极限概念的源流及其文化性探析

发布时间：2018-02-10 05:43

本文关键词： 极限概念无穷文化导数　出处：《华中师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：极限是数学中由常量到变量、有限到无限、近似到精确思想转变的重要概念,在数学发展史上占有重要地位.极限是大学数学课程中微积分的基础概念.随着新课程改革,导数(一种特殊的极限)的内容已进入高中数学教材.极限作为大学乃至中学数学中一种重要的数学思想方法,在整个数学学习中也是最基本的思想方法之一.因此,笔者想系统地探讨极限概念的发展过程,以利于教师与学生全面了解极限的历史、现状.然而,从数学史的角度进行研究是数学史家已做过的工作,价值不大且枯燥无味.而数学文化是探讨在数学的起源、发展、完善和应用方面对人的观念、思想和思维方式有所影响的内容,是研究极限概念的一个新视角,本文就试着从数学文化的视角探讨极限概念的发展历程,并考察高中数学课程中极限内容的设置,剖析其中蕴涵的文化性,给出教学建议,以期对高中数学教师的教学和学生的学习有所助益.本文主要采用文献法与历史溯源法,先综述了文化、数学文化的含义与特征,极限概念的研究视角分析,这是研究极限概念文化性的理论基础.接着,探讨极限概念在中西方数学中的发展过程,其中蕴涵着极限的文化性,最重要的是展现对人类思想观念的转变的影响与勇于探索进取的精神.在中国数学对极限概念的认识中,主要是追溯古代文化传统下的极限思想,以及对其它数学问题解决的影响.与之相对的是西方数学中极限的发展历程.由于极限概念在西方得到了最终的严密化、形式化的发展,所以从符号化程度高低的角度,我将西方数学中极限的发展分四个阶段进行探究,分别是萌芽阶段、神秘阶段、统一阶段和扩展阶段.最后,从数学课程标准、教材与教学的角度分析教育形态的极限的文化性.通过对极限概念的文化性探析过程,得到的主要结论与建议：1高中数学中的极限内容基本上体现出极限文化发展历程来进行编排的.2高中生虽然对形式化的极限定义理解有难度,但极限作为“导数及其应用”内容的基础性概念,教材中应给出极限的描述性定义.3在数学课程中更主动、有意识地渗透极限的本质---无穷思想.在极限概念之前可适当渗透无穷的思想,如对自然数个数的无穷性的体会,无理数概念中小数无限性的认识.
[Abstract]:Limit is an important concept in mathematics, which changes from constant to variable, from finite to infinite, approximate to precise thought, and occupies an important position in the history of mathematics development. Limit is the basic concept of calculus in the course of university mathematics. The content of derivative (a special limit) has entered high school mathematics textbook. Limit, as an important mathematical thought method in university and middle school mathematics, is also one of the most basic thought methods in the whole mathematics study. The author wants to discuss the development process of the limit concept systematically, so as to help teachers and students to understand the history and present situation of the limit in an all-round way. However, it is the work of mathematics historians to study from the point of view of the history of mathematics. Mathematics culture is a new angle of view to study the concept, thought and mode of thinking of people in the origin, development, perfection and application of mathematics. This paper attempts to explore the development of the concept of limit from the perspective of mathematical culture, and to investigate the setting of the limit content in the mathematics curriculum of senior high school, to analyze the cultural content contained therein, and to give some teaching suggestions. This paper mainly adopts the literature method and the historical traceability method, first summarizes the culture, the mathematics culture meaning and the characteristic, the limit concept research angle of view analysis, the article mainly uses the literature method and the history tracing method, first summarizes the culture, the mathematics culture meaning and the characteristic, the limit concept research angle of view. This is the theoretical basis for studying the cultural nature of the limit concept. Then, the development process of the limit concept in Chinese and Western mathematics is discussed, which implies the cultural nature of the limit. The most important thing is to show the influence on the change of human ideology and the spirit of being enterprising. In the understanding of the concept of limit in Chinese mathematics, it is mainly traced back to the limit thought under the tradition of ancient culture. And its influence on the solution of other mathematical problems. In contrast, it is the development of limit in western mathematics. Because the concept of limit has finally developed in the west, so from the perspective of the degree of symbolization, I divided the development of the limit in western mathematics into four stages, namely, the budding stage, the mysterious stage, the unification stage and the extension stage. Finally, from the mathematics curriculum standard, The cultural nature of the limit of educational form is analyzed from the perspective of teaching materials and teaching. The main conclusions and suggestions obtained are that the limit content in Senior High School Mathematics of 1: 1 basically reflects the development course of limit culture, though it is difficult for the senior middle school students to understand the formalized definition of the limit although it is difficult for them to understand the formalized definition of the limit. However, as the basic concept of "derivative and its application", the descriptive definition of limit should be given in the textbook. The idea of infinity can be properly infiltrated before the concept of limit, such as the understanding of the infinity of the number of natural numbers and the understanding of infinity of the concept of irrational number.
【学位授予单位】：华中师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：G633.6

【共引文献】