几何推理与代数推理的关系研究

发布时间：2018-03-31 15:49

本文选题：几何推理　切入点：代数推理　出处：《华中师范大学》2015年硕士论文

【摘要】：数学推理教育的本质意义,在于培养人良好的数学思维习惯以及极强的反应能力。几何推理与代数推理贯穿整个数学学习过程,所以说培养学生的几何推理与代数推理思维对其学好数学相当重要。学生的几何推理与代数推理能力的培养逐步受到国内外数学教育界的关注,本研究旨在讨论我国各层级阶段几何推理与代数推理学习能力的表现情形,依据几何推理与代数推理能力发展的认知先后顺序,指出了不同年级阶段的推理形式,小学阶段是对推理的初步认知,初中阶段几何推理占主导地位,高中阶段几何推理与代数推理能力已趋于成熟,此阶段着重培养对几何推理与代数推理的灵活运用能力,其中数形结合是连接这两种推理的主导思想。本文依据教材内容要求,阶段性分析推理的不同学习形式,结合数学家们的研究成果,来区分几何推理与代数推理之间的差异性及联系。首先,在对推理有了初步认识情况下,对推理能力的地位进行分析。了解推理与证明的区分,让学生明白推理与证明之间的关系,通过分析教材,结合教材内容,教学目标,从宏观上来认识几何推理与代数推理在数学发展中的先后顺序,分析出小学阶段,初中阶段,高中阶段几何推理与代数推理引入及着重引用的推理方式。其次,由于此方面的研究甚少,几何推理与代数推理并没有严格的概念性语言。笔者通过文献分析总结出几何推理与代数推理的概念及其特征,通过查阅期刊文献等等,了解到几何推理与代数推理的发展历程,结合教学实践了解推理在数学各内容方面的运用及其之间的关联。分析出哪些题目类型适合于几何推理,哪些类型必须用代数推理,又有哪些题目几何与代数推理结合起来更易于题目的解决。再次,指出只要有数学存在,就会有几何与代数,几何推理与代数推理能够很好的培养数学思维,是数学发展的推进剂。处理好两者之间的关系,能够帮助学生更好的学习数学。本研究获得的结论是： (1)促进几何与代数之间的联系,能够引导学生对推理能力进行顺利地转换； (2)关注学生推理能力的发展,注重推理之间的差异性,具体情况具体分析。为了之后能够更好的进行推理知识的传授,推理能力的培养,及时的发现它们之间更多的联系是相当有必要的； (3)教师自身的数学推理素养首先要得以提升,在对推理进行深刻理解的基础之上将推理知识渗透于教学中,因材施教； (4)要促进国内外数学知识之间的及时交流,教师之间的交流尤为重要,知识上的沟通融汇,能够促进教师及学生的专业性发展。
[Abstract]:The essential meaning of mathematical reasoning education lies in the cultivation of good mathematical thinking habits and strong reaction ability.Geometric reasoning and algebraic reasoning run through the whole process of mathematics learning, so it is very important to cultivate students' thinking of geometric reasoning and algebraic reasoning for them to learn mathematics well.The cultivation of students' ability of geometric reasoning and algebraic reasoning has been paid more and more attention to in the field of mathematics education at home and abroad. The purpose of this study is to discuss the performance of the learning ability of geometric reasoning and algebraic reasoning at different stages in China.According to the cognitive sequence of the development of geometric reasoning and algebraic reasoning, this paper points out the reasoning forms in different grades. The primary stage is the primary cognition of reasoning, and the junior middle school stage is dominated by geometric reasoning.The ability of geometric reasoning and algebraic reasoning has matured in senior middle school. In this stage, the flexible application of geometric reasoning and algebraic reasoning is emphasized, among which the combination of number and form is the leading idea linking the two reasoning.According to the content requirements of the textbook, this paper analyzes the different learning forms of reasoning by stages, and combines the research results of mathematicians to distinguish the differences and relations between geometric reasoning and algebraic reasoning.First of all, the status of reasoning ability is analyzed with a preliminary understanding of reasoning.The methods of introduction and citation of geometric reasoning and algebraic reasoning in primary, junior and senior middle school are analyzed.Secondly, there is no strict conceptual language for geometric reasoning and algebraic reasoning due to the lack of research in this field.The author summarizes the concepts and characteristics of geometric reasoning and algebraic reasoning through literature analysis, and finds out the development course of geometric reasoning and algebraic reasoning by consulting the periodical literature, etc.Combined with teaching practice to understand the application of reasoning in various aspects of mathematics and the relationship between them.It is found out which problem types are suitable for geometric reasoning, which types must be algebraic reasoning, and which problems can be solved more easily by combining geometry with algebraic reasoning.Thirdly, it is pointed out that as long as mathematics exists, there will be geometry and algebra, geometric reasoning and algebraic reasoning can train mathematical thinking very well and are propellants of mathematics development.Dealing with the relationship between the two can help students learn math better.The conclusions of this study are as follows: 1) promoting the relationship between geometry and algebra, which can guide students to transfer their reasoning ability smoothly; 2) paying attention to the development of students' reasoning ability, paying attention to the difference between reasoning and concrete analysis.In order to teach reasoning knowledge, cultivate reasoning ability, and find more connections between them in time, it is necessary to improve teachers' mathematical reasoning literacy.On the basis of deep understanding of reasoning, the reasoning knowledge is permeated into teaching and teaching in accordance with students' aptitude. (4) to promote the timely exchange of mathematics knowledge at home and abroad, the communication among teachers is particularly important.Can promote the professional development of teachers and students.
【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

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