地方本科院校大學代數(shù)系列課程教學的幾點體會.docx

一、引言大學代數(shù)課程主要包括數(shù)學專業(yè)的高等代數(shù)、近世代數(shù)課程及公共課線性代數(shù).這三門課程都具有高度的抽象化和形式化的特征，是被學生公認為比較難學又極其重要而基礎的專業(yè)課程.從大學代數(shù)課程的教學研究和實踐出發(fā)，對其教學內(nèi)容、教材建設、教學手段等方面進行有效的改革，從而提高教學質(zhì)量，同時培養(yǎng)學生的數(shù)學素質(zhì)與創(chuàng)新能力，使得學生從“知識教育”向“能力教育”逐漸轉(zhuǎn)變，這便是我們對代數(shù)課程進行相關探索和研究的主要目標.如何結(jié)合地方院校自身的特點，讓學生更容易、更有效率地學好這幾門專業(yè)課程，并讓學生盡量利用所學的代數(shù)思想方法應用于實踐，從而培養(yǎng)他們形成解決實際問題的能力，這便是我們進行相關探索和研究的重要內(nèi)容.目前，已有不少文獻探討了高等代數(shù)近世代數(shù)或線性代數(shù)課程的一些教學實施與體會，如可參看文獻1-4等.本文作者將結(jié)合自身在廣東省精品資源共享課程高等代數(shù)近世代數(shù)及線性代數(shù)這三門大學代數(shù)課程的教學研究及實踐的基礎上給出一些教學體會.二、大學代數(shù)課程教學的幾點嘗試與實踐（一）始終不渝地把握四個教學原則1.體現(xiàn)大學代數(shù)學的典型思想方法的原則培養(yǎng)學生系統(tǒng)地掌握代數(shù)研究問題的基本方法是代數(shù)課的教學目的之一.代數(shù)中有代表性的典型思想方法包括：公理化演繹的思想（如：向量空間、歐式空間等各類代數(shù)系統(tǒng)），分類的思想（如：矩陣的相似、合同、等價等等各種等價關系），相互關聯(lián)的思想（如：同態(tài)、同構(gòu)等各種形式的映射），矩陣的方法，初等變換的方法，抽象推理的方法等等.了解這些思想方法的具體含義和在代數(shù)中的具體應用對代數(shù)課程教學是十分有益的.文獻1，4也結(jié)合高等代數(shù)課程的教學體會，詳細地探究了嚴格的邏輯推理方法，公理化方法，結(jié)構(gòu)化方法，矩陣表示方法和等價分類方法等在教學中有效實施.2.體現(xiàn)與時俱進的原則參考國內(nèi)外最新的教材內(nèi)容，結(jié)合我們的教研、科研，把課程的前沿知識、研究現(xiàn)狀和發(fā)展趨勢，及時貫徹到教學過程中，常講常新.例如，我們可以在教學過程中把代數(shù)學家的一些故事、代數(shù)學界最近的研究現(xiàn)狀及所發(fā)生的一些事情帶入到課堂，介紹給學生，以此激發(fā)他們學習數(shù)學的興趣與熱情.3.體現(xiàn)現(xiàn)代教育理念的原則適當安排一些探索性內(nèi)容，擴展性內(nèi)容，構(gòu)建終身學習所需要的代數(shù)學的基礎.將現(xiàn)代化手段在數(shù)學課程教學中的應用將全面鋪開；從教學內(nèi)容的組織與安排看，課堂教學與課外延伸相結(jié)合，將知識傳授、能力培養(yǎng)、素質(zhì)教育融為一體，采用各種形象化的教學手段，使用投影儀和計算機輔助教學，增加教學的直觀性，化解數(shù)學的抽象和難點，促進教學質(zhì)量的提高.4.突出師范教育的特點惠州學院的數(shù)學與應用數(shù)學專業(yè)是師范專業(yè)，而高師數(shù)學專業(yè)培養(yǎng)的目標是中小學數(shù)學教師，我們努力在高等代數(shù)與近世代數(shù)課程的教學之中滲透教育學和數(shù)學課程教學論的思想，注重研究代數(shù)學課程對中學數(shù)學教學的指導，充分體現(xiàn)數(shù)學文化和數(shù)學美，培養(yǎng)學生的數(shù)學文化素養(yǎng)和未來數(shù)學教師的綜合素質(zhì)，適應基礎教育教學和改革的需要.（二）不斷嘗試各種教學理念和方法1.采用“本原教學法”進行教學高度的抽象化和形式化是代數(shù)學的基本特征，高等代數(shù)、近世代數(shù)及線性代數(shù)這三門大學代數(shù)課程是被學生公認為比較難學的數(shù)學課程.所謂“本原教學法”，就是教學中要返璞歸真，從源頭講起，講清楚問題產(chǎn)生和發(fā)展的過程，先講明道理，水到渠成，讓學生自己歸納定義或結(jié)論，再講推理，然后再抽象化和形式化.例如，在引入同構(gòu)概念之前，我們可以先讓學生回憶三角形全等的概念和判定方法.ABC與三角形ABC的全等實際上是建立兩個三角形的頂點和邊的一一對應.點的對應可以看成兩個集合S和T的元素的一一對應，即AA，邊可以看成兩個點所作用的結(jié)果，從而S和T的邊的對應可以是看成保持它們兩個點的運算結(jié)果.這樣一來，兩個代數(shù)系統(tǒng)的同構(gòu)其實就是這兩個代數(shù)系統(tǒng)間可以建立一個一一映射，并且該映射保持這兩個代數(shù)系統(tǒng)的所有運算.再例如在引入向量的線性相關的概念時，我們先從“平面向量的共線”及“空間向量的共面”入手，介紹一些具體的、學生熟悉的例子，最后歸納出線性相關的一般定義.教學實踐證明，這種教學方法學生易于接受，效果明顯.2.采用“研究性教學法”進行教學在自身開展科研的同時，我們經(jīng)常將所授課程的前沿知識，研究現(xiàn)狀和發(fā)展趨勢融入到教學過程中，將自己的研究實踐經(jīng)驗、思維創(chuàng)新方法、學科前沿動態(tài)介紹給學生，并適時適度提出一些問題供學生研究.例如我們在高等代數(shù)或線性代數(shù)課程教學中，可以提出如下問題給學生探究：矩陣表示方法的綜合體現(xiàn)、等價分類方法的滲透與應用、同構(gòu)思想的應用、分析學思想在代數(shù)學中應用等等.此外，我們也偶爾可以不從定義出發(fā)而從問題出發(fā)來組織和展開本課程的教學內(nèi)容和體系，即從重要的問題出發(fā)，根據(jù)需要引入概念，并總結(jié)出定理，引導學生去探索和發(fā)現(xiàn)知識，從而培養(yǎng)學生的創(chuàng)新思維.這一教學過程的主體是學生，主導是教師.3.利用類比法進行各代數(shù)系統(tǒng)相關內(nèi)容的教學類比法是數(shù)學發(fā)現(xiàn)中最常用、最有效的方法之一，它在科學發(fā)展史上起過重大作用.法國數(shù)學家拉普拉斯指出：甚至在數(shù)學里，發(fā)現(xiàn)真理的工具是歸納和類比，這也足以看出類比方法的重要性.類比是通過兩類不同對象A，B間的某些屬性的相似，從而從A具有某種其他屬性便猜想B也有這種屬性. 本科階段主要接觸的代數(shù)系統(tǒng)有向量空間、歐式空間、群、環(huán)和域等.由于這些代數(shù)系統(tǒng)之間具有一些屬性的相似，即都是一些帶有運算的集合，這即表明類比的數(shù)學思想方法可嘗試在這些代數(shù)課程的學習或教學中去運用.例如，我們在講授高等代數(shù)或線性代數(shù)時，可以利用類比法來講解向量空間與歐式空間、矩陣與線性變換的定義與性質(zhì)、聯(lián)系與區(qū)別等等.又例如，我們在講授近世代數(shù)時，可利用類比法來講解群環(huán)域等代數(shù)系統(tǒng)及其子系統(tǒng)的概念，講解代數(shù)系統(tǒng)的同態(tài)基本定理，講解一些特殊環(huán)（整環(huán)、除環(huán)與域）之間關系，講解一些特殊整環(huán)（唯一分解環(huán)、主理想環(huán)、歐氏環(huán)等）的關系等等.教學實踐證明，該方法教學效果明顯，而且可以培養(yǎng)學生如何發(fā)現(xiàn)新問題的科研興趣和能力.4.課堂精講、返講與自學相結(jié)合我們在代數(shù)系列課程的教學中，努力做到課堂精講、返講與自學相結(jié)合.課堂上，講重點，講知識的背景與形成過程，揭示知識的內(nèi)在聯(lián)系；對難點、重點內(nèi)容進行返講，使學生深刻理解抽象的理論，從怕學到愛學；自學是指有些教材內(nèi)容則采用學生自學為主，教師給出思考題，課后下班輔導及答疑.我們采取了一系列措施指導學生自主學習，主要做法是針對不同專業(yè)的學生建立不同層次的試卷庫，建立自測卷，同時，統(tǒng)一考試標準及要求，保證其公正、公平.5.以科技創(chuàng)新活動為突破口，激勵學生研究性學習（1）開發(fā)第二課堂通過講座，介紹代數(shù)發(fā)展歷史上的典型人物、典型事件、典型的思想方法，代數(shù)與相關學科的聯(lián)系、應用前景，提高學習代數(shù)學的興趣.指導學生去發(fā)現(xiàn)實踐中的數(shù)學問題，指導學生使用Matlab分析和解決問題；指導學生自主式學習、探究式學習，給他們布置一些難度不是很大的研究性問題，讓他們課外去找資料解決，并用規(guī)范論文的格式打印出來.這樣，一方面，我們可以讓所有學生學會如何撰寫數(shù)學專業(yè)論文，另一方面，我們也可以讓一部分寫得比較好的學生的論文拿去發(fā)表，從而達到一舉雙得的效果.此外，我們也提倡學生在數(shù)學的認識與實踐、數(shù)學教育學報、大學數(shù)學、高等數(shù)學研究、數(shù)學通報、中學數(shù)學研究等一些專業(yè)涉及知識不深的期刊中找適合自己的文章閱讀、報告和探討.（2）以學科競賽為平臺，提高學生協(xié)同創(chuàng)新能力我們的具體做法有：以全國和國際數(shù)學建模競賽為平臺，培養(yǎng)學生的解決實際問題的創(chuàng)新能力；以全國普通高校信息技術創(chuàng)新活動為載體，培養(yǎng)學生信息技術創(chuàng)新能力.數(shù)學建模對激勵學生學習數(shù)學的積極性、提高學生建立數(shù)學模型和運用計算機技術解決實際問題的綜合能力、推動大學數(shù)學教學內(nèi)容和方法的改革等方面均有重要意義.通過“一年兩賽”模式參加國內(nèi)和國際數(shù)學建模學科競賽，努力提高學生的應用能力與創(chuàng)新能力，提倡“以賽促教，以教育賽”，并將建模融入日常教學中；以數(shù)學建模競賽為切入點，努力培養(yǎng)學生的創(chuàng)新能力.（3）指導學生申報各類大學生科技創(chuàng)新項目，培養(yǎng)學生研究性學習的能力在教師的指導下制定研究課題，鼓勵學生自主申報并研究國家級、省級、校級大學生創(chuàng)新創(chuàng)業(yè)訓練項目、暑寒期社會實踐項目等各項課題，鼓勵學生踴躍向國內(nèi)外專業(yè)期刊投稿，以此來增強學生的科學研究及寫作能力.（4）鼓勵學生參加教師的課題，提高學生以及教師的科研創(chuàng)新能力教師是培養(yǎng)大學生科技創(chuàng)新能力的關鍵因素之一，倡導教師將學生納入自身的科研工作之中，根據(jù)學生的知識階段，指導學生完成力所能及的研究工作，努力提高學生的科研創(chuàng)新能力.三、結(jié)束語本文就高等代數(shù)近世代數(shù)及線性代數(shù)這三門大學代數(shù)課程的教學原則、教學理念、教學方法、教學研究及實踐等方面，給出了一些教學思考與體會.旨在強調(diào)探索和改進傳統(tǒng)的教學模式，不斷滲透數(shù)學思想和方法，對提高教學質(zhì)量，培養(yǎng)和發(fā)展學生數(shù)學思維能力具有非常重要的意義.因此，我們今后需不斷地對大學代數(shù)課程課堂的教學內(nèi)容、模式和方法進行有效改革，使得學生既感興趣地學到必要的數(shù)學知識和數(shù)學技能，又掌握了其中的數(shù)學思想和方法，好為他們將來更好地從事數(shù)學方面的相關工作打下良好的基礎.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 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 o