A Study On The Methodology And Strategy Facilitating Middle School Students’ Mathematical Understanding

It has always been practiced in mathematics teaching that the majority of teachers design their teaching according to their accumulated teaching experience, which makes it difficult to grasp the whole knowledge, methods and mathematics ideologies as well as students’ actual situation. Moreover, this practice asks students to do a good number of exercises to deepen their understanding, which makes teaching an exhausting job and learning a tiring process. Hence, this paper aims to solve the above problem to make teaching design more scientific, more complete and briefer and to help to improve the young teachers’ teaching.This paper starts with the literature review concerning the current studies both at home and abroad as well as the theorical bases of mathematical understanding. The foreign scholars conduct their research from the following three perspectives: definitions, levels and process of understanding. Understanding is meaning construction, information-infusion cognition and information process, which consists of a variety of levels. Meanwhile, some Chinese scholars also explore this field, for example, understanding in learning mathematical concepts by Li Shanliang, CPFS structure-understanding model by Yu Ping,6-dimensional understanding model by Lv Linhai and so on, which facilitate the teaching and learning research on mathematical understanding.The author of this paper thinks that mathematical understanding is to establish psychological meaning and association for mathematical objects. Psychological meaning can be obtained through internal representation and the link between internal and external representations can be got by conversion. And association is to get a complete and all-around meaning via infusing the object into the cognitive structure.Incorporating the strong points of different schools concerned and combining mathematical features with students’ cognitive characteristics, this paper, based on cognition and construction,Taxonomy of Bloom’s teaching objectives, proposes the four understanding levels of junior middle school students: experience-formality-structure-conception, and the six under－standing teps: perception-understanding-operation-organization-exploration, based on the Duality and APOS. In addition, understanding strategies are assumed as intuitive understanding,operational understanding, analysing understanding, explanatory understanding, summarizing understanding, organizing understanding, reflective understanding and exploratory one in terms of understanding process and methods.Furthermore, the objectives and strategies are listed in the2-dimensional tables in terms of content and process.Some case studies are used in this paper to illustrate on the application of mathematical understanding strategies to the2-dimensional tables to design teaching with the expectation to generate more public discussion.