This study investigates mathematics teaching in Hong Kong and Shanghai by considering the ways of handling Pythagoras' Theorem at eighth grade from the perspective of variation. It has three aims. Firstly, it attempts to identify the characteristics of the ways of handling the same topic in the two cities. Secondly, it tries to identify patterns of handling the topic in Hong Kong and Shanghai through making a comparison between different ways of handling the same topic. Thirdly, it seeks to interpret the findings from a cultural perspective and contribute to solving the paradox of Chinese mathematics teaching.

The Hong Kong data includes the following: eight videotaped lessons with the same topic of Pythagoras' Theorem and supplementary documents such as teachers' questionnaires, samples of text pages, worksheets, and other materials used in the lessons. All of the Hong Kong data were taken from the Third International Mathematics and Science Study Repeat Video Study. The Shanghai data consists of eleven videotaped lessons from different schools in which Pythagoras' Theorem was taught, with supplementary documents, matching those used in Hong Kong. Mainly based on the videotaped lessons and supplementary documents, the inter-and intra-city comparisons were made from the perspective of variation in order to identify the characteristics of the different ways of handling the same topic.

The following similarities have been observed: (1) the focuses and structures of the lessons are quite similar; (2) The teachers tend to emphasize exploration of the theorem; (3) the teachers seein to emphasize exercises with variation; and (4) the teachers control the classrooin activities but they still encourage students to engage well in the process of learning. However, some differences have also been identified. Firstly, the approaches to encouraging discovery of the theorem vary in that, the Hong Kong teachers prefer to employ physical manipulation while their Shanghai counterparts