Chapter 3: Mathematics Teaching in a Chinese Classroom : A Hybrid-Model Analysis of Opportunities for Students' Learning

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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.