数形结合思想在高中数学中的应用
摘要：数形结合思想是中学数学中很重要的一种思想方法，它主要是通过数与形之间的对应和转化来解决数学问题，它包含以形助数和以数解形两个方面。本文从这两个方面论述了数形结合思想在解题中的具体应用，包括：一、构造几何图形辅助代数问题；二、借助代数曲线解决相关问题；三、几何图形转化为代数问题，从而使复杂问题简单化，抽象问题具体化。同时，本文阐述了数形结合应用时应注意的几个问题：图形的存在性，准确性以及等价性。数、形是数学中两大基本概念，可以说全部数学大体上都是围绕这两个基本概念的提炼、演变、发展而展开的。数形结合是根据数学问题的条件与结论之间的内在联系，既分析其代数意义，又揭示其几何直观，使数量关系的精确刻划与空间形式的直观形象巧妙、和谐地结合在一起。数形结合是贯穿中高等数学教学始终的基本思想，同时在高等数学教学中它也有很大的益处。
关键词：数形结合；数学教学；数学思想
The Thinking of Combining Numbers with Shapes and It’s Application in Teaching
Abstract: Thinking of the combination of quantity and graph is a very important way of thinking method in middle school mathematics . It solves mathematical problem mainly through the correspondence and transformation between quantity and graph, which contains two aspects, graph helping quantity and quantity soluting graph. This paper discusses from three aspects the applications of the combination of quantity and graph on mathematical problem solving in middle school. First, construct geometry graph to help algebraic problems. Second, with the help of algebraic curve solute related question. At last, transform geometry graph into algebraic problems. So that we can simplify the complex problems, specify abstract problem .At the same time, this paper elaborates several issues be paid attention to when the combination of quantity and graph is utilized: Graphic existence, accuracy and equivalence.Number and shape are two basic concepts of mathematics，it can be said that the evolving of all the mathematic are generally surrounding the abstraction, evolution and development of the two basic concepts. Combining numbers with shapes is according to the intrinsic link between conditions and conclusions of mathematical problems, it can both analyze the meaning of algebra and reveal the intuitive of geometry which make a artful and harmonious combination between accurate depiction of the number-shape relationship and intuitionistic image of spacial modality. Combining numbers with shapes is basic thinking through mathematics teaching in primary and secondary schools all along, at the same time, it has a great benefit in higher mathematics teaching.
Key words: combining numbers with shapes; mathematics education; mathematics thinking
1 绪论 1
1.1 数形结合思想方法概述 2
1.2 数形结合思想方法历史演进 3
2 数形结合思想在高等数学教学中的应用 5
2.1 数形结合思想在高等数学教学中的应用 5
2.2 数形结合思想在高等数学教学中的应用 8
2.2.1 数形结合思想在高等数学教学中的地位 9
2.2.2 数形结合思想在高等数学教学的应用举例 9
2.3 数形结合思想在高中数学教学中的应用 11
2.3.1数形结合思想在高中数学教学中的地位 11
2.3.2 数形结合思想在高中数学教学的应用举例 13
2.3.3 数形结合思想的课堂灌输 16
3 应用数形结合时，要注意的主要事项 18
3.1 精确作图，避免潦草作图而导出的错误 18
3.2 注意转化过程要等价，避免定义域扩大或缩小 19
3.3 注意图形的存在合理性，不可“无中生有” 20
3.4 注意仔细观察图像，避免漏掉了一些可能的情形 20
4 结束语 22
致谢 24
参考文献 25
参考文献
[1] Angelika Bikner-Ahsbahs,Susanne Prediger. Diversity of theories in mathematics education-How can we deal with it?[J]. ZDM,Volume 38,Number 1,2006.2.
[2] 徐文龙.＂数形结合＂的认知心理[D]. 广西:广西师范大学,2005.
[3] 顾亚萍.数形结合思想方法之教学研究[D]. 南京:南京师范大学,2004.
[4] 朱家生.数学史[M]. 北京:高等教育出版社,2004.
[5] 刘加霞.“数形结合”思想及其在教学中的渗透[J].高等教学数学版.2008,5:44-46.
[6] 徐先荣.谈高等数学数形结合的教学策略[J].大教论坛. 2007.4:75.
[7] 邹良量.数、形结合法在解题中的应用[J].广西轻工业. 2008, 6:125-137.
[8] 周房安.巧用数形结合法解题[J].高中. 2006, 6:21-22.
[9] 刘会芳.浅谈数形结合思想的课堂灌输[J].安庆师范学院报.2003,3:118-119.
[10] 崔秋珍. 数形结合与高等数学问题处理[J]. 洛阳工业高等专科学校学报.1997,4:38-40.
[11] 朱殿利.数形结合法在高等数学中的应用再探析[J].岱宗学刊.1998, 4:23-25.
[12] Ross L.Finney. 托马斯微积分（英文版）[M]. 北京: 高等教育出版社,2004.
http://www.bysj360.com/ http://www.bysj360.com/html/5215.html http://www.bylw520.net
