Research on the validity and Effectiveness of Two-Axes Process Model of Understanding Mathematics

作为本研究的结果，我们发现了以下几点。首先，通过对小学数学课的个案研究，验证了数学理解“双轴过程模型”的有效性和有效性。第二，我们或许能够将这样一种使儿童深入而有意义地理解数学的数学教与学描述为儿童在课堂上个人和社会建构的辩证过程。为了实现这样的数学课堂，建议教师根据“双轴过程模型”制定数学教与学的计划，并结合目标和儿童的实际情况，用专题教材加以体现。教师还应该通过与儿童和儿童之间的讨论，在个人和社会建构的辩证过程中发挥促进者的作用。教师在数学教与学过程中的角色有两个重要特征。一是与儿童的个体建构有关，是创设儿童能够有自己的学习任务的问题情境，鼓励他们独立地拥有各种数学思想和方法。另一个是与儿童的社会建构有关的，鼓励和允许他们在课堂上以社交的方式解释、分享和讨论他们的各种数学思想和方法。

As a result of this research, we find out the followings. First, we can exemplify the validity and effectiveness of the "two-axes process model" of understanding mathematics by the case study of the elementary school mathematics class. Second, we might be able to characterize such a teaching and learning of mathematics that enables children to understand mathematics deeply and in their meaningful way as the dialectic process of children's individual and social constructions in their classroom. In order to realize such mathematics classroom, it is suggested that a teacher should make a plan of teaching and learning mathematics in the light of "two-axes process model", and embody it with teaching materials of a topic in due consideration both of the objectives and the actual state of children. The teacher also should play a role as a facilitator for the dialectic process of individual and social constructions through a discussion with children and among them.There are two important features of teacher's role in the process of teaching and learning mathematics. The one is related to children's individual construction and it is to set the problematic situation where children are able to have their own learning tasks and encourage them to have various mathematical ideas and ways individually. The other is related to children's social construction and it is to encourage and allow them to explain, share and discuss their various mathematical ideas and ways socially in the classroom.