Practical Research on Improving Primary School Mathematics Lessons Based on the Two-Axes Process Model of Understanding Mathematics

作为这项研究的结果，发现了以下几点。第一，通过对数学课程和教材的分析，体现数学理解的“双轴过程模型”，并将其用于小学数学课堂教学的改进。二是可以根据理解数学的“双轴过程模型”，确定并提炼出设计课程的三个原则和三种方法;原则1 [理解数学是一个复杂的动态过程]，原则2 [建立层次和学习阶段]，原则3 [强调个人和社会建设的活动]，方法1 [澄清理解的层次]，方法2 [评估学生的理解水平]和方法3 [体现学习阶段]。第三，这些原则和方法是教师改进数学课的有效指导，通过与教师长期深入的“小学习”，加深学生对数学概念、性质和关系的理解。第四，数学理解的“双轴过程模式”及其原理和方法可以应用于小学职前教师培训和在职教师教育，以改进小学数学课程。

As a result of this research, the followings are found out. First, the "two axes process model" of understanding mathematics can be embodied and used for improving primary school mathematics lessons as the teaching and learning mathematics in a classroom by analyzing mathematics curriculum and teaching materials. Second, we can identify and refine 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 students' level of understanding], and Method 3 [Embodying learning stages]. Third, these principles and methods are used as effective guidelines for a teacher when he/she improving mathematics lessons in order to deepen his/her students' understanding of mathematical concepts, properties and relations through the long-going and intensive "lessen study" with teachers in a school. Fourth, the "two axes process model" of understanding mathematics and its principles and methods can be applicable in pre-service teacher training and in-service teacher education at primary school level for improving their mathematics lessons.