A Study On The Development And Teaching Of 7～(th)-9～(th) Grader's Geometry Reasoning Ability

Geometry enjoys a long history in mathematics education. In different countries, however, there are significant differences in terms of the statuses enjoyed by geometry in the course of mathematics. The major reason lies in the great disputes about the recognition of the values of geometry education, and the issue of geometric reasoning proof is the focus of those disputes.The current geometric course reform in fundamental education, focusing on "proving" and "reasoning", partly diminishes the requirements of proof through geometric formal logic reasoning,. But teachers still feel confused about how to teach, and how far it can be taught. Therefore, curriculum reform calls for pertinent teaching researches based on empirical researches, and teachers in the frontline are eager for the guidance of operable and effective teaching paradigm.On the basis of the cognitive sequence of students' development of geometric reasoning, the present study points out the ways and technical traits of students' geometric reasoning, and the differences of the geometric reasoning ability development in students of different grades, and thereby proposes some thoughts on teaching designs based on different levels so as to mainly focus on the systematic development of students' geometric reasoning. The major findings of the research are as follows:First, the geometric reasoning modes of 7^th - 9^th graders can be summarized as follows: perceptive reasoning, descriptive reasoning, structure-related reasoning, and formal logical reasoning. Students demonstrate distinctive features in different reasoning modes, and significant differences in reasoning abilities of different reasoning modes.Second, the geometric reasoning abilities of 7^th - 9^th graders have a hierarchical nature among the 4 reasoning modes. Specifically speaking, the hierarchical nature can be found in the same graders among different reasoning modes and different graders in the same reasoning mode. Different graders have different features in the process of geometric reasoning.Third, the present study proposes a structure model of geometric reasoning levels according to the hierarchical development of 7^th - 9^th graders' geometric reasoning ability. The model of hierarchical structure of geometric reasoning levels reflects two ways of the development of reasoning ability: one is to develop comprehensive reasoning ability through the hierarchical sequence of perceptive reasoning, descriptive reasoning, structure-related reasoning, and formal logical reasoning; and the other is to develop students' formal logical reasoning ability through the developmental sequence of proof preparation, proof introduction, and proof development. The present study maintains to use the first way to design the teaching system of geometry, and attach importance to the second way at the same time. Effective teaching design should demonstrate the development of comprehensive reasoning ability and promote formal logical reasoning ability at the same time.Fourth, the present study proposes the general framework of the teaching design of geometric reasoning levels. Attention should be paid to the organization of course targets and course contents, problem situation and activity design, procedure assessment and feedback, thereby imbedding the structure model of geometric reasoning levels into the general framework, and conducting course construction through the two dimensions of "vertical organization" and "horizontal organization". On the one hand, it makes the whole framework exemplify the hierarchical nature in the horizontal and vertical organization; on the other hand, it betters planning development directions for the reasoning targets and activities at each level, so as to avoid the randomness that might exist in course organization.The thoughts on teaching design for different reasoning modes are as follows:(1) The teaching design for geometric perceptive reasoning can be organized through the procedure of imagery identification→experimental verification→intuitive perception. (2) The teaching design for geometric descriptive reasoning can be organized through the procedure of concept description→language shift→descriptive reasoning. (3) The teaching design for geometric structure-relating reasoning can be organized through the procedure of information reception→relation change→verification-review. (4) The teaching design for formal logical reasoning can be organized through the procedure of information reception→standard representation→relation transformation and restructuring→formal logical expression. The teaching of each reasoning mode has its distinctive rules.The present study also offers some thoughts on the teaching design based on research projects and experimental research. The teaching design for research projects can be organized through the procedure of problem situation→mathematization of empirical materials→logical organization→application→reflection. The results of the experimental research indicate that using the division of geometric reasoning modes and hierarchical developmental sequence to organize teaching can effectively change teachers' teaching concepts, develop teachers' ability to design, organize, and implement geometry teaching, change teachers and students' living status in the classroom of geometry teaching, and promote the development of students' geometric reasoning ability and upgrade levels..Fifth, the present study advances the following suggestions: (1) develop students' ability of geometric reasoning systematically; (2) think about and supervise the phenomenon of "division" in geometry learning multi-dimensionally; (3) consider the developmental process of perceptive reasoning levels systematically; (4) promote the harmonious development between geometric reasoning activities and expression; (5) emphasize the development of students' geometric structure-related reasoning ability; (6) catch the appropriate timing for the requirement of geometry "formalization"; and (7) emphasize the diversification of teaching contents in higher grades.The innovation of the present study is as follows: (1) based on the cognitive sequence of students' geometric reasoning ability development, it is proposed that the geometric reasoning modes of 7^th - 9^th graders can be divided into perceptive reasoning, descriptive reasoning, structure-related reasoning, and formal logical reasoning; (2) it points out that students show different technical traits in different modes of geometric reasoning, and distinct differences of geometric reasoning ability in different reasoning modes, and it conducts qualitative description and quantitative analysis on students' geometric reasoning characteristics and differences in levels ; (3) it advances the model of hierarchical structure of geometric reasoning levels, and provides thoughts on the design of geometric reasoning levels teaching.