The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers' perceptions of students' ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From the responses of the participating students (N = 815), it was found that students in schools with a higher level of academic achievement had better algebra test results than did those in schools with a lower level of such achievement. Moreover, it was found that a teacher's perception of a student's ability has a correlation with that student's level of achievement. Based on this finding, an instrument that measures teaching effectiveness is discussed. Last but not least, typical errors in algebra are identified, and some ideas for an instructional design based on these findings are discussed.
[Copyright of International Journal of Mathematical Education in Science and Technology is the property of Taylor & Francis. Full article may be available at the publisher's website:
http://dx.doi.org/10.1080/00207390903236434]

This study undertakes a new attempt in theory and practice in curriculum design in mathematics, which includes an operational model, its theoretical interpretation and experimentation.
The operational model, called Spiral bianshi (literally means variation) Curriculum, is extracted from the use of connecting problem clusters, one of the prominent features in current practice in mathematics instruction in Mainland China, the origin of which can be traced back to the Qingpu experience in Shanghai in the 1980s. Such a long history of practice was conceptualized according to the nature of mathematics and mathematics learning. The Spiral bianshi Curriculum comprises four components: The inductive bianshi which abstract mathematics notions from realistic situations, the deepening bianshi which lead one to new concepts and rules from already established one, the broadening bianshi which connects various concepts which are already acquired and the applicative bianshi which helps learners applying concepts and rules to more realistic situations.
Division of fraction was taken as the topic for experimentation. The corresponding bianshi curriculum was designed and experimented in 14 classes in three Primary schools in Hong Kong. The other 13 classes in three comparable schools were taken as control group.
The students' cognitive learning outcomes and self-efficacy were measured before and after the experimentation by means of a pre-test and a post-test. The students' approach to learning and other affective variables were measured too.
Significant positive results were obtained by the spiral bianshi curriculum. The post-test scores of learning outcomes in the experimental group were significantly higher than those in the control group while all these scores indicated no significant differences in the pre-tests among these two groups. The students in the experimental group reflected that they have clearer relevant concepts than their counterparts. They also understood

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