高中代數大多是基于代數運算和方程求解。而大學代數則是基于集合論定義和形式證明。由于群論概念在本科代數中的抽象性和表征形式,這種向形式化數學的過渡是不容易的。群論概念的抽象表征對學生的理解力來說是不小的挑戰(zhàn),而且影響他們是否能學好這門課程。本研究探討了阿富汗本科生對重要群論概念的理解、在學習群論基本概念時產生的錯誤觀念和遇到的困難、以及產生這些錯誤觀念和困難可能的來源或原因,相關的一些以前的概念,以及學生對于這些形式化構建的概念的定義在觀念上存在的有力和不利的方面。本研究利用大衛(wèi)·塔勒（David Tall）的“數學的三個世界”理論和學生的問題解決能力,作為測評學習者在解決基礎群論概念相關問題的理解水平的一種方法。對112名學生以書面問卷的形式收集的有關基礎群論概念相關問題的數據分析表明學生對基礎群論概念的理解存在幾個突出問題。本研究表明總體上隨著群論概念變得越復雜,阿富汗本科生對這些概念的理解變得越困難。本研究還表明阿富汗本科生在理解群論概念方面存在新的錯誤觀念和許多困難。尤其是,本研究發(fā)現首要困難與群論中概念定義的符號表征有關。與具體形式化數學世界中概念的文字定義相比,學生更難理解數... 

【文章來源】：西南大學重慶市 211工程院校 教育部直屬院校

【文章頁數】：242 頁

【學位級別】：博士

【文章目錄】：
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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