Which roles of mathematics in physics do physicists employ when instructing high-school teachers?

Research highlights two roles of mathematics in physics: a technical role and a structural role. Learners of physics usually perceive the role of mathematics as technical, e.g., for solving equations. Physicists, by contrast, use mathematics as a structure guiding their reasoning. Which aspects of the structural role can we teach at school, and how? Teacher education plays a crucial role in this endeavor. We study the question empirically in the context of an MSc program for physics teachers. The program's uniqueness is in the equal time it dedicates to both advanced physics courses, taught by active physicists, and to pedagogical content courses, taught by science education researchers. We held semi-structured interviews with the physics instructors, focusing on elementarization processes that they employed during instruction. Elementarization, derived from the model of educational reconstruction, entails focusing on the elements at the foundation of the scientific practice, reducing their complexity, and presenting them in different contexts making them accessible to learners. Thematic analysis of the interviews exposed several elements of the interplay between mathematics and physics that physicists deem important for instruction: conflict as a mechanism for discovery, symmetries constraining the form of physical laws, computational modeling showing coherence between the microscopic and macroscopic scale, and others. We propose how our findings could enrich future curricular design.