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

通过这项研究，我们发现了以下几点。首先，我们可以通过对小学数学课堂的个案研究来验证理解数学的“双轴过程模型”的有效性。其次，我们可以把这种让孩子们深刻地、有意义地理解数学的数学教学描述为孩子们在课堂上的个体建构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”制定数学的教与学计划，并结合幼儿的目标和实际情况，用某一主题的教材来体现。教师还应通过与儿童以及儿童之间的讨论，在个体和社会建构的辩证过程中发挥推动者的作用。在数学教与学的过程中，教师的角色有两个重要特征。一种与儿童的个体建构有关，即设置问题情境，使儿童能够有自己的学习任务，鼓励他们个体地产生各种数学观念和方法。另一个与儿童的社会建构有关，鼓励和允许他们在课堂上社会性地解释、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.