The paper reports about a study on the knowledge of prospective mathematics teachers from China (Hong Kong and Mainland China) with regard to argumentation and proof as well as beliefs about using preformal and formal proofs in lessons based on a study design developed by Schwarz (2013). The study pointed out that the prospective teachers were generally strong in algebraic reflections and operations, but they lacked reflections on mathematical argumentation and proofs and their adequate teaching, for example, how to provide adequate feedback about proving approaches. Furthermore, differences concerning the acceptance of preformal proofs between prospective teachers from Hong Kong and Mainland China were found out in this study, which may be related to differences in the mathematics curriculum in these two regions. Implications for teaching proofs in geometry and algebra are discussed.
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This study investigated the conceptual understanding of fraction division of 72 Hong Kong pre-service teachers by asking them to pose a real life story problem. Analyses of the problems posed show that many of these pre-service teachers had incorrectly interpreted the division concept in their problems as sharing rather than measurement. A short intervention shows significant improvement in posing problems in terms of mathematical correctness as related to real life situations. This study particularly shows an increase of the correct use of the measurement concept from the pre-test to the post-test. The study also shows that problem posing could be used to measure conceptual understanding. Additional studies should be conducted to show effects of different teaching interventions regarding improving the conceptual understanding of divisions of fractions.
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