高中代數(shù)大多是基于代數(shù)運(yùn)算和方程求解。而大學(xué)代數(shù)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(shù)中的抽象性和表征形式,這種向形式化數(shù)學(xué)的過(guò)渡是不容易的。群論概念的抽象表征對(duì)學(xué)生的理解力來(lái)說(shuō)是不小的挑戰(zhàn),而且影響他們是否能學(xué)好這門(mén)課程。本研究探討了阿富汗本科生對(duì)重要群論概念的理解、在學(xué)習(xí)群論基本概念時(shí)產(chǎn)生的錯(cuò)誤觀念和遇到的困難、以及產(chǎn)生這些錯(cuò)誤觀念和困難可能的來(lái)源或原因,相關(guān)的一些以前的概念,以及學(xué)生對(duì)于這些形式化構(gòu)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(shù)學(xué)的三個(gè)世界”理論和學(xué)生的問(wèn)題解決能力,作為測(cè)評(píng)學(xué)習(xí)者在解決基礎(chǔ)群論概念相關(guān)問(wèn)題的理解水平的一種方法。對(duì)112名學(xué)生以書(shū)面問(wèn)卷的形式收集的有關(guān)基礎(chǔ)群論概念相關(guān)問(wèn)題的數(shù)據(jù)分析表明學(xué)生對(duì)基礎(chǔ)群論概念的理解存在幾個(gè)突出問(wèn)題。本研究表明總體上隨著群論概念變得越復(fù)雜,阿富汗本科生對(duì)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(cuò)誤觀念和許多困難。尤其是,本研究發(fā)現(xiàn)首要困難與群論中概念定義的符號(hào)表征有關(guān)。與具體形式化數(shù)學(xué)世界中概念的文字定義相比,學(xué)生更難理解數(shù)... 

【文章來(lái)源】：西南大學(xué)重慶市 211工程院校 教育部直屬院校

【文章頁(yè)數(shù)】：242 頁(yè)

【學(xué)位級(jí)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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