Felix Klein's notion of double discontinuity between university mathematics and secondary school mathematics has persisted in mathematics teacher education. Situating this study in Hong Kong, we investigated three prospective secondary teachers' experiences in higher education and secondary mathematics classrooms, including their figured worlds, identities, and mathematical knowledge constructed through those experiences. Our analysis revealed their contrasting experiences in these two contexts, adding to our understanding of the phenomenon of double discontinuity in terms of not only how it is relevant to Hong Kong but also how it may manifest in the specific cultural context. Moreover, we found coherency and continuity during the prospective teachers' secondary-tertiary transition; their learning experiences in both environments served as critical resources upon which they reflected to envision their future mathematics teaching. We conclude with implications and suggestions for mathematics teacher education to support prospective teachers' (and their future students') transitions between high school and university mathematics. Copyright © 2022 Springer.

Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers' professional mathematical modeling competencies in three different regions—Germany, Mainland China, and Hong Kong—where educational and cultural traditions differ, including the role of mathematical modeling. In total, 232 pre-service mathematics teachers from the three regions completed a modeling task covering mathematics content knowledge (MCK) of modeling and mathematical pedagogical content knowledge (MPCK) of modeling. The results show that pre-service teachers from Germany demonstrated the strongest MCK and MPCK of mathematical modeling; by contrast, pre-service mathematics teachers from Mainland China and Hong Kong demonstrated relatively weaker MCK and MPCK of mathematical modeling. MCK and MPCK of mathematical modeling were also found to be unevenly developed at different competence levels for the three regions. These differences may be attributed to the history of mathematical modeling in mathematics curricula, teacher education, and teaching culture in these three regions. Copyright © 2021 The Author(s).