Most European countries face severe teacher shortages, particularly in mathematics, as well as an increased awareness of teachers’ self-perceived need for professional development (PD) in teaching mathematics. In Germany the lack of specialist teachers has led to a high percentage of out-of-field teachers, resulting in an additional need for PD (Eurydice, 2021). At the same time the average student’s mathematics achievements in many European countries, especially in Germany, have decreased (Mullis et al., 2020). Accordingly, students lack basic concepts that are necessary for the cumulative process of acquiring knowledge and competence in mathematics. Against this background, a PD program was implemented, aimed at improving teachers’ expertise for fostering students’ understanding of basic mathematical concepts (Prediger et al., 2023).

Evaluating whether a PD program promotes teachers’ expertise is challenging when the success of the PD program is not only measured in terms of the teachers’ increase in knowledge but also regarding the effect on teachers’ teaching practices. The latter often remains unclear when PD programs are evaluated by pre-post knowledge test results or teacher self-reports. Evaluating how teaching practice has been improved requires an ecologically valid instrument (Krolak-Schwerdt et al., 2018). To approximate teaching practice as much as possible, we developed a vignette-based approach that places teachers in an authentic teaching situation in which classroom discussions are conducted to foster students’ understanding of basic concepts in mathematics. Our instrument not only serves to measure whether teachers benefit from the PD, but also provides a tool to assess teachers’ abilities to foster students’ understanding. While extensive research on teachers’ diagnostic abilities and how they can be facilitated exists (Chernikova et al., 2020), evidence on how teachers foster students’ understanding is rare.

Previous evidence on how students gain conceptual understanding provides insights into conditions of supportive learning environments. Accordingly, using multiple presentations such as visual, numerical, and symbolical representations, and elaborating on the relationship between them is an effective strategy to support students in attaining conceptual understanding (Hunt & Little, 2014; O-Dwyer et al., 2015; Tzur et al., 2020). Further studies have shown that prompting students for explanations and justifications (Booth et al., 2015; Jansen et al., 2017) of their mathematical reasoning or what they have learned can support them to gain deeper conceptual understanding. Studying and reflecting on incorrect mathematical work in addition to or in combination with correct work are other ways to support students’ conceptual understanding that have been confirmed by evidence (Siegler & Chen, 2008). Also, encouraging students to verbalize their thinking and enhancing communication on mathematical aspects amongst students is a prerequisite (Erath et al., 2021) to enable students to gain a deeper conceptual understanding.

Therefore, teachers are assumed to foster students’ conceptual understanding if they provide a learning environment that considers the following supportive conditions:

• Connecting representations,
• Prompting explanations or justifications,
• Prompting student-initiated error analysis,
• Encouraging verbalization of students’ thinking,
• Encouraging communication among the students.

Starting from these assumptions we examined the following research questions.

1a. Do teachers choose a supportive learning environment to foster students’ conceptual understanding?

1b. Do teachers justify their choice of learning environment by identifying supportive conditions?

2a. Does the accuracy of choosing a supportive learning environment increase during the PD?

2b. Does the quality of teachers’ justifications of choice increase during the PD?

Sample In sum, 75 teachers attended the PD program, with 62 of them agreeing at the kick-off meeting to participate in our study. Among them were 46 female and 14 male teachers, two did not provide any gender specification. The teachers had an average age of 44.6 years, and 11.5 years of teaching experience, (SD=9.7, range=0.5 to 35). At the end of the PD program, 46 teachers who participated in the last session of the PD, filled in the post evaluation. However, ten of the teachers did not attend the first PD meeting, leading to an overlap between the two measurements of 36 teachers. Instrument Mathematical tasks, matching the content of the PD, with three student solutions, and a conversation between the three students about their solutions were provided to the teachers. The teachers were asked to choose one of three continuations of the conversations, which, to different extents, provided a supportive learning environment by taking implicitly into account the supportive conditions named above. The teachers were further asked to justify their choice. Nine experts in the field confirmed that the tasks, the students’ solutions, and the conversations are well suited to examine teachers’ abilities to foster students’ understanding and that the presented continuations of conversations represent the intended levels of supportive learning environments. Data analyses Research question 1a was answered by relative frequencies of teachers who chose the most supportive learning environment. The open-ended teachers’ justifications of their choice were coded collectively by a team of three researchers resulting in a consensual intercoder agreement to examine 1b. The supportive conditions mentioned above served as deductively derived categories. According to the extent to which the categories were mentioned in teachers’ justifications they were assigned to three different levels of quality. Level 0 comprised justifications that did not mention any of the supportive conditions, and level 1 contained justifications that mentioned at least one of the conditions generally, but without reference to an action of the teacher. Level 2 represents justifications that mention at least one of the supportive conditions with reference to a teacher's action and explain why it is supportive for a student to gain conceptual understanding. Research questions 2a and b were examined by applying Wilcoxon-Tests for related samples comparing the accuracy of choosing the supportive learning environment and the level of justifications between the two measurement instances.

Results Out of the 62 teachers participating in the first measurement, 58% chose the most supportive environment for fostering students’ understanding. In their justifications, 36% of the teachers mentioned supportive conditions with explanation (e.g. “By having student x explain exactly why he used (…), the teacher puts a lot of emphasis on explaining ideas and formulating a justification”). The remaining teachers just mentioned one of the conditions (30%, e.g. Thanks to the material, student y understands.”) or none of them (24%, e.g. “It depends on the child.”). The accuracy of choosing a supportive learning environment increased significantly during the PD (z=-2.32, p<.020) with an almost large effect size (r=.39). While 58% of the teachers in the first measurement chose the most supportive environment, in the second measurement 83% did so. Also, the level of justifications significantly increased between the two measurements (z=-2.91, p<.004) with a large effect size (r=.41). After the PD, a lower number of teachers’ justifications was at level 0 or 1, and a higher number at level 2 (63% after, 36% at the beginning of the PD). Contribution to research and practice Using an approximation of practice approach, firstly, we gained insights into how teachers would foster students’ understanding of mathematics. Secondly, we obtained indications of the improvement of teachers’ expertise in choosing and justifying supportive learning environments during the PD. We thereby enriched the extensive research on how teachers diagnose students’ understanding and narrowed the research gap on how teachers foster students’ understanding. Moreover, we developed an ecologically valid instrument, which sensitively measures teachers’ improvement of expertise to foster students’ understanding, that can be adapted to different mathematical content. Particularly, teachers’ justifications for their choice of learning environments allowed for deeper insights into the improvement of expertise during the PD.

Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A. J., Barbieri, C., Augustine, A., & Koedinger, K. R. (2015). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530–551. Chernikova, O., Heitzmann, N., Fink, M.C. et al. Facilitating Diagnostic Competences in Higher Education—a Meta-Analysis in Medical and Teacher Education. Educ Psychol Rev 32, 157–196 (2020). https://doi.org/10.1007/s10648-019-09492-2 Erath, K., Ingram, J., Moschkovich, J. et al. Designing and enacting instruction that enhances language for mathematics learning: a review of the state of development and research. ZDM Mathematics Education 53, 245–262 (2021). https://doi.org/10.1007/s11858-020-01213-2 European Commission / EACEA / Eurydice, 2021. Teachers in Europe: Careers, Development and Well-being. Eurydice report. Luxembourg: Publications Office of the European Union. Hunt, J. H., & Little, M. E. (2014). Intensifying Interventions for Students by Identifying and Remediating Conceptual Understandings in Mathematics. TEACHING Exceptional Children, 46(6), 187-196. https://doi.org/10.1177/0040059914534617 Jansen, A., Berk, D., & Meikle, E. (2017). Investigating alignment between elementary mathematics teacher education and graduates’ teaching of mathematics for conceptual understanding. Harvard Educational Review, 87(2), 225-250. Krolak-Schwerdt, S., Hörstermann, T., Glock, S., & Böhmer, I. (2018). Teachers' assessments of students' achievements: The ecological validity of studies using case vignettes. Journal of Experimental Education, 86(4), 515–529. https://doi.org/10.1080/00220973.2017.1370686 Mullis, I. V. S., Martin, M. O., Foy, P., Kelly, D. & Fishbein, B. (2020). TIMSS 2019 international results in mathematics and science. Chestnut Hill, MA: TIMSS & PIRLS International Study Center. Boston College. O’Dwyer, L.M., Wang, Y. & Shields, K.A. Teaching for conceptual understanding: A cross-national comparison of the relationship between teachers’ instructional practices and student achievement in mathematics. Large-scale Assess Educ 3, 1 (2015). https://doi.org/10.1186/s40536-014-0011-6 Prediger, S., Dröse, J., Stahnke, R. et al. Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth. J Math Teacher Educ 26, 481–508 (2023). https://doi.org/10.1007/s10857-022-09538-3 Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM, 49(4), 599–611. Siegler, R.S. & Chen, Z. (2008). Differentiation and integration: guiding principles for analyzing cognitive change. Developmental Science, 11(4), 433–453. Tzur, R., Johnson, H. L., Hodkowski, N. M., Nathenson-Mejia, S., Davis, A., & Gardner, A. (2020). Beyond getting answers: Promoting conceptual understanding of multiplication. Australian Primary Mathematics Classroom, 25(4), 35–40.