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一、引言大学代数课程主要包括数学专业的高等代数、近世代数课程及公共课线性代数.这三门课程都具有高度的抽象化和形式化的特征，是被学生公认为比较难学又极其重要而基础的专业课程.从大学代数课程的教学研究和实践出发，对其教学内容、教材建设、教学手段等方面进行有效的改革，从而提高教学质量，同时培养学生的数学素质与创新能力，使得学生从“知识教育”向“能力教育”逐渐转变，这便是我们对代数课程进行相关探索和研究的主要目标.如何结合地方院校自身的特点，让学生更容易、更有效率地学好这几门专业课程，并让学生尽量利用所学的代数思想方法应用于实践，从而培养他们形成解决实际问题的能力，这便是我们进行相关探索和研究的重要内容.目前，已有不少文献探讨了高等代数近世代数或线性代数课程的一些教学实施与体会，如可参看文献1-4等.本文作者将结合自身在广东省精品资源共享课程高等代数近世代数及线性代数这三门大学代数课程的教学研究及实践的基础上给出一些教学体会.二、大学代数课程教学的几点尝试与实践（一）始终不渝地把握四个教学原则1.体现大学代数学的典型思想方法的原则培养学生系统地掌握代数研究问题的基本方法是代数课的教学目的之一.代数中有代表性的典型思想方法包括：公理化演绎的思想（如：向量空间、欧式空间等各类代数系统），分类的思想（如：矩阵的相似、合同、等价等等各种等价关系），相互关联的思想（如：同态、同构等各种形式的映射），矩阵的方法，初等变换的方法，抽象推理的方法等等.了解这些思想方法的具体含义和在代数中的具体应用对代数课程教学是十分有益的.文献1，4也结合高等代数课程的教学体会，详细地探究了严格的逻辑推理方法，公理化方法，结构化方法，矩阵表示方法和等价分类方法等在教学中有效实施.2.体现与时俱进的原则参考国内外最新的教材内容，结合我们的教研、科研，把课程的前沿知识、研究现状和发展趋势，及时贯彻到教学过程中，常讲常新.例如，我们可以在教学过程中把代数学家的一些故事、代数学界最近的研究现状及所发生的一些事情带入到课堂，介绍给学生，以此激发他们学习数学的兴趣与热情.3.体现现代教育理念的原则适当安排一些探索性内容，扩展性内容，构建终身学习所需要的代数学的基础.将现代化手段在数学课程教学中的应用将全面铺开；从教学内容的组织与安排看，课堂教学与课外延伸相结合，将知识传授、能力培养、素质教育融为一体，采用各种形象化的教学手段，使用投影仪和计算机辅助教学，增加教学的直观性，化解数学的抽象和难点，促进教学质量的提高.4.突出师范教育的特点惠州学院的数学与应用数学专业是师范专业，而高师数学专业培养的目标是中小学数学教师，我们努力在高等代数与近世代数课程的教学之中渗透教育学和数学课程教学论的思想，注重研究代数学课程对中学数学教学的指导，充分体现数学文化和数学美，培养学生的数学文化素养和未来数学教师的综合素质，适应基础教育教学和改革的需要.（二）不断尝试各种教学理念和方法1.采用“本原教学法”进行教学高度的抽象化和形式化是代数学的基本特征，高等代数、近世代数及线性代数这三门大学代数课程是被学生公认为比较难学的数学课程.所谓“本原教学法”，就是教学中要返璞归真，从源头讲起，讲清楚问题产生和发展的过程，先讲明道理，水到渠成，让学生自己归纳定义或结论，再讲推理，然后再抽象化和形式化.例如，在引入同构概念之前，我们可以先让学生回忆三角形全等的概念和判定方法.ABC与三角形ABC的全等实际上是建立两个三角形的顶点和边的一一对应.点的对应可以看成两个集合S和T的元素的一一对应，即AA，边可以看成两个点所作用的结果，从而S和T的边的对应可以是看成保持它们两个点的运算结果.这样一来，两个代数系统的同构其实就是这两个代数系统间可以建立一个一一映射，并且该映射保持这两个代数系统的所有运算.再例如在引入向量的线性相关的概念时，我们先从“平面向量的共线”及“空间向量的共面”入手，介绍一些具体的、学生熟悉的例子，最后归纳出线性相关的一般定义.教学实践证明，这种教学方法学生易于接受，效果明显.2.采用“研究性教学法”进行教学在自身开展科研的同时，我们经常将所授课程的前沿知识，研究现状和发展趋势融入到教学过程中，将自己的研究实践经验、思维创新方法、学科前沿动态介绍给学生，并适时适度提出一些问题供学生研究.例如我们在高等代数或线性代数课程教学中，可以提出如下问题给学生探究：矩阵表示方法的综合体现、等价分类方法的渗透与应用、同构思想的应用、分析学思想在代数学中应用等等.此外，我们也偶尔可以不从定义出发而从问题出发来组织和展开本课程的教学内容和体系，即从重要的问题出发，根据需要引入概念，并总结出定理，引导学生去探索和发现知识，从而培养学生的创新思维.这一教学过程的主体是学生，主导是教师.3.利用类比法进行各代数系统相关内容的教学类比法是数学发现中最常用、最有效的方法之一，它在科学发展史上起过重大作用.法国数学家拉普拉斯指出：甚至在数学里，发现真理的工具是归纳和类比，这也足以看出类比方法的重要性.类比是通过两类不同对象A，B间的某些属性的相似，从而从A具有某种其他属性便猜想B也有这种属性. 本科阶段主要接触的代数系统有向量空间、欧式空间、群、环和域等.由于这些代数系统之间具有一些属性的相似，即都是一些带有运算的集合，这即表明类比的数学思想方法可尝试在这些代数课程的学习或教学中去运用.例如，我们在讲授高等代数或线性代数时，可以利用类比法来讲解向量空间与欧式空间、矩阵与线性变换的定义与性质、联系与区别等等.又例如，我们在讲授近世代数时，可利用类比法来讲解群环域等代数系统及其子系统的概念，讲解代数系统的同态基本定理，讲解一些特殊环（整环、除环与域）之间关系，讲解一些特殊整环（唯一分解环、主理想环、欧氏环等）的关系等等.教学实践证明，该方法教学效果明显，而且可以培养学生如何发现新问题的科研兴趣和能力.4.课堂精讲、返讲与自学相结合我们在代数系列课程的教学中，努力做到课堂精讲、返讲与自学相结合.课堂上，讲重点，讲知识的背景与形成过程，揭示知识的内在联系；对难点、重点内容进行返讲，使学生深刻理解抽象的理论，从怕学到爱学；自学是指有些教材内容则采用学生自学为主，教师给出思考题，课后下班辅导及答疑.我们采取了一系列措施指导学生自主学习，主要做法是针对不同专业的学生建立不同层次的试卷库，建立自测卷，同时，统一考试标准及要求，保证其公正、公平.5.以科技创新活动为突破口，激励学生研究性学习（1）开发第二课堂通过讲座，介绍代数发展历史上的典型人物、典型事件、典型的思想方法，代数与相关学科的联系、应用前景，提高学习代数学的兴趣.指导学生去发现实践中的数学问题，指导学生使用Matlab分析和解决问题；指导学生自主式学习、探究式学习，给他们布置一些难度不是很大的研究性问题，让他们课外去找资料解决，并用规范论文的格式打印出来.这样，一方面，我们可以让所有学生学会如何撰写数学专业论文，另一方面，我们也可以让一部分写得比较好的学生的论文拿去发表，从而达到一举双得的效果.此外，我们也提倡学生在数学的认识与实践、数学教育学报、大学数学、高等数学研究、数学通报、中学数学研究等一些专业涉及知识不深的期刊中找适合自己的文章阅读、报告和探讨.（2）以学科竞赛为平台，提高学生协同创新能力我们的具体做法有：以全国和国际数学建模竞赛为平台，培养学生的解决实际问题的创新能力；以全国普通高校信息技术创新活动为载体，培养学生信息技术创新能力.数学建模对激励学生学习数学的积极性、提高学生建立数学模型和运用计算机技术解决实际问题的综合能力、推动大学数学教学内容和方法的改革等方面均有重要意义.通过“一年两赛”模式参加国内和国际数学建模学科竞赛，努力提高学生的应用能力与创新能力，提倡“以赛促教，以教育赛”，并将建模融入日常教学中；以数学建模竞赛为切入点，努力培养学生的创新能力.（3）指导学生申报各类大学生科技创新项目，培养学生研究性学习的能力在教师的指导下制定研究课题，鼓励学生自主申报并研究国家级、省级、校级大学生创新创业训练项目、暑寒期社会实践项目等各项课题，鼓励学生踊跃向国内外专业期刊投稿，以此来增强学生的科学研究及写作能力.（4）鼓励学生参加教师的课题，提高学生以及教师的科研创新能力教师是培养大学生科技创新能力的关键因素之一，倡导教师将学生纳入自身的科研工作之中，根据学生的知识阶段，指导学生完成力所能及的研究工作，努力提高学生的科研创新能力.三、结束语本文就高等代数近世代数及线性代数这三门大学代数课程的教学原则、教学理念、教学方法、教学研究及实践等方面，给出了一些教学思考与体会.旨在强调探索和改进传统的教学模式，不断渗透数学思想和方法，对提高教学质量，培养和发展学生数学思维能力具有非常重要的意义.因此，我们今后需不断地对大学代数课程课堂的教学内容、模式和方法进行有效改革，使得学生既感兴趣地学到必要的数学知识和数学技能，又掌握了其中的数学思想和方法，好为他们将来更好地从事数学方面的相关工作打下良好的基础.One, the introductionWith specialized mathematics of the university algebra course mainly includes the advanced algebra, modern algebra course and general course linear algebra. The three courses are highly abstract and formalized characteristics, is recognized by students to compare the difficult and extremely important and basic professional course. Starting from the teaching research and practice of college algebra course, the teaching content, teaching means, teaching material construction for effective reform, so as to improve the teaching quality, cultivate the students mathematics quality and innovation ability at the same time, make the students from the knowledge education to capability education gradually shift, this is our related exploration and research of algebra course. How to combine the characteristics of local colleges and universities themselves, let the student much easier and more efficient to learn the professional course, and let the students try to use what they have learned algebra thought method is applied to the practice, so as to cultivate their ability to solve practical problem formation, this is the important content of related exploration and research.At present, there have been many literatures discussed the advanced algebra modern algebra or linear algebra courses teaching and experience, such as 1-4 can see literature. In this paper, the author will combine their own fine sharing resources in guangdong province advanced algebra modern algebra and the three college algebra course linear algebra teaching research and practice on the basis of given some teaching experience.Second, college algebra course teaching of try and practice(a) unswervingly hold four teaching principles1. Typical ideology that embodies the college algebra method principleTo cultivate students to systematically master the basic method of algebra research problem is one of the teaching purpose of algebra class. Representative typical thought in algebra method includes: axiomatic deductive thinking (such as vector space, European space and other kinds of algebraic system), the classification of ideas (such as: similar matrix, contracts, equivalence and so on all sorts of equivalence relation), interrelated ideas (e.g. homomorphism and isomorphism, etc all kinds of mapping), the matrix method, the method of elementary transformation, abstract reasoning method, and so on. Understand the specific meaning of these thought methods and in the application of algebra to algebra course is very useful. The literature 1, 4 also combined with the teaching of higher algebra course experience, explores in detail the strict logical reasoning methods, axiomatic method, structured method, matrix representation method and equivalent classification methods in the teaching effectively.2. The principle of keeping pace with The TimesReference at home and abroad the latest teaching material content, combined with our teaching, scientific research, the forefront of the course knowledge, research status and development trend, timely carry out the teaching process, often speak often new. For example, we can in the teaching process of the generation of mathematicians, algebra research of recent situation and some story of what happened something into the classroom, introduced to students, to motivate them to learn mathematics interest and enthusiasm.3. Reflect the principle of modern education conceptProperly arrange some exploratory content, content of extensibility, needed to build life-long learning algebra. The basis of applying modern methods in the teaching of mathematics course will be carried out; From the organization and arrangement of teaching content, classroom teaching and extra-curricular activities, the combination of knowledge, ability training and quality education, adopt various visual teaching method, using a projector and computer assisted instruction, increase the visual teaching, dissolve the mathematical abstraction and difficulty, promote the improvement of teaching quality.4. Highlight the characteristics of normal educationHuizhou university mathematics and application mathematics specialty is normal, and goals of mathematics in normal universities is the primary and secondary school teachers of mathematics, we work hard in the advanced algebra and modern algebra teaching pedagogy theory and mathematic teaching ideas, pay attention to study algebra course for middle school mathematics teaching guidance, fully embody the mathematical culture and mathematical beauty, cultivate students mathematical literacy and future math teachers comprehensive quality, to adapt to the needs of the elementary education and teaching reform.(2) keep trying various teaching ideas and methods1. primitive teaching method is adopted to improve the teachingHigh level of abstraction and formalization are the basic characteristics of algebra, advanced algebra, modern algebra and linear algebra the three college algebra course is recognized by students is difficult to learn math courses. The so-called primitive teaching method, is want to in the teaching, from the source, clear problems and the development process, first to truth, for us, lets student inductive definition or conclusion, another reason, and then abstract and formalized.Before introducing concept of isomorphism, for example, we can let the students to recall the concept and determination methods of triangles are congruent. Delta ABC with triangle delta A B C congruent is actually set up two triangle vertices and edges. Corresponding collection can be seen as two point S and T the elements of one to one correspondence, namely AA, and can be as A consequence of the two points, thereby corresponding can be seen as the edge of S and T keep them the result of two points. As A result, the isomorphism of two algebraic system is in fact the two may establish A one-to-one mapping between algebraic system, and the mapping to keep the two algebraic system of all operations.Again, for example, the introduction of the concept of vector linear correlation, we start with collinear plane vector and space vectors coplanar, introduces some examples of concrete, students are familiar with, and finally concludes the general definition of linear correlation. Teaching practice proves that this method of teaching students easy to accept, the effect is obvious.2. research-oriented teaching method is adopted to improve the teachingIn conducting scientific research at the same time, we will often taught courses at the forefront of knowledge, research into the teaching process, present situation and development trend of research on the practical experience, the way of thinking innovation, discipline frontier dynamic introduced to students, and puts forward some problems timely appropriate for students to study. For example, we in the advanced algebra or linear algebra courses teaching, the probe can give students the following questions: integrated embodiment of its matrix representation method, equivalence and classification methods of infiltration and application, the application of isomorphic thought, the analysis thought in algebra application and so on. In addition, we also can occasionally not starting from the definition and starting from the problem to organize and expand teaching of the course content and system, namely, starting from the important issues, according to the need to introduce concept, and sums up the theorem, guide students to explore and discover knowledge, to cultivate students innovative thinking. This is the body of the teaching process students, leading is a teacher.3. The use of analogy method for the teaching of the algebraic system related contentAnalogy method is commonly used in the mathematical discovery, one of the most effective way, it played a major role in the scientific history. The French mathematician Laplace said: even in mathematics, discover the truth of the tool is the induction and analogy, this would be enough to see the importance of analogy method.Analogy is through two kinds of different objects A, B between some of the properties of similar, so from A certain other attributes then guess B also has this property. The algebraic system of undergraduate major contact space vector space, european-style, group, ring and domain, etc. As A result of these algebraic system has some of the properties between similar, which is A collection of with operation, which shows that mathematical thinking method of analogy to try on these algebra course in the teaching or learning to use.For example, we in the teaching of advanced algebra or linear algebra, the analogy method can be used to explain the vector space and the European space, the definition and properties of matrix and linear transformation, relation and distinction between, and so on., for example, we are in the teaching of modern algebra, can use analogy to explain group of ring domain such as the concept of algebra system and its subsystems, homomorphism fundamental theorem of algebra system, explain some special ring (the ring, in addition to the ring and domain), the relationship between explain some special order (the only decomposition ring, principal ideal ring, Euclidean ring, etc.) of the relationship, and so on. Teaching practice proves that the method of teaching effect is obvious, and can cultivate the students how to discover new problems of scientific interest and ability.4. The class earnestly, back to speak with self-studyWe in algebra courses teaching, strive to class earnestly, back to speak with self-study. Class, focus, knowledge background and formation process and reveals the internal relation of knowledge and Returned to the emphasis and difficulty of the content, make students understand the abstract theory, learn from fear of love to learn; Self-study is to point to some students self-study, the teaching material content is used teachers questions, guidance and answering questions www.maka7.com www.zuoxuanroujian3.com from work after class. We adopted a series of measures to guide students autonomous learning, is the main approach for different students to establish different levels of test paper library, establish a self-test volume, at the same time, the unified exam standards and requirements, ensure the impartiality and fairness.5. With scientific and technological innovation activities as the breakthrough point, motivate students inquiry learning(1) to develop the second classroomThrough lectures, introduction to algebra typical character, typical events in the history of the development, the typical thought method, connection, the application prospect of algebra and related disciplines, improve the learning interest in algebra. Guide students to discover mathematical problems in practice, to guide students to use the Matlab analysis and solve problems; To guide students autonomous learning, inquiry learning, gave them some difficulty is not a lot of research problem, let them go to outside information, print it out and use standard paper format. In this way, on the one hand, we can let all students to learn how to write a thesis math major, on the other hand, we also can make some of the better students written papers published to, so as to achieve every double effect.In addition, we also advocate students mathematics knowledge and practice, journal of mathematics education, the university mathematics, higher mathematics, bulletin, mathematics studies of middle school mathematics involved such as some professional knowledge is not deep find suits own journal articles, reports, and discussed in this paper.(2) subject contest as the platform, to improve students innovation ability togetherOur particular way is: with the national and international mathematical modeling competition platform, cultivate students innovative ability to solve practical problems; With the national ordinary university information technology innovation activities as the carrier, cultivate students information technology innovation ability.Mathematical modeling to inspire students enthusiasm to study mathematics, improving students mathematical model is established and the comprehensive ability to use computer technology to solve practical problems, promote the reform of college mathematics teaching content and method, etc all have important significance. Through two year model to participate in domestic and international mathematical contest in modeling disciplines, efforts to improve the students application ability and innovation ability, advocate to promote teaching, to education, and the modeling into their daily teaching; In mathematical modeling competition as the breakthrough point, trying to cultivate students innovation ability.(3) to guide students to declare all kinds of college students of science and technology innovation projects, cultivate students ability of inquiry learningUnder the guidance of teachers to develop research, encourage students applied and on the national, provincial and university college students innovative entrepreneurial training project, heat cold period of the social practice project, various issues such as encouraging students to contribute to the domestic and international professional journals, in order to enhance the students ability of scientific research and writing.(4) encourage students to participate in the teachers subject, to improve students and teachers scientific research innovation abilityTeacher is one of the key factors of cultivating college students science and technology innovation ability, advocate the teachers bring students into their scientific research work, according to the students knowledge stage, to guide students to complete its work, efforts to improve the students scientific research innovation ability.Third, the conclusionIn this paper, the advanced algebra, modern algebra and the three college algebra course linear algebra teaching principle, teaching concept, teaching method, teaching research and practice, thinking www.yulematou.com www.14se.net and offers some teaching experience. Emphasis on exploration and to improve the traditional teaching mode, the continuous infiltration mathematical thought and method, to improve the teaching quality, cultivate and develop students mathematical thinking ability has very important significance. Therefore, we need to continually in the future to college algebra course classroom teaching content, mode and method for effective reform, so that half of the students interested in learning the necessary mathematical knowledge and mathematical skills, and grasp the mathematical thought and method of these, well for their future work of mathematics are better to lay a good foundation.
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