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.

Mathematical disagreements occur when students challenge each other’s ideas related to a mathematical concept. In this research, we examined Hong Kong and U.S. elementary teachers’ perceptions of mathematical disagreements and their resolutions using a video-stimulated survey. Participants were directed to give particular attention to the mathematics embedded within the disagreement and approaches for resolving the disagreement. Results revealed incomplete knowledge structures for the SELECTed mathematical topic for both Hong Kong and U.S. participants. Differences existed between the two groups regarding the resolution of the disagreement, as Hong Kong participants focused on the content within the resolution process while U.S. participants focused on the form of the resolution process. Both groups found value in mathematical disagreements but for different reasons. Hong Kong participants indicated that the mathematical disagreements supported them in identifying students’ misunderstandings. In contrast, U.S. participants reported that mathematical disagreements helped to identify areas in which students lacked prerequisite knowledge. Implications for future work are provided. Copyright © 2016 International Journal of Education in Mathematics, Science and Technology.