面向教育的三维动态几何关键技术研究
|
摘要
将信息技术深入应用到具体学科,以构建体现学科特点、学生认知特点和教学规律的数字化学习环境、学习工具和学习资源,是改变传统教学方式、实施创新人才培养的一条有效途径,也是教育信息化的趋势与潮流。数学在基础教育阶段处于核心地位,因此,在信息技术与学科整合研究领域,它是最受关注、研究最深入、也是应用最成功的一门学科,其主要标志是动态几何技术的出现和动态几何软件的广泛应用。自第一款动态几何软件《几何画板》问世以来,动态几何的教育价值得到了世界各国教师和教育专家的充分肯定。
几何是中学数学的重点,立体几何则是重点中的难点。与平面动态几何相比,三维动态几何系统的设计与实现难度更大,主要表现在三维动态几何的实现机制、空间图形及空间关系的呈现、用户与空间图形的交互等方面的设计与实现上。目前,国内外已有一百多款动态几何软件,其中绝大部分仅涉及与平面几何相关的知识点,只有少数几款具有立体几何教学功能。并且,这几款三维动态几何软件中较为成熟的都是国外开发的,无法满足我国立体几何教学的实际需求。因此,研究三维动态几何相关的实现机制和核心技术,为我国立体几何教学量身定做一款动态几何系统,具有重要理论意义和现实价值。
本文对三维动态几何的实现机制和核心技术进行了深入系统的研究,并面向我国立体几何教学需求设计开发了一款三维动态几何系统,最后对该系统的教育应用价值进行了分析和讨论。主要研究工作和创新点包括：
(1)研究了动态几何的实现机制,提出了用有向无循环图的数据结构来表示三维空间中的几何图形和几何关系,研究了基于数值计算来求解几何约束问题的实现方法,并基于有向无循环图和数值计算设计了一种混合式几何约束求解算法；基于有向无循环图的数据结构,设计了一套高效更新机制,能在图形运动变化中实时保持几何体之间的固有几何关系不变。
(2)提出了一种满足“数形结合”设计思想的参数化模型,突破了以“父子链表”为核心数据结构的传统动态几何设计框架。基于参数化模型,能通过参数的变化来更加准确地控制图形的运动和变化,并能通过参数在多个图形之间建立联系并统一控制。此外,参数化模型可方便地将几何属性和代数运算联系起来,以呈现复杂多变的动态效果。与传统的基于鼠标的驱动方式相比,参数驱动极大地丰富了动态几何的交互方式,扩展了动态几何的功能。
(3)针对我国立体几何教学的实际需求,设计了三维动态几何软件的系统架构,实现了一款具有立体几何图形绘制、几何关系动态保持、图形拾取、动画、轨迹、跟踪、测量、迭代和代数运算等功能的三维动态几何系统,并面向立体几何教学制作了一系列教学案例资源。
(4)根据立体几何教学的具体需求,将本文研发的三维动态几何系统与国外知名三维动态几何软件《Cabri3D》进行了功能对比,验证了本系统在功能、知识点覆盖,操作舒适性与便捷性等方面的优势。此外,通过教学案例资源对本系统的教育价值进行了分析和论证,表明本文系统能够有效引导学生经历“直观感知——操作确认——思辨论证——度量计算”这一过程来学习和探索立体几何知识,有助于培养学生的空间想象能力和几何直观能力。
本文系统阐述了三维动态系统从设计、实现到教育应用的完整过程,为信息技术与学科整合研究提供了实证性的参考和依据,对开发类似学科工具有一定的借鉴作用,对促进教育信息化向纵深发展具有一定的推动作用。
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.
几何是中学数学的重点,立体几何则是重点中的难点。与平面动态几何相比,三维动态几何系统的设计与实现难度更大,主要表现在三维动态几何的实现机制、空间图形及空间关系的呈现、用户与空间图形的交互等方面的设计与实现上。目前,国内外已有一百多款动态几何软件,其中绝大部分仅涉及与平面几何相关的知识点,只有少数几款具有立体几何教学功能。并且,这几款三维动态几何软件中较为成熟的都是国外开发的,无法满足我国立体几何教学的实际需求。因此,研究三维动态几何相关的实现机制和核心技术,为我国立体几何教学量身定做一款动态几何系统,具有重要理论意义和现实价值。
本文对三维动态几何的实现机制和核心技术进行了深入系统的研究,并面向我国立体几何教学需求设计开发了一款三维动态几何系统,最后对该系统的教育应用价值进行了分析和讨论。主要研究工作和创新点包括：
(1)研究了动态几何的实现机制,提出了用有向无循环图的数据结构来表示三维空间中的几何图形和几何关系,研究了基于数值计算来求解几何约束问题的实现方法,并基于有向无循环图和数值计算设计了一种混合式几何约束求解算法；基于有向无循环图的数据结构,设计了一套高效更新机制,能在图形运动变化中实时保持几何体之间的固有几何关系不变。
(2)提出了一种满足“数形结合”设计思想的参数化模型,突破了以“父子链表”为核心数据结构的传统动态几何设计框架。基于参数化模型,能通过参数的变化来更加准确地控制图形的运动和变化,并能通过参数在多个图形之间建立联系并统一控制。此外,参数化模型可方便地将几何属性和代数运算联系起来,以呈现复杂多变的动态效果。与传统的基于鼠标的驱动方式相比,参数驱动极大地丰富了动态几何的交互方式,扩展了动态几何的功能。
(3)针对我国立体几何教学的实际需求,设计了三维动态几何软件的系统架构,实现了一款具有立体几何图形绘制、几何关系动态保持、图形拾取、动画、轨迹、跟踪、测量、迭代和代数运算等功能的三维动态几何系统,并面向立体几何教学制作了一系列教学案例资源。
(4)根据立体几何教学的具体需求,将本文研发的三维动态几何系统与国外知名三维动态几何软件《Cabri3D》进行了功能对比,验证了本系统在功能、知识点覆盖,操作舒适性与便捷性等方面的优势。此外,通过教学案例资源对本系统的教育价值进行了分析和论证,表明本文系统能够有效引导学生经历“直观感知——操作确认——思辨论证——度量计算”这一过程来学习和探索立体几何知识,有助于培养学生的空间想象能力和几何直观能力。
本文系统阐述了三维动态系统从设计、实现到教育应用的完整过程,为信息技术与学科整合研究提供了实证性的参考和依据,对开发类似学科工具有一定的借鉴作用,对促进教育信息化向纵深发展具有一定的推动作用。
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.
引文
[1]何克抗.对国内外信息技术与课程整合途径与方法的比较分析[J].中国电化教育,2009(9)：7-16.
[2]何克抗.迎接教育信息化发展新阶段的挑战[J].中国电化教育,2006(8)：5-11.
[3]张景中,江春莲,彭翕成.《动态几何》课程的开设在数学教与学中的价值[J].数学教育学报,2007,16(3)：1-5.
[4]张景中,江春莲,彭翕成.基于《超级画板》开设《动态几何》课程的实践与思考[J].数学教育学报,2008,17(5)：1-5.
[5]The Geometer's Sketchpad[EB/OL]. http://www.dynamicgeometry.com.
[6]毛雪琴,何妹珊.动态几何在中学数学教学中的应用及其价值[J].高等函授学报(自然科学版),2008(3)：57-58.
[7]裘红明,吴道春.动态几何软件在几何证明教育中的利与弊[J].广西教育学院学报,2007(6)：97-98.
[8]唐卫宁,齐晓波.动态几何与计算机辅助教学[J].昌吉师专学报,2000(2)：46-47.
[9]沈捷.几何画板的应用价值——21世纪的动态几何[J].二十一世纪教育思想文献,2007(1)：428-430.
[10]张景中.超级画板在高中数学教学中的应用[J].高等函授学报(自然科学版),2008(1)：3-8.
[11]方海光,张景中.教育软件可用性评测研究[J].电化教育研究,2008(2)：63-67.
[12]徐素霞,徐丽芬.数学教学好帮手——Z+Z智能教育平台之超级画板[J].现代教育技术,2006(5)：65-69.
[13]唐彩斌,彭翕成.基于超级画板的小学数学教学实践研究[J].中国电化教育,2009(11)：92-95.
[14]邹军华,黄涛,刘清堂.基于本体构建面向课程管理系统的数学知识元[J].中国电化教育,2010(12)：116-120.
[15]Cabri-Geometry[EB/OL]. http://www.cabri.com.
[16]Cinderella[EB/OL]. http://www.cinderella.de.
[17]江建国,张景中.基于Rete算法的几何自动推理系统[J].四川大学学报(工程科学版),2006(03)：135-138.
[18]张景中,李永彬.几何定理机器证明三十年[J].系统科学与数学,2009(09)：1155-1168.
[19]邹宇,郑焕,张景中.仿射质点几何的可读机器证明[J].计算机应用,2010(07)：1899-1902.
[20]Hauptman H. Enhancement of spatial thinking with Virtual Spaces 1.0[J]. Computers & Education,2010,54(1):123-135.
[21]Martin S, Rubio R. Parallax cues in the design of graphics used in technical education to illustrate complex spatial problems[J]. Computers & Education,2009,53(2):493-503.
[22]张景中,葛强,彭翕成.教育技术研究要深入学科[J].电化教育研究,2010(2)：8-13.
[23]梁维高.高中数学教学中运用《几何画板》的实践与思考[J].中国教育信息化,2010(24)：57-58.
[24]王竹.介绍《几何画板》软件在中学教学中的应用[J].太原教育学院学报,2001(1)：32-37.
[25]王琳.新课程理念下信息技术与初中数学教学整合的实践与思考——例谈使用几何画板的教学感想[J].中国教育信息化,2011(8)：54-56.
[26]李琼.简谈《几何画板》在高中数学教学中的应用[J].山西师范大学学报(自然科学版),2011(S1)：17-19.
[27]何立特.几何画板在探究轨迹问题中的应用[J].河南教育学院学报(自然科学版),2009(3)：65-66.
[28]张景中,彭翕成.三款数学教育软件的比较与设计思想分析[J].中国电化教育,2010(1)：107-113.
[29]张景中.从PPT到动态几何与超级画板[J].高等数学研究,2007(2)：57-61.
[30]张景中.从PPT到动态几何超级画板(续)[J].高等数学研究,2007(3)：42-48.
[31]张景中.超级画板的初步认识和体验[J].数学通讯,2007(1)：7-8.
[32]马复,王鹏远.超级画板与数学新课程[M].北京：科学出版社,2005.
[33]Kortenkamp U. Foundations of Dynamic Geometry[D]. SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH,1999.
[34]Geometry Expressions[EB/OL]. http://www.geometryexpressions.com.
[35]Ye Z, Chou S C, Gao X S. Visually Dynamic Presentation of Proofs in Plane Geometry Part 1. Basic Features and the Manual Input Method[J]. Journal of Automated Reasoning, 2010,45(3):213-241.
[36]Ye Z, Chou S C, Gao X S. Visually Dynamic Presentation of Proofs in Plane Geometry Part 2. Automated Generation of Visually Dynamic Presentations with the Full-Angle Method and the Deductive Database Method[J]. Journal of Automated Reasoning,2010,45(3):243-266.
[37]Ling R, He Y J, Deng K R. Automatic generation of geometric base sequences, Shanghai, 2010[C]. Scientia Sincia.
[38]Kapur D. Geometry theorem proving using Hilbert's Nullstellensatz:Proceeding SYMSAC'86 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation, New York, 1986[C].
[39]Chou S C, Gao X S, Zhang J Z. Automated generation of readable proofs with geometric invariants, Ⅱ. Theorem proving with full-angles[J]. Journal of Automated Reasoning, 1996,17(3):325-347.
[40]周咸青,高小山,张景中.基于前推法的几何搜索系统[J].计算机学报,1996,10(19)：721-727.
[41]Chou S C, Gao X S, Zhang J Z. A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering[J]. Journal of Automated Reasoning,2000,25(3):219,
[42]叶征.平面几何的动态可视证明研究[D].浙江大学,2010.
[43]Van Labeke N. Mutiple External Represntations in Dynamic Geometry:a Domain-Informed Design:AIED2001-10th International Conference on Artificial Intelligence in Education, San antonio, Texas,2001 [C].
[44]Archimedes Geo3D[EB/OL]. http://raumgeometrie.de/drupal.
[45]刘郑,陈矛.中学立体几何智能教育软件的设计与实现[J].中国电化教育,2011(5)：109-115.
[46]Math3.0[EB/OL]. http://www.microsoft.com/education/en-us/teachers/guides/Pages/Mathematics-guide.aspx.
[47]Mathmetica[EB/OL]. http://www.wolfram.com/mathematica.
[48]Maya[EB/OL]. http://www.maya.com.cn.
[49]AutoCAD[EB/OL]. http://usa.autodesk.com/autocad.
[50]欧几里德,兰纪正,朱恩宽.几何原本[M].西安：陕西科学技术出版社,2003.
[51]张景中,彭翕成.动态几何教程[M].北京：科学出版社,2007.
[52]蒋鲲.几何自动作图[M].哈尔滨：哈尔滨工业大学,2007.
[53]Sutherland I E. Sketchpad,a man-machine graphical communication system:Proc.of the Spring Joint Comp. Conference, Detroit, Michigan,1963[C].
[54]Dasgupta B, Mruthyunjaya T S. The Stewart platform manipulator:a review[J]. Mechanism and Machine Theory,2000,35(1):15-40.
[55]高小山,张桂芳,杨伟强.几何约束求解与复杂连杆机构的模拟[J].计算机辅助设计与图形学学报,2003,15(5)：517-522.
[56]刘惠林,张同庄,丁洪生.3-RPR平面并联机构正解的吴方法[J].北京理工大学学报,2000,20(5)：565-569.
[57]高小山,王定康,裘宗燕,等.方程求解与机器证明-基于MMP的问题求解[M].北京科学出版社.
[58]刘芳.对尺规作图教学的三个思考[J].中学数学杂志(初中版),2009(5)：11-13.
[59]乐嗣康,崔雪芳,张奠宙.尺规作图教学的现代意义[J].中学数学月刊,2005(12)：7-9.
[60]蒋鲲,高小山,岳晶岩.参数化模型欠、过和完整约束的判定算法[J].软件学报,2003(12)：2092-2097.
[61]Kondo K. Algebraic method for manipulation of dimensional relationships in geometric models[J]. Computer-Aided Design,1992,24(3):141-147.
[62]高小山,蒋鲲.几何约束求解研究综述[J].计算机辅助设计与图形学学报,2004(4)：385-396.
[63]吴文俊.几何定理证明的基本原理[M].北京：科学出版社,1984.
[64]Podgorelec D. A new constructive approach to constraint-based geometric design[J]. Computer-Aided Design,2002,34(11):769-785.
[65]周济,陈立平,向文,等.几何约束系统推理研究[J].华中理工大学学报,1995(6)：70-74.
[66]罗浩,陈立平,周济.基于归约的几何约束推理研究[J].计算机辅助设计与图形学学报,1997(3)：65-71.
[67]王小平.变量化设计中尺寸约束的倒推理[J].兰州铁道学院学报,1998(1)：96-98.
[68]Gao X S, Chou S C. Solving geometric constraint systems. I. A global propagation approach[J]. Computer-Aided Design,1998,30(1):47-54.
[69]Kramer G A. A geometric constraint engine[J]. Artificial Intelligence,1992,58(1-3):327-360.
[70]刘斌,张林煊,王新龙,等.二维草图变量化设计中的混合法求解[J].机械科学与技术,1999(4)：137-140.
[71]庄晓,周雄辉.三维装配约束求解的解析方法[J].计算机辅助设计与图形学学报,1999,11(6)：497-499.
[72]陈立平,王波兴,彭小波,等.一种面向欠约束几何系统求解的二部图匹配优化处理方法[J].计算机学报,2000,23(5)：523-530.
[73]Winroth H. Dynamic Projective Geometry[D]. STOCKHOLMS UNIVERSITET,1999.
[74]Atiyah M. Mathematics in the 20th Century[J]. American Mathematical Monthly, 2001,108(7):654-666.
[75]Roanes-Lozano E, Roanes-Macias E, Villar-Mena M. A bridge between dynamic geometry and computer algebra[J]. Mathematical and Computer Modelling,2003,37(9-10):1005-1028.
[76]Roanes-Lozano E, van Labeke N, Roanes-Macias E. Connecting the 3D DGS Calques3D with the CAS Maple[J]. Mathematics and Computers in Simulation,2010,80(6):1153-1176.
[77]GeoGebra[EB/OL]. http://www.geogebra.org.
[78]Microsoft Visual Studio[EB/OL]. http://en.wikipedia.org/wiki/Microsoft_Visual_Studio.
[79]C++[EB/OL]. http://en.wikipedia.org/wiki/C%2B%2B.
[80]OpenGL[EB/OL]. http://www.opengl.org.
[81]MFC[EB/OL]. http://en.wikipedia.org/wiki/MFC.
[82]QT[EB/OL]. http://qt.nokia.com/products.
[83]舍利.计算机图形学[M].北京：人民邮电出版社,2007.
[84]陈家新,周纬杰.动态光线跟踪算法与实现[J].系统仿真学报,2001(S2)：136-137.
[85]王文玺,肖世德,孟文,等.一种基于八叉树空间剖分技术的光线跟踪算法[J].计算机应用,2008,28(3)：656-658.
[86]张全贵,王普,闫健卓,等.基于光线跟踪方法的全局光照研究[J].计算机科学,2010,37(4)：27-30.
[87]林素仙,谭文安.基于软件生命周期模型的网络课件的开发方法与实现过程[J].计算机时代,2007(9)：25-27.
[88]彭翕成,江春莲.动态性是信息技术辅助数学教学的灵魂[J].数学通讯：教师阅读,2008(10)：10-12.
[89]弗罗伊登塔尔.作为教育任务的数学[M].上海：上海教育出版社,1995.
[90]李志厚,王嘉毅.教学知识工程化：一个理论见之于实践的研究领域[J].教育理论与实践,2009(10)：35-38.
[91]Lindquist, Montgomery M, Shulte. Learning and Teaching Geometry, K-12.1987 Yearbook[M]. National Council of Teachers of Mathematics,1987.
[92]Van de Walle J A. Elementary and Middle School Mathematics:Teaching Developmentally[M]. Boston:Allyn&Bacon,2003.
[93]霍华德·加德纳,沈致隆.关于多元智能的对话[J].江苏教育研究：C版,2009(3)：37-40.
[94]Evans C W, Leija A, Falkner T R. Math Links:Teaching the NCTM 2000 Standards Through Children's Literature [M]. USA:Libraries Unlimited,2001.
[95]孙晓天.数学课程发展的国际视野[M].北京：高等教育出版社,2003.
[96]王奋平.高师数学专业开设《几何画板》课的必要性研究[J].琼州学院学报,2011,18(5)：56-58.
[97]李传中,张景中.智能知识平台的构想及其实现[J].世界科技研究与发展,2001(6)：1-6.
[98]张景中,李传中.自动推理与教育软件智能平台[J].广州大学学报：综合版,2001,15(2)：1-6.
[99]赵生初,杜薇薇,卢秀敏.《几何画板》在初中数学教学中的实践与探索[J].中国电化教育,2012(3)：104-107.
[100]张景中.超级画板自由行[M].北京：科学出版社,2006.
[2]何克抗.迎接教育信息化发展新阶段的挑战[J].中国电化教育,2006(8)：5-11.
[3]张景中,江春莲,彭翕成.《动态几何》课程的开设在数学教与学中的价值[J].数学教育学报,2007,16(3)：1-5.
[4]张景中,江春莲,彭翕成.基于《超级画板》开设《动态几何》课程的实践与思考[J].数学教育学报,2008,17(5)：1-5.
[5]The Geometer's Sketchpad[EB/OL]. http://www.dynamicgeometry.com.
[6]毛雪琴,何妹珊.动态几何在中学数学教学中的应用及其价值[J].高等函授学报(自然科学版),2008(3)：57-58.
[7]裘红明,吴道春.动态几何软件在几何证明教育中的利与弊[J].广西教育学院学报,2007(6)：97-98.
[8]唐卫宁,齐晓波.动态几何与计算机辅助教学[J].昌吉师专学报,2000(2)：46-47.
[9]沈捷.几何画板的应用价值——21世纪的动态几何[J].二十一世纪教育思想文献,2007(1)：428-430.
[10]张景中.超级画板在高中数学教学中的应用[J].高等函授学报(自然科学版),2008(1)：3-8.
[11]方海光,张景中.教育软件可用性评测研究[J].电化教育研究,2008(2)：63-67.
[12]徐素霞,徐丽芬.数学教学好帮手——Z+Z智能教育平台之超级画板[J].现代教育技术,2006(5)：65-69.
[13]唐彩斌,彭翕成.基于超级画板的小学数学教学实践研究[J].中国电化教育,2009(11)：92-95.
[14]邹军华,黄涛,刘清堂.基于本体构建面向课程管理系统的数学知识元[J].中国电化教育,2010(12)：116-120.
[15]Cabri-Geometry[EB/OL]. http://www.cabri.com.
[16]Cinderella[EB/OL]. http://www.cinderella.de.
[17]江建国,张景中.基于Rete算法的几何自动推理系统[J].四川大学学报(工程科学版),2006(03)：135-138.
[18]张景中,李永彬.几何定理机器证明三十年[J].系统科学与数学,2009(09)：1155-1168.
[19]邹宇,郑焕,张景中.仿射质点几何的可读机器证明[J].计算机应用,2010(07)：1899-1902.
[20]Hauptman H. Enhancement of spatial thinking with Virtual Spaces 1.0[J]. Computers & Education,2010,54(1):123-135.
[21]Martin S, Rubio R. Parallax cues in the design of graphics used in technical education to illustrate complex spatial problems[J]. Computers & Education,2009,53(2):493-503.
[22]张景中,葛强,彭翕成.教育技术研究要深入学科[J].电化教育研究,2010(2)：8-13.
[23]梁维高.高中数学教学中运用《几何画板》的实践与思考[J].中国教育信息化,2010(24)：57-58.
[24]王竹.介绍《几何画板》软件在中学教学中的应用[J].太原教育学院学报,2001(1)：32-37.
[25]王琳.新课程理念下信息技术与初中数学教学整合的实践与思考——例谈使用几何画板的教学感想[J].中国教育信息化,2011(8)：54-56.
[26]李琼.简谈《几何画板》在高中数学教学中的应用[J].山西师范大学学报(自然科学版),2011(S1)：17-19.
[27]何立特.几何画板在探究轨迹问题中的应用[J].河南教育学院学报(自然科学版),2009(3)：65-66.
[28]张景中,彭翕成.三款数学教育软件的比较与设计思想分析[J].中国电化教育,2010(1)：107-113.
[29]张景中.从PPT到动态几何与超级画板[J].高等数学研究,2007(2)：57-61.
[30]张景中.从PPT到动态几何超级画板(续)[J].高等数学研究,2007(3)：42-48.
[31]张景中.超级画板的初步认识和体验[J].数学通讯,2007(1)：7-8.
[32]马复,王鹏远.超级画板与数学新课程[M].北京：科学出版社,2005.
[33]Kortenkamp U. Foundations of Dynamic Geometry[D]. SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH,1999.
[34]Geometry Expressions[EB/OL]. http://www.geometryexpressions.com.
[35]Ye Z, Chou S C, Gao X S. Visually Dynamic Presentation of Proofs in Plane Geometry Part 1. Basic Features and the Manual Input Method[J]. Journal of Automated Reasoning, 2010,45(3):213-241.
[36]Ye Z, Chou S C, Gao X S. Visually Dynamic Presentation of Proofs in Plane Geometry Part 2. Automated Generation of Visually Dynamic Presentations with the Full-Angle Method and the Deductive Database Method[J]. Journal of Automated Reasoning,2010,45(3):243-266.
[37]Ling R, He Y J, Deng K R. Automatic generation of geometric base sequences, Shanghai, 2010[C]. Scientia Sincia.
[38]Kapur D. Geometry theorem proving using Hilbert's Nullstellensatz:Proceeding SYMSAC'86 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation, New York, 1986[C].
[39]Chou S C, Gao X S, Zhang J Z. Automated generation of readable proofs with geometric invariants, Ⅱ. Theorem proving with full-angles[J]. Journal of Automated Reasoning, 1996,17(3):325-347.
[40]周咸青,高小山,张景中.基于前推法的几何搜索系统[J].计算机学报,1996,10(19)：721-727.
[41]Chou S C, Gao X S, Zhang J Z. A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering[J]. Journal of Automated Reasoning,2000,25(3):219,
[42]叶征.平面几何的动态可视证明研究[D].浙江大学,2010.
[43]Van Labeke N. Mutiple External Represntations in Dynamic Geometry:a Domain-Informed Design:AIED2001-10th International Conference on Artificial Intelligence in Education, San antonio, Texas,2001 [C].
[44]Archimedes Geo3D[EB/OL]. http://raumgeometrie.de/drupal.
[45]刘郑,陈矛.中学立体几何智能教育软件的设计与实现[J].中国电化教育,2011(5)：109-115.
[46]Math3.0[EB/OL]. http://www.microsoft.com/education/en-us/teachers/guides/Pages/Mathematics-guide.aspx.
[47]Mathmetica[EB/OL]. http://www.wolfram.com/mathematica.
[48]Maya[EB/OL]. http://www.maya.com.cn.
[49]AutoCAD[EB/OL]. http://usa.autodesk.com/autocad.
[50]欧几里德,兰纪正,朱恩宽.几何原本[M].西安：陕西科学技术出版社,2003.
[51]张景中,彭翕成.动态几何教程[M].北京：科学出版社,2007.
[52]蒋鲲.几何自动作图[M].哈尔滨：哈尔滨工业大学,2007.
[53]Sutherland I E. Sketchpad,a man-machine graphical communication system:Proc.of the Spring Joint Comp. Conference, Detroit, Michigan,1963[C].
[54]Dasgupta B, Mruthyunjaya T S. The Stewart platform manipulator:a review[J]. Mechanism and Machine Theory,2000,35(1):15-40.
[55]高小山,张桂芳,杨伟强.几何约束求解与复杂连杆机构的模拟[J].计算机辅助设计与图形学学报,2003,15(5)：517-522.
[56]刘惠林,张同庄,丁洪生.3-RPR平面并联机构正解的吴方法[J].北京理工大学学报,2000,20(5)：565-569.
[57]高小山,王定康,裘宗燕,等.方程求解与机器证明-基于MMP的问题求解[M].北京科学出版社.
[58]刘芳.对尺规作图教学的三个思考[J].中学数学杂志(初中版),2009(5)：11-13.
[59]乐嗣康,崔雪芳,张奠宙.尺规作图教学的现代意义[J].中学数学月刊,2005(12)：7-9.
[60]蒋鲲,高小山,岳晶岩.参数化模型欠、过和完整约束的判定算法[J].软件学报,2003(12)：2092-2097.
[61]Kondo K. Algebraic method for manipulation of dimensional relationships in geometric models[J]. Computer-Aided Design,1992,24(3):141-147.
[62]高小山,蒋鲲.几何约束求解研究综述[J].计算机辅助设计与图形学学报,2004(4)：385-396.
[63]吴文俊.几何定理证明的基本原理[M].北京：科学出版社,1984.
[64]Podgorelec D. A new constructive approach to constraint-based geometric design[J]. Computer-Aided Design,2002,34(11):769-785.
[65]周济,陈立平,向文,等.几何约束系统推理研究[J].华中理工大学学报,1995(6)：70-74.
[66]罗浩,陈立平,周济.基于归约的几何约束推理研究[J].计算机辅助设计与图形学学报,1997(3)：65-71.
[67]王小平.变量化设计中尺寸约束的倒推理[J].兰州铁道学院学报,1998(1)：96-98.
[68]Gao X S, Chou S C. Solving geometric constraint systems. I. A global propagation approach[J]. Computer-Aided Design,1998,30(1):47-54.
[69]Kramer G A. A geometric constraint engine[J]. Artificial Intelligence,1992,58(1-3):327-360.
[70]刘斌,张林煊,王新龙,等.二维草图变量化设计中的混合法求解[J].机械科学与技术,1999(4)：137-140.
[71]庄晓,周雄辉.三维装配约束求解的解析方法[J].计算机辅助设计与图形学学报,1999,11(6)：497-499.
[72]陈立平,王波兴,彭小波,等.一种面向欠约束几何系统求解的二部图匹配优化处理方法[J].计算机学报,2000,23(5)：523-530.
[73]Winroth H. Dynamic Projective Geometry[D]. STOCKHOLMS UNIVERSITET,1999.
[74]Atiyah M. Mathematics in the 20th Century[J]. American Mathematical Monthly, 2001,108(7):654-666.
[75]Roanes-Lozano E, Roanes-Macias E, Villar-Mena M. A bridge between dynamic geometry and computer algebra[J]. Mathematical and Computer Modelling,2003,37(9-10):1005-1028.
[76]Roanes-Lozano E, van Labeke N, Roanes-Macias E. Connecting the 3D DGS Calques3D with the CAS Maple[J]. Mathematics and Computers in Simulation,2010,80(6):1153-1176.
[77]GeoGebra[EB/OL]. http://www.geogebra.org.
[78]Microsoft Visual Studio[EB/OL]. http://en.wikipedia.org/wiki/Microsoft_Visual_Studio.
[79]C++[EB/OL]. http://en.wikipedia.org/wiki/C%2B%2B.
[80]OpenGL[EB/OL]. http://www.opengl.org.
[81]MFC[EB/OL]. http://en.wikipedia.org/wiki/MFC.
[82]QT[EB/OL]. http://qt.nokia.com/products.
[83]舍利.计算机图形学[M].北京：人民邮电出版社,2007.
[84]陈家新,周纬杰.动态光线跟踪算法与实现[J].系统仿真学报,2001(S2)：136-137.
[85]王文玺,肖世德,孟文,等.一种基于八叉树空间剖分技术的光线跟踪算法[J].计算机应用,2008,28(3)：656-658.
[86]张全贵,王普,闫健卓,等.基于光线跟踪方法的全局光照研究[J].计算机科学,2010,37(4)：27-30.
[87]林素仙,谭文安.基于软件生命周期模型的网络课件的开发方法与实现过程[J].计算机时代,2007(9)：25-27.
[88]彭翕成,江春莲.动态性是信息技术辅助数学教学的灵魂[J].数学通讯：教师阅读,2008(10)：10-12.
[89]弗罗伊登塔尔.作为教育任务的数学[M].上海：上海教育出版社,1995.
[90]李志厚,王嘉毅.教学知识工程化：一个理论见之于实践的研究领域[J].教育理论与实践,2009(10)：35-38.
[91]Lindquist, Montgomery M, Shulte. Learning and Teaching Geometry, K-12.1987 Yearbook[M]. National Council of Teachers of Mathematics,1987.
[92]Van de Walle J A. Elementary and Middle School Mathematics:Teaching Developmentally[M]. Boston:Allyn&Bacon,2003.
[93]霍华德·加德纳,沈致隆.关于多元智能的对话[J].江苏教育研究：C版,2009(3)：37-40.
[94]Evans C W, Leija A, Falkner T R. Math Links:Teaching the NCTM 2000 Standards Through Children's Literature [M]. USA:Libraries Unlimited,2001.
[95]孙晓天.数学课程发展的国际视野[M].北京：高等教育出版社,2003.
[96]王奋平.高师数学专业开设《几何画板》课的必要性研究[J].琼州学院学报,2011,18(5)：56-58.
[97]李传中,张景中.智能知识平台的构想及其实现[J].世界科技研究与发展,2001(6)：1-6.
[98]张景中,李传中.自动推理与教育软件智能平台[J].广州大学学报：综合版,2001,15(2)：1-6.
[99]赵生初,杜薇薇,卢秀敏.《几何画板》在初中数学教学中的实践与探索[J].中国电化教育,2012(3)：104-107.
[100]张景中.超级画板自由行[M].北京：科学出版社,2006.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |