Narrow Results

This chapter consists of two parts. First, based on a longitudinal experiment aiming at effective mathematics teaching and learning in China, a theory, called teaching with variation, is summarized by adopting two concepts of variation, i.e. conceptual variation and procedural variation. Secondly, it is demonstrated that the Chinese theory is strongly supported by several well-known Western theories of learning and teaching. Particularly, the Marton's theory offers an epistemological foundation and conceptual support for the Chinese theory. Moreover, the authors argue that the teaching with variation characterizes the mathematics teaching in China and by adopting teaching with variation, even with large classes, students still can actively involve themselves in the process of learning and achieve excellent results.

This chapter presents the findings of a study on the mathematics classrooms in Hong Kong and Shanghai in an attempt to explore a so-called Paradox of Chinese Learners. Eight Hong Kong lessons and eleven Shanghai lessons in which Pythagoras' theorem was taught were examined in great detail from the perspective of variation. It was found that in both cities the teachers (1) tended to emphasize exploration of the theorem, (2) seemed to emphasize exercises with variation, and (3) controlled the classroom activities but they still encouraged students to engage well in the process of learning. The findings suggest that good teaching seems to take place in the Chinese classrooms despite their large class size, and further challenge the very idea of the paradox of Chinese learners. Furthermore, they demonstrate that exploration of Chinese mathematics pedagogy should be done with caution because of intra-cultural differences.

This chapter analyzes mathematics teaching in Chinese classrooms by articulating opportunities for learning (cognitive change) created for students. A hybrid model consisting of a tripartite theoretical lens is presented and used: Reflection on Activity-Effect Relationship (Ref*AER), Hypothetical Learning Trajectory (HLT), and Teaching with Bridging and Variation. The analysis examines how teachers use the latter two strategies to (a) tie goals for students' learning with their extant knowledge, (b) create a need for exploring the new mathematics, and (c) provide situations for action and reflection that promote achieving the learning objects. This analysis inspires a three-tiered model for examining and guiding mathematics instruction. At a macro tier, HLT guides setting learning goals, designing mental activity sequences, and articulating cognitive reorganization processes. At an intermediate tier, teaching with bridging and variation provides tools for the deliberate design of problem situations and tasks within a specific HLT to create opportunities for the intended reorganization and thus achieving goals for students' learning—interrelated conceptual and procedural understandings. At a micro tier, Ref*AER provides a lens to link situations/tasks with changes in students' conceptions.

Dramatic changes in mathematics education in Chinese mainland have taken place since the new mathematics curriculum standard was implemented in 2001. What new features do exemplary lessons appear under the context of the curriculum reform? This chapter will answer this question by presenting a case study of 13 elementary mathematics lessons that were evaluated as excellent exemplary lessons by mathematics educators in China (mainland). It finds that, consistent with the ideas advocated by the new curriculum, the selected lessons demonstrated the features of emphasizing on student's overall development, connecting mathematics to real-life, providing students the opportunities for inquiring and collaborating, and teachers' exploiting various resources for teaching. Meanwhile, the selected lessons also shared other common features in the lesson structure, interaction between the teacher and students, and classroom discourse. The results reveal that the exemplary lessons have practiced the advocated ideas of the current reform, while they also embodied some elements that might be the stable characteristics of Chinese mathematics education.