中学数学竞赛中的构造性思维研究

发布时间：2018-05-13 18:13

本文选题：构造 + 构造性思维　；参考：《深圳大学》2017年硕士论文

【摘要】：奥林匹克数学竞赛作为国际性的数学联赛,有很多教育家、研究者们对数学竞赛中的思想方法进行了系统的论述.其中构造性的思想方法作为重要的数学方法,在奥林匹克竞赛的命题和解题中都有着广泛的应用.已有的有关数学“构造性思维”的研究主要是针对一些具体题目的构造性解法,对构造性思维的本质、竞赛数学中涉及到的构造性思维解题分类缺乏系统的研究.本文通过往年中学数学竞赛中涉及到的构造法习题进行实例分析,从不同的题型如何分类、以及构造法在不同数学学科中的运用两个角度出发研究当今背景下中学数学竞赛中涉及到的的构造性思维.本文首先概述了构造性思维的研究背景、国内外研究历史和现状及构造性思维的理论依据.其次结合以往数学竞赛的题目对构造法解题的原则、策略和特例进行探讨,同时配合竞赛题目对运用构造法解题提出自己的观点.然后通过对以往研究文献和竞赛题目的整理分析提出了不同数学竞赛题型中的几类构造法应用,包括存在性命题的结论构造、否定命题的反例构造和转化问题形式的推演构造.最后通过对以往竞赛题目的整理分析,将初等数论、代数、几何、组合数学中运用构造法解题的题目进行分析说明,详细的探讨如何在中学数学竞赛中根据具体的题目条件或结论特征恰当的运用构造性思想方法.文章的最后论述了构造性思维的教学误区和培养建议,包括怎样运用构造法解题、构造法解题的误区、构造性思维的培养建议,希望对竞赛数学的教学和构造性思维的培养提供有价值的参考.
[Abstract]:As an international mathematics league, there are many educators and researchers who systematically discuss the thought and method of the Olympic mathematics competition. As an important mathematical method, the constructive thinking method is widely used in the proposition and problem solving of the Olympic competition. The existing research on "constructive thinking" in mathematics is mainly aimed at the constructive solution of some specific topics, but there is no systematic study on the nature of constructive thinking and the classification of structural thinking problems involved in contest mathematics. In this paper, by analyzing the construction exercises involved in mathematics competitions in middle schools in previous years, how to classify different types of questions is given in this paper. And the application of the construction method in different mathematics subjects from two angles to study the constructional thinking involved in the middle school mathematics competition under the background of the present day. This paper first summarizes the research background of constructive thinking, the research history and present situation at home and abroad, and the theoretical basis of constructive thinking. Secondly, this paper discusses the principles, strategies and special cases of solving problems by structural method in combination with the topics of previous mathematical competitions, and at the same time puts forward its own views on the use of structural methods in solving problems with competition topics. Based on the analysis of previous research papers and competition topics, this paper puts forward several kinds of constructional application of different mathematical contest question types, including the conclusion construction of existence proposition, the counterexample construction of negation proposition and the deductive construction of transformation problem form. Finally, through the analysis of the previous competition topics, the problems in elementary number theory, algebra, geometry and combinatorial mathematics are analyzed and explained by using the construction method to solve the problems. This paper discusses in detail how to apply the constructive thinking method according to the specific subject condition or the conclusion characteristic in the middle school mathematics contest. At the end of the article, the author discusses the teaching misunderstandings and training suggestions of constructive thinking, including how to solve problems by constructional method, and how to cultivate constructive thinking. It hopes to provide valuable reference for the teaching of competitive mathematics and the cultivation of constructive thinking.
【学位授予单位】：深圳大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：G633.6

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