阿富汗本科生对基础群论概念的理解研究

发布时间：2021-07-10 21:51

　　高中代数大多是基于代数运算和方程求解。而大学代数则是基于集合论定义和形式证明。由于群论概念在本科代数中的抽象性和表征形式,这种向形式化数学的过渡是不容易的。群论概念的抽象表征对学生的理解力来说是不小的挑战,而且影响他们是否能学好这门课程。本研究探讨了阿富汗本科生对重要群论概念的理解、在学习群论基本概念时产生的错误观念和遇到的困难、以及产生这些错误观念和困难可能的来源或原因,相关的一些以前的概念,以及学生对于这些形式化构建的概念的定义在观念上存在的有力和不利的方面。本研究利用大卫·塔勒（David Tall）的“数学的三个世界”理论和学生的问题解决能力,作为测评学习者在解决基础群论概念相关问题的理解水平的一种方法。对112名学生以书面问卷的形式收集的有关基础群论概念相关问题的数据分析表明学生对基础群论概念的理解存在几个突出问题。本研究表明总体上随着群论概念变得越复杂,阿富汗本科生对这些概念的理解变得越困难。本研究还表明阿富汗本科生在理解群论概念方面存在新的错误观念和许多困难。尤其是,本研究发现首要困难与群论中概念定义的符号表征有关。与具体形式化数学世界中概念的文字定义相比,学生更难理解数... 

【文章来源】：西南大学重庆市 211工程院校 教育部直属院校

【文章页数】：242 页

【学位级别】：博士

【文章目录】：
英文摘要
中文摘要
ACKNOWLEDGEMENTS
LIST OF PUBLICATIONS
I.INTRODUCTION
    1.1 Background to the Problem
    1.2 Rationale of the Study
    1.3 Research Questions of the Study
    1.4 Significance of the Study
    1.5 Some Necessary Definitions for the Study
    1.6 Outline of the Study
II.THE ORETICAL UNDERPINNINGS AND LITERATURE REVIEW
    2.1 Introduction
    2.2 Theoretical Underpinnings of the Study
        2.2.1 Mathematical Concept,Concept Definition and Concept Image
        2.2.2 Understanding Mathematical Concept
    2.3 Theoretical and Conceptual Framework of the Study
        2.3.1 Theoretical Framework of the Study
        2.3.2 Conceptual and Analytic Framework
    2.4 Literature Review
        2.4.1 Studies on the Learning of Modern Algebra
        2.4.2 Studies on the Teaching of Modern Algebra
    2.5 Deducing Results from Definitions and Axioms
    2.6 The Role of Modern Algebra in Undergraduate Mathematics
    2.7 Summary
III.RESEARCH DESIGN AND METHODOLOGY
    3.1 Introduction
    3.2 The Research Design and Method
        3.2.1 The Case Study Design
    3.3 Research Site
    3.4 Participants
    3.5 Instruments
        3.5.1 Assessment of the Course
        3.5.2 Reliability and Validity
    3.6 Data Collection Procedure
    3.7 Modern Algebra Course Materials
        3.7.1 Modern Algebra Booklet and Reference Book
        3.7.2 Teaching Methods Employed in Modern Algebra Course
    3.8 Summary
IV.THE ANALYSIS OF DATA AND RESULTS
    4.1 Introduction
    4.2 The Data Analysis
    4.3 Algebraic Structures
    4.4 Defining Semi Group,Monoid,Group and Abelian Group
    4.5 Groups,Group Properties,Subgroups,Cyclic Groups and Order of an Element
    4.6 Homomorphism and Isomorphism of Groups
    4.7 Cosets and Lagrange’s Theorem
    4.8 Symmetric Group,Alternating Group,Normal Subgroup and Factor Group
    4.9 Summary
V.FINDINGS AND DISCUSSION
    5.1 Introduction
    5.2 Students’Conceptual Understanding,Difficulties and Possible Sources of Their Difficulties in Learning Group Theory Concepts
        5.2.1 Students’Conceptual Understanding and Difficulties in Binary Operation and Algebraic Structure
        5.2.2 Students’Conceptual Understanding and Difficulties in Group Concept
        5.2.3 Students’Conceptual Understand ing and Difficulties in Subgroup Concept
        5.2.4 Students’Conceptual Understanding and Difficulties in Properties of Group Concept
        5.2.5 Students’Conceptual Understanding and Difficulties in Isomorphic Binary Structures and Groups
        5.2.6 Students’Conceptual Understanding and Difficulties in Cosets and Lagrange’s Theorem
        5.2.7 Students’Conceptual Understanding and Difficulties in Symmetric and Alternating Groups
        5.2.8 Students’Conceptual Understanding and Difficulties in Normal Subgroup and Factor Group
    5.3 Summary
VI.CONCLUSIONS AND SUGGESTIONS
    6.1 Introduction
    6.2 The Main Findings of the Study
        6.2.1 Students’Understanding of Elementary Group Concepts
        6.2.2 Students’Difficulties in Learning Elementary Group Concepts
        6.2.3 Linking Students’Difficulties to Its Possible Sources
    6.3 Implications of the Study
    6.4 Limitations of the Study
    6.5 Concluding Remarks
REFERENCES
APPENDICES



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