Research on Designing Lessons Based on the "Two-Axes Process Model" of Understanding Mathematics

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As a result of this research, the followings are found out. First, the "two-axes process model" of understanding mathematics based on constructivist perspective can be complemented by sociocultural and interactionism perspectives. Second, the two axes of hierarchical levels and learning stages can be embodied by analyzing mathematics curriculum and teaching materials. Third, the model can be used as a framework for the teacher who are willing to encourage students to understand mathematics deeply and meaningfully in the dialectic process of students' individual and social construction of mathematics in a classroom. Fourth, we can identify the three principles and three methods for designing lessons based on the "two-axes process model" of understanding mathematics ; Principle 1 [Understanding mathematics as a complex and dynamic process], Principle 2 [Setting up hierarchical levels and learning stages], Principle 3 [Emphasizing both activities of individual and social construction], Method 1 [Clarifying hierarchical levels of understanding], Method 2 [Assessing student's level of understanding], Method 3 [Embodying learning stages].