Mathematics teacher education has long been the focus of educational research. Much attention has been paid to university studies and training in schools in the early years (e.g. TEDS-M, Döhrmann et al., 2012). In recent years, the continuing development of teachers already working in schools has also become increasingly important. Professional development (PD) courses are crucial for advancing mathematics education, enriching teachers’ skills, and overall enhancing mathematics instruction (Prediger et al., 2022). Facilitators play a pivotal role in this process, significantly influencing teacher learning (e.g., Borko et al., 2011). Notably, facilitator expertise, particularly their ability to engage in noticing during PD courses, is instrumental in supporting teachers’ effective learning. Following van Es and Sherin's (2002) framework, noticing includes recognizing the significance of a situation, establishing connections between interactions and broader teaching principles, and using contextual knowledge to reason about interactions.

Facilitator expertise in PD settings, essential for navigating complex instructional scenarios, has been explored by Zaslavsky and Leikin (1999). However, recent research tends to overlook content-specific considerations while focusing mostly on generic aspects, relevant for different subjects (Prediger et al., 2022). To address this gap, our study focuses on content-related PD, examining facilitators' categories like pedagogical content knowledge on the PD level (PCK-PD) and general pedagogical knowledge on the PD level (GPK-PD). Particularly, we analyze how facilitators categorize a fictional PD situation.

The concept of noticing for teachers (van Es & Sherin, 2002) is also transferable to facilitators and PD settings. This transferability can be seen when aligning this framework with the PID-model proposed by Kaiser et al. (2015), which highlights situation-specific skills such as the perception of events, interpretation of activities, and making of decisions. In a PD setting, similarly to in a classroom situation, perception, interpretation, and decision-making are central processes of a diagnostic competence and skilled navigation of facilitation and teaching (Hoth et al., 2016).

In considering the expertise that facilitators bring to PD settings, frameworks for examining teachers’ expertise have been lifted to the facilitator level (Prediger et al., 2022). Specific to mathematics content-related facilitator expertise, Prediger et al. (2022) applied a content-related framework for teacher expertise (Prediger, 2019) to the facilitator level. The framework includes jobs as typical and complex situational demands that are connected to the facilitation of specific mathematics PD content. Furthermore, practices are seen as recurring patterns of facilitators’ utterances and actions for handling the jobs and are influenced by underlying categories, pedagogical tools, orientations, and situative goals. In regards to the knowledge that determines the facilitators’ categories or categorial perception and thinking that impacts the facilitators’ practices, this knowledge is connected to their pedagogical content knowledge for teachers’ professional development (PCK-PD). Such PCK-PD consists of the knowledge the facilitators have relative to teachers’ learning. More general pedagogical knowledge on the PD level (GPK-PD) refers to the knowledge facilitators have in relation to the management and instruction of PD courses such as related to motivational aspects (Prediger et al., 2022)

In considering these underpinning aspects concerning noticing and facilitator expertise, we pursued the following research questions as a way of examining the expertise and needs of the facilitators in a mathematics PD and qualification program, and specifically, as a means of examining the PCK-PD and GPK-PD that facilitators demonstrate when noticing:

1) How do experienced facilitators engage in noticing of a fictional PD situation?

2) How do less experienced facilitators engage in noticing of a fictional PD situation?

Sample In total, 156 mathematics facilitators participated in the PD qualification program for facilitators that lasts the first year of a ten year program for the qualification and professional development of mathematics facilitators and teachers. Seventy-two of the facilitators had experience leading 10 or more mathematics PD courses, while 84 of the facilitators had facilitated fewer than 10 PD courses. Instrument The facilitators were provided with a situated instrument, containing a fictional dialogue of three mathematics teachers in a PD who discuss the use of a learning application (app) as a means of supporting student learning. The teachers in the fictional dialogue champion less productive ways of using apps, by emphasizing short-term and motivational benefits of the app, while not reflecting on the lack of construction of conceptual understanding, means of promoting cognitive activation, ways to monitor students’ learning progress, and the development of a learning environment that stimulates communication. The facilitators were asked: 1) Briefly describe what stands out to you about this discussion amongst teachers in a PD? 2) How would you interpret the statements of the three teachers? 3) As a facilitator, how would you respond? Questions 1 and 2 prompted the facilitators to perceive and interpret the situation while question 3 provided the facilitators with the opportunity to show their decision-making concerning the situation. Data analysis The experienced and less experienced facilitators’ responses to questions 1, 2, and 3 were analyzed in terms of their categories for perceiving, interpreting, and deciding to act upon the situation. Thereby, the facilitators’ general pedagogical knowledge (GPK-PD) and pedagogical content knowledge (PCK-PD) were distinguished (Prediger et al., 2022). The five principles of the PD were deductively coded with respect to the facilitators’ PCK-PD: conceptual focus, cognitive demand, student focus and adaptivity, longitudinal coherence, and enhanced communication. Each code was rated as 0 (category not addressed in response) or 1 (category was addressed in response). In considering the facilitators’ GPK-PD, their responses were inductively coded, yielding six categories: atmospheric argumentation, general digital media focus, methodological individualization, short-term success, affective-motivational aspects, and general description. The different PCK-PD and GPK-PD subcategories were assigned to facilitators’ perception/interpretation of the PD situation (questions 1 and 2), as well as to their decision-making (question 3). After several rounds of discussion, a Cohen’s (1960) kappa between k = .88 and k = .94 for inter-rater reliability for the coding was reached by the research team.

Results Concerning the first research question, it can be observed that the experienced facilitators refer to both PCK-PD and GPK-PD in both perception/interpretation and decision-making (P/I-PCK-PD: M=0.77(0.86); P/I-GPK-PD: M=1.37(1.09); D-PCK-PD: M=0.82 (1.08); D-GPK-PD: M=1.14(0.94)). The second research question reveals that the less experienced facilitators also use both PCK-PD and GPK-PD. Here, too, it can be seen that GPK-PD is used more frequently than PCK-PD when noticing the situation. However, the difference between these two is greater than with the experienced facilitators (P/I-PCK-PD: M=0.52(0.77); P/I-GPK-PD: M=1.53(1.05); D-PCK-PD: M=0.64(1.06); D-GPK-PD: M=1.36(0.89)). An experienced facilitator statement exemplifies what a strong PCK-PD focus for perception/interpretation encompasses: “Ms. M. and Mr. M. are pleased that the app contains tasks at different levels and adapts to the individual learning level of the children. No one makes a statement about whether the app works in an understanding-oriented way (e.g. with representations) and whether the levels of representation are interlinked, but this does not seem to be the case. This should be discussed with the teachers.” The statement from a less experienced facilitator exhibits their focus on GPK-PD for decision-making: “I would try to emphasize the positive aspects of the app and assuage potential fears or motivate people to try it out.” Contribution By uncovering differences in how facilitators with varying levels of experience engaged in approximated noticing of a PD situation, this research, in using a situated approach, provides insights into designing qualification programs as based on facilitators’ needs. Thereby, an emphasis on PCK-PD can be integrated into the qualification program so that less experienced facilitators can be better prepared for leading PD. However, it also shows that even experienced facilitators need support. By aligning the design of qualification programs with the needs of the participating facilitators, facilitators will be better prepared to offer PD for teachers.

Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM Mathematics Education, 43(1), 175-187. https://doi.org/10.1007/s11858-010-0302-5 Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20(1), 37-46. Döhrmann, M., Kaiser, G., & Blömeke, S. (2012). The conceptualisation of mathematics competencies in the international teacher education study TEDS-M. ZDM Mathematics Education, 44(3), 325-340. Hoth, J., Döhrmann, M., Kaiser, G., Busse, A., König, J., & Blömeke, S. (2016). Diagnostic competence of primary school mathematics teachers during classroom situations. ZDM Mathematics Education, 48(1), 41-53. Kaiser, G., Busse, A., Hoth, J., König, J., & Blömeke, S. (2015). About the complexities of video-based assessments: Theoretical and methodological approaches to overcoming shortcomings of research on teachers’ competence. International Journal of Science and Mathematics Education, 13(2), 369-387. Prediger, S. (2019). Promoting and investigating teachers’ pathways towards expertise for language-responsive mathematics teaching. Mathematics Education Research Journal, 31, 367-392. https://doi.org/10.1007/s13394-019-00258-1 Prediger, S., Roesken-Winter, B., Stahnke, R., & Pöhler, B. (2022). Conceptualizing content-related PD facilitator expertise. Journal of Mathematics Teacher Education, 25, 403-428. https://doi.org/10.1007/s10857-021-09497-1 van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10(4), 571-596. Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms. Review of Educational Research, 78(3), 516-551. https://doi.org/10.3102/0034654308320292 Zaslavsky, O. & Leikin, R. (1999). Interweaving the training of mathematics teacher educators and the professional development of mathematics teachers. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 143-158). PME.