高中数学竞赛中的函数方程问题研究
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摘要
函数方程是一个历史悠久、内容丰富、应用极其广泛的数学分支。20世纪以来函数方程常常出现在国际数学奥林匹克竞赛中,成为数学竞赛的一个重要组成部分,函数方程问题以其求解的技巧丰富和创新越来越受到各类数学竞赛命题者的青睐,并引起国内外数学教育界的广泛关注。
本研究采用文献分析法,以波利亚的怎样解题为理论依据,首先系统地介绍了国内外数学竞赛发展的现状及比赛层次,并对数学竞赛中的函数方程问题的定义进行界定,指出函数方程问题在现有的数学竞赛中日趋受到重视。其次,对国内外数学竞赛中的函数方程问题进行汇编、分析、整理和统计,发现不同类型的数学竞赛中的函数方程问题的特征和区别,进一步说明函数方程问题解法的丰富与多样,说明函数方程问题必须具体问题具体分析,问题的类型可难可易,多数情况下考查函数方程问题时会融合其他知识点的考查等特征。第三,提出数学竞赛中的函数方程问题可能用到的解法,论述了解决函数方程问题的基本策略,举例阐释了换元法、赋值法、柯西法、待定系数法、数学归纳法等常见的函数方程问题的解题方法。第四,借鉴已有的函数方程问题的成功命题案例与个别失败案例剖析了函数方程问题命题应遵循的基本原则及命题的一般方法,包括改造法、组合法等,并统计分析了吴伟朝的函数方程问题命题实例。最后,提出若干数学竞赛中的函数方程问题的应用。
基于本文的研究,对目前高中数学竞赛中的函数方程问题的解题与命题提出建议:教师在培训学生过程中应多研究试题的结构和本质,呈现解题思维过程,提高学生解函数方程问题的能力;建立完善的命题制度,提高数学竞赛中的函数方程问题命题质量,正确认识解题与命题的关系,以更好地促进学生数学素质的发展。
Functional equation has a long history and is rich in content. As a branch of mathematics, it has been widely applied in practice. Since the 20th century, functional equation often appears in the International Mathematical Olympiad, which has become an important part in mathematics competitions. Due to its various and creative skills in solving problems, functional equation has won the designers’favor in different kinds of competitions and received more and more attention of the mathematics science field at home and abroad.
Based on G. Polya’s How to Solve It: A New Aspect of Mathematic Method as theoretical basis, this essay employs the method of analyzing documents. The Introduction of this essay focuses on the present situation of international mathematics competition, as well as competition levels. With the definition of functional equation, it points out that functional equation has received more and more attention in mathematics competitions. Chapter Two is aiming to compile functional equation and analyze the statistics in order to find out the characteristics and differences of functional equation in various types of mathematics competitions. What’s more, it demonstrates that in the process of tackling problems of functional equation, the competitors should make a concrete analysis of concrete problems, though there are abundant solutions. Functional equation in the competitions may be difficult or easy while in most circumstance it is characterized by the combination of other knowledge points. Chapter Three proposes the possible solutions and the basic strategies for functional equation and illustrates the typical methods by examples such as substitution method, assignment method, Cauchy method, method of undetermined coefficients and mathematical induction. Through the analysis of successful and failed cases, Chapter Four analyzes the basic principles and the general designing methods of functional equation, including the transformation of the existing functional equation, the adaptation and the preparation of such problems. Furthermore, it analyzes Wu Weichao’s designing problems of functional equation. The last chapter puts forwards to the application of functional equation in mathematics competitions.
Based on this research, this essay proposes the following suggestions about the solution and design problems of functional equation in mathematics competitions for the current high school. In the process of training the students, the teachers should pay more attention to the structure and nature of the questions and improve the students’capacity of solving functional equation through the illustration of the problem-solving thinking process. On the other hand, it is useful for promoting the quality for the students to establish a sound designing system for functional equation from the perspective of solving-problems, which can not only improve the quality of functional equation but also discover the correct relationship between problem-solving and problem-designing.
本研究采用文献分析法,以波利亚的怎样解题为理论依据,首先系统地介绍了国内外数学竞赛发展的现状及比赛层次,并对数学竞赛中的函数方程问题的定义进行界定,指出函数方程问题在现有的数学竞赛中日趋受到重视。其次,对国内外数学竞赛中的函数方程问题进行汇编、分析、整理和统计,发现不同类型的数学竞赛中的函数方程问题的特征和区别,进一步说明函数方程问题解法的丰富与多样,说明函数方程问题必须具体问题具体分析,问题的类型可难可易,多数情况下考查函数方程问题时会融合其他知识点的考查等特征。第三,提出数学竞赛中的函数方程问题可能用到的解法,论述了解决函数方程问题的基本策略,举例阐释了换元法、赋值法、柯西法、待定系数法、数学归纳法等常见的函数方程问题的解题方法。第四,借鉴已有的函数方程问题的成功命题案例与个别失败案例剖析了函数方程问题命题应遵循的基本原则及命题的一般方法,包括改造法、组合法等,并统计分析了吴伟朝的函数方程问题命题实例。最后,提出若干数学竞赛中的函数方程问题的应用。
基于本文的研究,对目前高中数学竞赛中的函数方程问题的解题与命题提出建议:教师在培训学生过程中应多研究试题的结构和本质,呈现解题思维过程,提高学生解函数方程问题的能力;建立完善的命题制度,提高数学竞赛中的函数方程问题命题质量,正确认识解题与命题的关系,以更好地促进学生数学素质的发展。
Functional equation has a long history and is rich in content. As a branch of mathematics, it has been widely applied in practice. Since the 20th century, functional equation often appears in the International Mathematical Olympiad, which has become an important part in mathematics competitions. Due to its various and creative skills in solving problems, functional equation has won the designers’favor in different kinds of competitions and received more and more attention of the mathematics science field at home and abroad.
Based on G. Polya’s How to Solve It: A New Aspect of Mathematic Method as theoretical basis, this essay employs the method of analyzing documents. The Introduction of this essay focuses on the present situation of international mathematics competition, as well as competition levels. With the definition of functional equation, it points out that functional equation has received more and more attention in mathematics competitions. Chapter Two is aiming to compile functional equation and analyze the statistics in order to find out the characteristics and differences of functional equation in various types of mathematics competitions. What’s more, it demonstrates that in the process of tackling problems of functional equation, the competitors should make a concrete analysis of concrete problems, though there are abundant solutions. Functional equation in the competitions may be difficult or easy while in most circumstance it is characterized by the combination of other knowledge points. Chapter Three proposes the possible solutions and the basic strategies for functional equation and illustrates the typical methods by examples such as substitution method, assignment method, Cauchy method, method of undetermined coefficients and mathematical induction. Through the analysis of successful and failed cases, Chapter Four analyzes the basic principles and the general designing methods of functional equation, including the transformation of the existing functional equation, the adaptation and the preparation of such problems. Furthermore, it analyzes Wu Weichao’s designing problems of functional equation. The last chapter puts forwards to the application of functional equation in mathematics competitions.
Based on this research, this essay proposes the following suggestions about the solution and design problems of functional equation in mathematics competitions for the current high school. In the process of training the students, the teachers should pay more attention to the structure and nature of the questions and improve the students’capacity of solving functional equation through the illustration of the problem-solving thinking process. On the other hand, it is useful for promoting the quality for the students to establish a sound designing system for functional equation from the perspective of solving-problems, which can not only improve the quality of functional equation but also discover the correct relationship between problem-solving and problem-designing.
引文
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