Research Of Education-oriented Key Technologies On Three-dimensional Dynamic Geometry

The application of information technology to specific subjects, to construct the digital learning environment, learning tools and learning resources with subject characteristics, students’cognitive characteristics and teaching law, is a effective way of changing the traditional teaching and learning methods, and also a way for training the innovative personnel and promoting the education informationization. Mathematics is in the key position of the basic education, so in the research areas of the integration of information technology and subjects, it is the most talked about, most in-depth research and most successful one, and the main sign is the dynamic geometry system(DGS). The first DGS is The Geometer’s Sketchpad appeared in1987, up till now, there are dozens of DGS all over the world, the international education community has reached a consensus about the positive impact of dynamic geometry(DG) on education and the DGS has proven to be an excellent resource for teachers and students.Geometry education, including the plane geometry and space geometry, is one of the difficult and key content of the mathematics curriculum in the secondary school. However, most current DGS mainly target the teaching and learning of plane geometry. Currently, there are many two-dimensional DGS which reached a high level of development, while the development of three-dimensional(3D) DGS is at its beginning stage and there are only a few3D DGS existed for solid geometry education, especially in China. Compared with two-dimensional (2D) DGS, the development of a3D DGS is much more difficult and complex in the design and implementation, for example, the core dynamic geometry mechanism, the rendering of space objects and the interaction with geometric objects are all difficult problems for3D DGS. Therefore, how to design and implement a3D DGS is a very meaningful thing and is the main content of this paper.In this paper, first we researched the implementation mechanism and core technologies of3D DGS, then designed a3D DGS for space geometry, at last we discussed the value of geometry education with this system. The main research contents include following topics:(1) This paper researched the realization mechanism of3D DGS, and proposed a directed acyclic graph(DAG) to represent the three-dimensional space geometry and geometric relationship. To solve the geometric constraint problem in geometry education, a geometric constraint solving algorithm based on numerical methods and DAG was proposed. Based on the data structure of DAG, we designed a real-time efficient updating mechanism for keeping the constraint relation unchanged.(2) This paper proposed a parameterized model to solve the problem that the algebra related functions were missed in many3D DGS, and implemented the parameterized model in the3D DGS. With this model, the movement of geometric objects could be manipulated automatically and precisely, the synchronous changes of several geometry objects could be obtained, and simple algebra calculation could be realized, as a result, both the algebra and geometry were embodied in the DGS and also the algebra and geometry were connected closely.(3) This paper proposed a requirement analysis of the system and presented the system architecture. Then, we designed and implemented a3D DGS. The results showed that geometric objects could be plotted and manipulated easily in three dimensions, the common functions for a DGS such as animation, transformation, locus, iteration and measurement were realized for the spatial objects, and the updating process was very smooth, which means that the system was useful in space geometry class.(4) According to the specific needs of the space geometry teaching, we compared the functionality with several similar software to verify the advantages of our3D DGS in the areas of knowledge covering, operation comfort and convenience. In addition, through the analysis and argumentation of the teaching cases, it demonstrated the3D DGS could effectively guide the students through the process "intuitive perception-operation confirmation-speculative demonstration-metric calculation", thus it could be used to train the space imagination and geometric intuitive abilities for students.This paper formulated the complete process of design, implementation and application of3D DGS, which provided a empirical reference and basis for integration of subjects with information technology. It could also help the teaching of space geometry by visual and dynamic illustration of abstract geometric concepts and spatial geometry shapes and could promote the development of educational informationization.