related to representational changes of elementary functions and, in particular, assessed which changes are especially easy or difficult for students. Moreover, possible school track and gender differences were investigated by performing DIF analyses within the framework of Rasch modeling. The data were gathered using a paper–pencil test administered after the students had completed the teaching unit on linear functions in their mathematics lessons. Results Altogether, students were found to have limited competencies related to representational changes of elementary functions. There was no clear pattern regarding the types of representational change that were difficult or easy for them. Moreover, girls performed better on purely mathematical tasks, whereas boys did better at a complex modeling and problem-solving task. Classes from the academic track produced better results in tasks with a situational context compared to their peers from non-academic tracks, who performed relatively strongly on purely mathematical tasks. Conclusions These findings imply that various representations and representational changes should be included in lessons on functions to support students in building a rich concept of function and flexible problem-solving skills, thus fulfilling curricular requirements and responding to didactical considerations. In particular, the teaching of functions should be more balanced by mixing tasks with and without a situational context and the corresponding representational changes. These findings should motivate teachers, in particular those teaching non-academic tracks, to give a more prominent role to situational contexts in their lessons on functions in order to foster their students’ learning and build a bridge between mathematics and real-world situations.

Research on evidence-based argumentation shows that (pre-service) teachers have difficulties in orienting their actions to existing theories and empirical evidence. This article addresses the knowledge content needed for this and presents a vignette-based procedure. Within each vignette, two different theoretical perspectives are addressed. The behavior of a teacher can be either suitable or unsuitable from both perspectives or more or less suitable depending on the perspective. In study 1, the procedure is piloted and in study 2, an intervention on a specific area of knowledge takes place. The results show that participants differentiate the vignettes as expected. The intervention leads to corresponding increases in knowledge, which likely relates to a change in the evaluations. The presented approach is discussed with regard to possible applications in the context of research on evidence-based argumentation.