Dạy học toán cao cấp theo hướng phát triển tư duy phân tích cho sinh viên đại học khối ngành kinh tế, kĩ thuật tt tien anh

MINISTRY OF EDUCATION AND TRAINING VIETNAM NATIONAL INSTITUTE OF EDUCATIONAL SCIENCES NGUYEN THI DUNG TEACHING ADVANCED MATHEMATICS TOWARDS DEVELOPING ANALYTICAL THINKING FOR UNIVERSITY STUDENTS MAJORING IN ECONOMICS AND ENGINEERING SUMMARY OF DOCTORAL THESIS IN EDUCATIONAL SCIENCE Major: Code: Theory and methodology of Mathematics teaching 14 01 11 HANOI, 2020 The thesis is completed at Vietnam National Institute of Educational Sciences Supervisor: Dr Tran Dinh Chau Ass Prof Dr Do Tien Dat Reviewer 1: ………………………………………………………… …………………………………………………………… Reviewer 2: ………………………………………………………… …………………………………………………………… Reviewer 3: ………………………………………………………… …………………………………………………………… This thesis will be reported in front of the Council of thesis evaluation level Institute at the Vietnam National Institute of Educational Science, 101 Tran Hung Dao, Hanoi On…Date………………………………………………………… The thesis can found at - The National Library - The Library of Vietnam National Institute of Educational Sciences INTRODUCTION Reason for choosing the topic  The necessity of analytical thinking development In recent years, learning and developing thinking has been getting more and more attention from researchers, learners and working people who want effective thinking to apply in all aspects of life By referring to the literature on teaching and learning methodologies, teaching and learning theories at universities, etc., we can find certain parts related to analytical thinking, highlighting the importance of analytical thinking for students In fact, analytical thinking requires a clear, detailed and insightful understanding of the problem This is necessary when you want to classify and use information, when self-study to help remembering, judging, choosing to make decisions, and avoiding making mistakes Dividing complex problems into many small problems creates one of the methods to solve the problem For those who work in the engineering industry, they need to be careful, meticulous, pay attention to details, interactions between components in the system of machinery or construction, network system, etc For those who work in economics, they need to collect and select the correct information, categorize, compare, evaluate and explain causes, investigate thoroughly and make judgments, forecasts, etc There have been examples of applying analytical thinking in considering market needs, analyzing financial statements, analyzing causes of failure of a company or a team, etc Clear and logical analytical skills can help each person be more successful in his or her career As pointed out by several recent studies around the world that analytical thinking is necessary for students in the 21st century that will lead to the development of critical thinking, creative thinking and problem solving competence It helps improve thinking, create a habit of thinking and asking questions in every aspect of life Analytical thinking and critical thinking contribute to reducing the unemployment rate of students after graduation Today, there are many domestic and foreign universities that really appreciate the practice of analytical thinking for students In 2009, the Posts and Telecommunications Institute of Technology announced output standards for students, in which students need to have analytical thinking  The role of students’ analytical thinking in teaching Advanced Mathematics Advanced Mathematics consists of a number of subjects during the first academic year of students at their universities For freshmen, most of them are like fish out of water to the new life and methods of learning and research at the university level They will find that the Advanced Mathematics has more abstract contents than those of mathematics at the high schools In order to handle the vast amount of knowledge, they need to know how to self-study, and have a high level of creative and independent thinking Analytical thinking will help them learn the Advanced Mathematics more easily By asking analytical questions, for example, they can orient themselves on the process of thinking and solving problems independently Such orientation helps them to see and state propositions in a number of different ways, making it easier for them to understand the problem and present it more clearly This also helps them analyze and choose how to solve problems, avoiding just sticking in one direction which can lead to deadlock Paying attention to relationships helps them create a habit of linking Advanced Mathematics’s knowledge with the reality and their major, etc Thus, analytical thinking plays an important role for students in studying Advanced Mathematics Advanced Mathematics teaching towards developing student analytical thinking is considered necessary In addition, we believe that in the Advanced Mathematics teaching process, it is necessary to create many opportunities to develop analytical thinking for students, thereby contributing to train them some skills according to the output standards At present, we have not found any comprehensive and systematic research on analytical thinking and Advanced Mathematics teaching towards developing analytical thinking for students Those are the reasons making us decide to choose the topic: “Teaching Advanced Mathematics towards developing analytical thinking for university students majoring in economics and engineering” Within the scope of content and time frame of the subject, we only provide some illustrative examples which have not covered such specific and rich situations like the real life However, from which students can find suggestions in order to partially relate to the specific cases they will encounter later Research objectives Based on theoretical and practical research on analytical thinking and Advanced Mathematics teaching towards developing analytical thinking for university students majoring in economics and engineering, there are a number of proposed measures for teaching Calculs towards developing analytical thinking for university students majoring in economics and engineering Research overview So far, there have been studies on analytical thinking of many authors such as Saccacop, Koliagin, Ayman Amer, Ben Johnson, etc These studies have provided some conceptions of analytical thinking; the importance of analytical thinking; the relationship between analytical thinking and some other types of thinking; some tools and strategies which are often used for analytical thinking As argued by some authors: The use of problems with parameters, graphs, min, max functions, etc., in Calculs teaching can help develop analytical thinking for students Research object and subject - Research object: Advanced Mathematics teaching process for university students majoring in economics and engineering - Research subject: Advanced Mathematics teaching process towards developing analytical thinking for university students majoring in economics and engineering Scientific hypothesis When manifestations of analytical thinking of university students majoring in economics, engineering are identified in Advanced Mathematics teaching, the development and implementation of a number of appropriate measures in teaching Advanced Mathematics in the direction mentioned above can contribute to develop analytical thinking for students while improving the Advanced Mathematics teaching effectiveness Research tasks and scope Research tasks: - Conduct research on theoretical and practical basis of analytical thinking and Advanced Mathematics teaching towards developing analytical thinking for university students majoring in economics and engineering - Identify Advanced Mathematics teaching methods towards developing analytical thinking for university students majoring in economics and engineering - Conduct research on pedagogical experimentation in order to initially test the feasibility and effectiveness of proposed measures Research scope: - The scope of Advanced Mathematics in this research is only referredto Calculus - Experimental will be conducted for freshmen at the Posts and Telecommunications Institute of Technology Research methods Theoretical research, survey, observation, expert method, case study, experience summary, pedagogical experimentation, mathematical statistics are methods to be used Issues that are brought up for defense - Some manifestations of analytical thinking of university students majoring in economics and engineering in Advanced Mathematics teaching - Feasibility and effectiveness of some pedagogical methods in Advanced Mathematics teaching methods towards developing analytical thinking for university students majoring in economics and engineering Contributions of the dissertation - Concepts of analytical thinking and Advanced Mathematics teaching methods towards developing analytical thinking for university students majoring in economics and engineering - Some manifestations of analytical thinking of university students majoring in economics and engineering in Advanced Mathematics teaching - Some pedagogical methods in Advanced Mathematics teaching towards developing analytical thinking for university students majoring in economics and engineering 10 Dissertation structure The dissertation consists of chapters, in addition to the Introduction, Conclusions and recommendations, List of scientific works by authors related to the dissertation, Reference and Appendix CHAPTER 1: THEORETICAL AND PRACTICAL BACKGROUND 1.1 General information about thinking In this section, some of the issues are presented: Concepts of thinking, characteristics of thinking, thinking processes, basic elements of thinking processes, thinking manipulations, and types of thinking Richard Paul and Linda Elder stated that every thinking include these elements: Setting goals, raising questions, using information, using concepts, creating inferences, making assumptions, raising implications and containing a perspective 1.2 Analytical thinking 12.1 Concepts of analysis This section covers some concepts of analysis by Bloom, Sardakov, Chu Cam Tho, etc Bloom gave some common errors in analysis: Failure to see the relationship between the elements and the meaning that makes up the whole; Analysis without quality, without insight, etc 1.2.2 Concept of analytical thinking From the analysis, summarization of some concepts of analytical thinking of the authors Koliagin, Sardakov, etc., we boldly express our opinion of analytical thinking as follows: Analytical thinking is the type of thinking which clearly and thoroughly understand the subject The thinking process takes place based on a close examination of elements of the subject and their relationship to each other, the whole and the external elements Then reflection, judgement and reasonable conclusion are made based on logical background and reasoning 1.2.3 Characteristics of analytical thinking Analytical thinking has a number of following basic characteristics: Preferring analysis when approaching the subject; Frequently raising and answering questions, especially questions which break down problems, explain and reasoning; Examining the subject clearly and thoroughly; Using thinking manipulations, judgments and reasoning; Problem-based method basing on information, evidence and logics 1.2.4 The relationship between analytical thinking and some other types of thinking and problem-solving skill Analytical thinking is related to many other types of thinking (synthesis thinking, critical thinking, creative thinking, logical thinking, etc.) and to the problem-solving competency 1.3 Characteristics of Advanced Mathematics Advanced Mathematics of universities majoring in economics and engineering generally includes Mathematical analysis and Linear Algebra Depending on each discipline, each period, the content of this subject is adjusted by each university management in detail as appropriate The modules of Advanced Mathematics are usually taught to the freshmen For Mathematical analysis, the students majoring in engineering at the Posts and Telecommunications Institute of Technology study Mathematical analysis and Mathematical analysis Students in the field of Economics study Advanced Mathematics with the following main contents: - Mathematical analysis (45 periods): Chapter 1: Set of numbers, limit of a sequence Chapter 2: Differential calculus of single-variable function Chapter 3: Integral calculus Chapter 4: Theory of series - Mathematical analysis (45 periods): Chapter 1: Differential calculus of multiple variable function Chapter 2: Multiple Integrals Chapter 3: Line integrals and surface integrals Chapter 4: Differential equation and systems of differential equations - Advanced Mathematics (30 periods): Chapter 1: Functions and limits Chapter 2: Derivative and differential Chapter 3: Integral calculus Chapter 4: Multiple variable function Chapter 5: Differential equation The Advanced Mathematics teaching process does not go into proving math problems but focuses on creating foundation knowledge so that students can study specialized subjects and apply a part of that knowledge in the future Although there are a number of theorems not proven in the subject content, students are still required to understand the concepts, theorems and their practical applications The Advanced Mathematics has more abstract contents than those of mathematics at the high schools The content of knowledge in the program is systematic, rich and profound The students must acquire a lot of knowledge during an hour of study, understand the nature, think independently and learn more than they did when studying maths in high schools There are many situations in Advanced Mathematics teaching suitable for practicing logical reasoning and reasoning; helping to express clear, careful and insightful thoughts; finding out any relationships to reality or specialized issues 1.4 Some characteristics of university students majoring in economics and engineering In this section, we will present some characteristics of thinking development of university students compared to that of high school students, especially independence and creativity; as well as some difficulties for the freshmen We also point out some characteristics of people working in economics and engineering sectors (and many students who love these fields may have some of the same); some characteristics of thinking that university students majoring in economics and engineering need to practice during their study process to contribute to the development of career skills 1.5 Manifestations of analytical thinking of university students majoring in economics and engineering in Advanced Mathematics teaching For the purpose of identifying manifestations of analytical thinking of university students majoring in economics and engineering in Advanced Mathematics teaching, we rely on: - Basic elements of thinking: Setting goals, raising questions, using information, using concepts, creating inferences, making assumptions, raising implications and containing a perspective (As provided in the chapter 1.1.3, according to Richard Paul and Linda Elder) - Concepts and basic characteristics of analytical thinking: Thinking clearly and deeply; dividing subjects, finding out any relationships, making judgments, deducing, making logical arguments, etc - Characteristics of Advanced Mathematics; characteristics of university students majoring in economics and engineering; activities of university students majoring in economics and engineering when studying Advanced Mathematics For examples: The students need to determine the direction and choose the appropriate solution for the problem; understand and use the meaning of mathematical concepts to model mathematics in solving practical problems (or specialized problems); select, organize, classify information during the course of scientific research or problem presentation, etc These manifestations are also considered through activities that show the connection between analytical thinking and some other types of thinking and with the problem-solving skill and activities towards contributing to training vocational skills for students Since then, we believe that some typical manifestations of analytical thinking those university students in economics and engineering in Advanced Mathematics teaching should have are: 1.5 Determining the purpose of thinking, breaking down the goal The students should identify goals when solving problems (when solving problems, reading textbooks, etc.), then break that goal down into many meaningful goals, and regularly adjust their thinking to that goal 1.5.2 Determining the direction of view The students need to determine the direction when solving problems When solving problems, they need to divide a number of solutions and prilimilarily envisage a number of steps for each direction, as well as advantages and disadvantages of each direction When reading the material, they recognize which the direction and views the author presents They may explain a problem or re-state the proposition in a different way, looking at some of the problems in a way that is relevant to the reality or their major When understanding information, they review from multiple sources They realize structure of a document, etc For example, when studying about definite integrals: b  a n f ( x)dx  lim maxxi 0 ( x1 , x2 , , xn  f (i )xi i 1 is the arbitrary points on  a, b  ,  i is any point on  xi 1, xi  , xi  xi  xi1, i  1, n) The students majoring in engineering can relate to problems involving area and volume calculations and they have to consider xi (i  1, n) very small The students majoring in economics can relate to functions and variables that are commonly used in economics and they consider xi to be at least 1.5.3 Finding out information clearly and deeply The information may be used to solve the problem The students should explore information (assumptions of problems, facts, etc.) in a detailed, clear and insightful manner; with information interpretation; investigate any connection between information with each other and with the problem to be solved; organize, classify and synthesize information; recognize the correct information, wrong information, important information, missing information; correct and supplement information as well They can learn information and materials related to Advanced Mathematics and apply it in practice and specialized subjects Use relevant information to address the issue 1.5.4 Understanding concepts (theorems, propositions, rules, methods) clearly and deeply and using them to solve problems The students can interpret and present clear and detailed concepts (theorems, etc.); identify any relationships between concepts (theorems, etc.) with each other and with the problems to be solved, problems in the reality or their major It is possible to use concepts (theorems, etc.) to solve related problems, make some meaningful comments in the process of in-depth study of concepts (theorems, etc.) 1.5.5 Making a grounded judgment, in relation to the question posed The students need to make judgments related to the problem posed (predict a solution, give a hypothesis, guess the author's ideas, etc.) based on following evidences: consideration of relationships, use of information and thinking manipulations: specialization, generalization, analogization, overturning, etc Some judgments can be made for solving practical or discipline-related issues They often have to review judgments carefully 1.5.6 Deducing clearly and thoroughly, step by step, based on grounds related to the issue Students should make clear inference step by step Such inference is based on grounds and evidence, make in-depth inference through a number of several steps and directions They often make logical, meaningful conclusions that relate to each other and either practical or major-related issues They often flip back and forth the problem, scrutinize the whole process of inference and draw comments and lessons 1.5.7 Raising analytical questions The students often raise questions that require detailed explanations, identification of relationships (the relationship between the elements with each other and with the problem to be solved, with the reality or majors); pose questions about purpose, questions about vision, information, bases, concepts, theorems, propositions, experiences, related solved problems; questions about judgment and reasoning; questions about clarity, deep thinking, etc These questions are often broken down  Analytical thinking of students is manifested through the process of carrying out activities To give the manifestation degrees, we rely on consideration that the students can perform in the manifestation, as well as the hierarchy of activities as introduced by Nguyen Ba Kim The hierarchy is applied based on the following grounds: complex nature of the operating object; abstract and general nature of the object; contents of any activity; activity complexity; activity quality; coordination in many aspects serve the basis for activity hierarchy We propose the following levels: Level 1: Students can partly perform requirements of each above-mentioned manifestation with lecture suggestion; Level Students can partly perform requirements of each above-mentioned manifestation without lecture suggestion; Level 3: Students have above-mentioned manifestation without lecture suggestion In addition, at each level, under appropriate circumstances, smaller levels may be considered based on the complexity ofoperating object; abstraction and generalization of the object, content of the activity or complexity of the activity For example: When applying the Maclaurin expansion of the function to find the limit, the lecturer states that:  x3  x  x   o(x )   3! x  sin x lim  lim (1)  x  x (1  cos x ) x 0    x  x 1  1   o(x ) 2!    x3  o(x ) (2)  lim x 0 x  o(x ) x3  lim  (3) x 0 x 3 and raises questions to the students: “Do you have any question? Whether you can explain the solution above or not?” - Student A: Why you that? When that question is commented by the lecturer as too general, students should ask more specific questions in the part that is not understood For example, at step (1), it is necessary to compare two sides, and ask: Have you replaced sin x with x3 x  o(x ) ?”, Student A continues to ask: In step (1), have you replaced cos x with 3! x2 1  o(x ) ? 2! x3  o(x ) ? - Student B: At step (2), why don't we write lim 36 x 0 x  x o(x ) From (2) to (3) have we used higher order infinitesimals omission rule? Student C: “I think that in order to understand this solution and work it out when doing another task, we need to answer the following questions: + Whether it is x o(x )  o(x ) or not? Is o(x ) also is o(x ) or not? + From (2) to (3) we have removed some terms, have we used higher order infinitesimals omission rule? x3 + Why we write sin x  x   o(x ) in stead of sin x  x  o(x ) 3! x x4 cos x     o(x )? What happens if we write Similarly, why don't we write 2! ! like that? In addition, to solve this problem on our own, we must first ask and answer the questions: What knowledge does this exercise involve? It is related to Maclaurin development and limit, but Maclaurin development is related to infinitesimal So, are there any properties and comments related to the infinitesimal and limit parts? For examples: Replace equivalent infinitesimals, use the higher order infinitesimals omission x3  o(x ) rule, etc., in addition, we can ask that: If we need to find out lim 36 , can the x 0 x  x o(x ) computer it, and we need to pay attention to the typing of this limit search command? In the above case, student A shows a better ability to ask questions than student B 11 generally, the university students in economics, engineering currently show little analytical thinking Many of them are not active in thinking; are not willing to find out the details of details about concepts and theorems; fail to state the meaning of the concept to solve practical problems or specialized problems They not pay much attention to finding out and correcting any false information; as well as raising incorrect questions, or general questions Some students make judgments or conclusions without any solid basis In general, they rarely use comparative, similar and special manipulations in making any judgments Many of them fail to know how to use diagrams to make the deduction process become clearer and more coherent They not create a habit of reviewing, anticipating a few solutions to the problem and analyzing any advantages and disadvantages of those solutions Few students make a habit of drawing any comments or lessons for themselves, etc Although each lecturer has a true awareness of the importance of analytical thinking and has taught Advanced Mathematics towards developing analytical thinking for students, he or she still fails to take sufficient measures and specific methods in the teaching process As the lecturers have not yet fully envisioned necessary manifestations of analytical thinking, they fail to select many suitable examples, design activities corresponding to the manifestations, and provide their students with methods often used for analytical thinking CONCLUSION OF CHAPTER The above theoretical and practical studies help us draw some conclusions: Analytical thinking has general characteristics of thinking, which requires analytical manifestations and division into small parts as well as finding out any relationship The analysis process requires insight From the basic ideas above, it is possible to conceive that: Analytical thinking is the type of clear and deep thinking about understanding the subject The thinking process takes place on the basis of full consideration of parts of the subject and any relationship between them, with the whole and the external elements From this point, it is possible to think, judge, draw logical conclusions based on logical bases and reasoning The concept of analytical thinking mentioned above shares something in common with some conceptions given by other authors and is consistent with the students’ thinking in the course of studying Advanced Mathematics and analyzing some problems in the life, as well as studying some specialized subjects Analytical thinking is closely associated with the general thinking, creative thinking and problem solving Based on basic elements of thinking (Setting goals, raising questions, using information, using concepts, creating inferences, making assumptions, giving rise to implications and containing a perspective) combined with the characteristics of analytical thinking (breaking down, finding out any relationships, clarity, insight, etc.) and the characteristics of Advanced Mathematics, students’ activities majoring in economics, engineering when studying Advanced Mathematics, we are able to determine manifestations of analytical thinking of university students majoring in economics and engineering when studying as follows: Determining the purpose of thinking, breaking down the goal; Determining the direction of view; Finding out information clearly and 12 deeply to solve problems; Understanding concepts (theorems, propositions, rules, methods) clearly and deeply and using them to solve problems; Making a sound judgment, in relation to the question posed; Deducing clearly and deeply, in steps, based on grounds related to the issue; Raising analytical questions Analytical thinking of students is manifested through the process of implementing activities Based on the hierarchy of activities and any parts shown in the requirements of the stated manifestation, we provide a number of levels of analytical manifestation with illustrative examples in teaching Advanced Mathematics Level 1: Students can show a part of requirements of each of the above manifestation but need the lecturer's suggestion Level 2: Students can show a part of requirements of each of the above manifestation without the lecturer's suggestion Level 3: Students can show the above manifestation without the lecturer's suggestion In addition, for each of the aforementioned levels, under appropriate circumstances, smaller levels may be considered based on the complexity of the operating object; the abstraction and generalization of the object, the content of the activity or the complexity of the activity By referring to some concepts related to the development of thinking , students’ thinking in teaching, analytical thinking for students, we argue that teaching Advanced Mathematics towards developing analytical thinking for university students majoring in economics and engineering is understood as the type of teaching that help students to improve their analytical thinking to higher levels, on the basis of providing the students with a combination of solid foundations of knowledge and activities that are appropriate for them In term of teaching in this direction, the lecturers are recommended relying on the characteristic manifestations of analytical thinking of students in Advanced Mathematics teaching to select any contents, teaching methods, lesson design in a sequence of activities so that the product produced by students performing those activities represents their level of analytical thinking From results obtained in the course of surveying the current situation, comparing with necessary manifestations of analytical thinking of students in teaching Advanced Mathematics and concepts of teaching Advanced Mathematics towards developing analytical thinking for students, it is found that analytical thinking of students is limited Many lecturers fail to identify the manifestations of analytical thinking of students in Advanced Mathematics, making them unable to select the appropriate contents and methods in order to develop the students’ analytical thinking Research results on theoretical and practical basis above are considered the basis for proposing a number of Advanced Mathematics teaching methods towards developing the analytical thinking for students CHAPTER 2: ADVANCED MATHEMATICS TEACHING METHODS TOWARDS DEVELOPING ANALYTICAL THINKING FOR UNIVERSITY STUDENTS MAJORING IN ECONOMICS AND ENGINEERING 2.1 Orientations to determine the methods There are some orientations to determine the methods: The methods given should be designed to ensure the principle of university teaching, affect the development of analytical thinking, and must be feasible 2.2 Advanced Mathematics teaching methods towards developing analytical 13 thinking for university students majoring in economics and engineering 2.2 Method 1: To enhance activities about dividing, understanding each part and finding out relationship to be implemented by students a) Purpose This method is aimed at helping students form habits and be able to successfully perform activities of dividing, understanding each part and finding out relationship during the course of using information, raising questions, identifying directions, judging, making inferences, solving problem; relating Advanced Mathematics knowledge to reality or specialized issues b) Basis of method Dividing and finding out any relationships are characteristic of analytical thinking and included in the analytical manifestations of students but currently they have not well performed these activities c) How to apply - The lecturer shows students that division depends on the characteristics of the object, purpose of the activity and each specific case The students must look at the whole at first to choose how to divide and interpret each part They are required to understand the meaning of words, phrases, explain each part, express in many ways, illustrate or give examples, and make any comments - The lecturer may give the following questions: Why? How is it understood? What does it mean? What's your opinion? Please analyze - The lecturer should design situations that require students to go deeply into details - Exercises which are changed in terms of some properties of concepts and properties may be used - Exercises containing parameters may be given, because this type of exercise requires students to divide the solution into cases and provide appropriate arguments for each case In terms of pointing out the relationship, according to the Textbook of Basic Principles of Marxism-Leninism, the relationship in dialectics is used to refer to the regulation, the interaction and the mutual transformation between things, phenomena, or between sides, elements of each thing, phenomena in the world - Pointing out the relationship between elements is a large-scale problem, requiring students to pay attention to the situations and contents they have learned When solving problems, they are required to relate to many relationships, then screen and select any relationship that are appropriate for the situation The relationship between A and B can be manifested in many forms, for example: A  B (2) + A  B (1) + A is the condition to have B (3) + A combines with C to obtain B (4) + A combines with B to obtain X (5) + A, B combine with D to obtain Y (6) + B is property of A (7) + A and B have the same properties C (8) + For finding (proving) A you have to find (prove) B (9) + A and B are bound by a certain expression or rule (10) + A, B and C are bound by a certain expression or rule (11) (A, B can be clauses, formulas, etc.) 14 The relationships (4), (5), (6) often appear in the course of solving problems, at which considering these relationships is the use of synthesized operations Identifying the relationship (3) facilitates the process of ordering when you want to present, understand the structure and make logical arguments -Students need to perform activities related to classification, systematization, document arrangement (finding logical relationships, structural relationships) When analyzing and teaching mathematics, people often pay attention to some relationships such as "To find (prove) A, we must find (prove) B", causal relationship, relationship about the common - the private, - For the purpose of clearly seeing the relationship such as "To find (prove) A, we must find (prove) B", students should use the forward and reverse diagrams -In order to easily identify the causal relationships, in Advanced Mathematics, students need to pay attention to the relationship    A1  A1     A  B ; A  B1   Bn  B; A  B1   Bn  B;     B; A    ,     An    An   Thus, they are required to understand, memorize and correctly use the concepts, clauses, theorems, logical laws; know how to combine various theorems to deduce in many steps It also helps students make better deductive inferences When applying the common-the private relationship, students are recommended asking and answering the following questions: What A and B share in common? What attributes (characteristics, etc.) are common in these objects? Which objects have the same properties (characteristics, etc.) with object A? What is the particular (special) case of this? Is this content a particular (special) case of what? What would be more general results? If we want to know if the predictions about the common is true or not, we can try specialization If the specialization operation produces any false results, the hypothesis (prediction) is eliminated If the specialization operation produces any true results, we find a proof When provable, it is possible to state in the general case After that, another common thing can be predicted based on the analogization operation Finding the relationship between the common and the private leads to similar inferences and inductive inferences Because the common appears in many private things, but it may not be in every individual one, it is necessary to predict which objects have the same in common This involves the use of comparative and similar operations (If the object X has properties A, B, C, D and the object Y has properties A, B, C then Y also has property D) In addition, it is also necessary to use the abstract manipulation when considering certain common properties in the private without regard to other properties In some cases, when predicting the common, it is possible to analyze the private into elements, view each element in many directions, or make any general statement about it Summary: * Breaking down, understanding each part and finding out relationships help students understand information, concepts, theorems; consider in a number of directions; break down the goal; make clear, insightful inferences; make sound judgments; provide 15 logical arguments; and help them remember and solve problems * In order to help students identify any relationship, the lecturers should suggest and create opportunities for them to regularly engage in activities: - Pointing out some common relationships, selecting any appropriate ones in specific situations - Sorting and classifying (finding out the relationship between ideas, the structural relationships) - Remembering and correctly using concepts, clauses, theorems, logical laws; combining theorems to find out any transitive relationships - Answering the questions: "Why ?", "What will lead to this?", "What will this entail?", "What we look for to find A?", etc - Paying much attention to find the common, similar things, differences between the knowledge contents Combining comparative, analogy, specialization, generalization, abstraction manipulations (to detect and manipulate the common-private relationship, leading to similar inference, inductive inference, knowledge creation) 2.2.2 Method 2: To design, organize activities that demonstrate a clear and insightful way of thinking for students a) Purpose This is aimed at helping students think clearly and deeply when defining goals and objectives; determining the perspective; using information; raising questions; using concepts (theorems, propositions, rules, methods), judgments and reasoning; relating to the reality or the major b) Basis of method - Thinking clearly and deeply is a characteristic of analytical thinking, included in requirements of the manifestations - This affects the awareness of students' thinking process, thereby developing thinking in general and analytical thinking in particular c) How to apply The lecturers should make suggestions and create opportunities for students to carry out the following activities: + Write summary contents, write detailed contents, detect errors and correct, compare, classify + Present, explain the problem clearly, answer questions "why?, use illustrative examples and counter-examples + Look back to see the knowledge which is still unknown, record and think more or ask others + Constantly supplement newly discovered aspects to the existing knowledge system and adjust the thoughts + Pay attention to the gaps of knowledge not clarified or mentioned in the materials to think and find out any comments and conclusions + Search for many elements that are related to a problem + Take full consideration a problem, see it in different ways; make transitive inference in many steps; Complete, supplement, expand or relate other issues; Dissatisfy with the general answers but seek any deeper ones It is possible to find out many problems, relationships related to an element under consideration; make comments, pay full attention, put in additional clauses, and think deeply about the problem 16 (It should be suggested by the lecturers that these comments are often derived from the course of generalizing, specializing, similarizing, comparing, viewing counterpropositions, paying attention to mistakes, etc.) 2.2.3 Method 3: To provide students with some common methods for analytical thinking Some commonly used methods for analytical thinking include: - Understanding the subject with full consideration: Whole-Part-Whole - Using the question Answering a question with a question - Using types of diagrams, graphs, outline - Using SQR4 reading strategy" a) Purpose This method is intended to help students to use analytical thinking methodically; limit common mistakes in the analysis process; orientation, stimulating thinking; make the thinking process clear, coherent, deeper and more effective b) Basis of method Several studies have shown that the above methods are aimed at providing students with the strategies, techniques, and tools often used for analytical thinking, thus helping them to develop analytical thinking However, many students still not know how to use these methods c) How to apply - Understanding the subject with full consideration of Whole-Part-Whole + The lecturers should suggest to students the "Whole-Part-Whole strategy", and take specific examples In the process of solving math problems, at the first step when considering the whole, students need to understand all the factors and relationships in a rough way, visualize the relevant knowledge area to determine the analytical direction (However, the effectiveness at this step is highly dependent on whether students often solve math problems or not, because this will help them gain experience and have a better feeling when determining the analytical direction) The lecturers should also use this strategy in presenting their lessons + They may build forms of exercises that include many ideas, in which the ideas have little change but require students to solve the problem in a different direction - Using the question, answering a question with a question + The lecturers should ask students to learn about the importance of asking questions, answering a question with a question + The lecturers may show students the importance of using questions and answering a question with a question: guide the thinking process, stimulate thinking, etc It is possible to pose any split questions, explain and find out any relationships, raise questions about purpose, information, concepts (theorems, rules, methods), perspective, logical reasoning, judgment, deduction, clarity, depth, etc + The lecturers may suggest several types of questions, such as: Questions when reading theory; Question when solving problems; Questions after solving problems + The lecturers often pose questions, answer students' questions with a question and suggest them to the same - Using types of diagrams, graphs, outline: + The students are suggested by their lecturers studying about some types of diagrams themselves: mind map; conceptual map; forward diagram, reverse diagram 17 The lecturer then gives students additional hints about the above diagrams (usage, advantages, etc.) and provides illustrative examples during the teaching process (or gives documents to students for reference when studying at home) + They may design exercises and situations for students to perform activities that use diagrams and graphs: Read the diagram, comment on a diagram; draw a conceptual map showing the relationship between concepts; draw mind maps when summarizing lessons, indicating problem solutions; organizing, classifying and using information and planning; draw any shape; use information technology when drawing any shapes - Using SQR4 reading strategy: + The students are suggested by their lecturer studying about SQR4 reading strategy by themselves + The lecturers give students additional suggestions for SQR4 reading strategy and suggest using it when reading materials at home, which can be in the form of exercises, for example: "Show SQR4 reading strategy when you read the surface integral type two" + During the class teaching, it is possible for the lecturers to check some steps in this reading strategy, for example: "When you read the surface integral type two and use the SQR4 strategy, in step (survey), what key points you need to cover?” 2.2.4 Method 4: To design, organize activities that combine analysis with synthesis, creativity and problem-solving skills a) Purpose This is aimed at helping students often use analytical thinking combined with other types of thinking, in addition to using thinking types in a more flexible and effective manner b) Basis of method - In the teaching process, in addition to developing analytical thinking, it is necessary to develop other types of thinking, as well as the problem-solving skills for students - Analytical thinking is closely associated with the general thinking, creative thinking and problem solving - The integrated power may be created from the combination of training many types of thinking c) How to apply - Combination of analysis and synthesis: Students should be asked to summarize, prepare an outline, point out basic steps in a solution, and briefly such steps after analysis Exercises should be provided in a way that combines a variety of knowledge - Combination with creative thinking: The lecturers should encourage students to imagine what is about to read (or listen to) and analyze after reading (together with comparison) The lecturers suggest to students that: In oder to develop analytical and creative thinking, they should know how to use phrases such as: "I have a hunch on this topic ", "I can imagine it working in this way if ", "This reminds me of when I ", "I still wonder about the question about ”, “When looking at the whole thing, I think the key point here is…”; Using exercises with many solutions, silent exercises, open exercises, other exercises, etc., is also recommended - Combination of analysis and problem solving skills: The lecturers can introduce and let students solve problems in four steps of Polya, with close attention to 18 requirements on the manifestation of analytical thinking in each solution step 2.2 Method 5: To strengthening the use of analytical thinking during the selfstudy process a) Purpose This method helps students train their ability to self-study and develop analytical thinking by regularly conducting thinking activities on their own b) Implementation basis Self-study plays an important role in the students’ study process Self-study, especially when reading and solving problems, often involves elements of analytical thinking However, it is shown by the reality that students' ability to self-study is very limited In addition, as the detailed analysis takes time, it cannot only be done in the classroom, but need to focus on the student's self-study process Moreover, it is necessary for students to have a high level of independent thinking, so there is no need for the lecturers to analyze problems like those at high school level Therefore, the students must analyze themselves during the self-study process in order to understand knowledge c) How to apply In teaching Advanced Mathematics in this direction, the lecturer may: - Provide brief introduction for students to understand more about self-study methods at university - Develop a system of questions and exercises using analytical thinking and assigning students to complete by self-study The assignments must be appropriate for their qualifications, with higher education and consistence to the above methods Attention should be paid to some forms of exercises suitable for being done at home For examples: Drawing specific comments and comments (after reading, completing assignments, etc.); Drawing maps; Making classification, summary, etc.; Presenting how to solve the problem in four steps of Polya; Large, thematic exercises; computer-based exercises; practical or specialized application exercises, case studies; Using SQR4 reading technique, etc - Asking students to self-study to raise their own questions and exercises (suggest them how to ask these types of questions) - It is possible for students to study in groups at home and present, discuss in their class - Examining more frequently (including checking new lessons before class and old ones after studying) is required The test is not exactly the same as the exercise already in the learning materials, with more or less changes to require students to have inference - It is possible to use KWL teaching techniques CONCLUSIONS OF CHAPTER In this chapter, we have elaborated five methods based on the principles of university teaching and examining manifestations of students' analytical thinking when studying Advanced Mathematics, combined with findings in theoretical and practical basis, including: To design, organize activities that combine analysis with synthesis, creativity and problem-solving skills; To design, organize activities that demonstrate a clear and insightful way of thinking for students; To provide students with some common methods for analytical thinking; To design, organize activities that combine analysis with 19 synthesis, creativity and problem-solving skills; To strengthening the use of analytical thinking during the self-study process For method 1, the students carry out activities of division, interpretation, understanding each detail, finding out any relationships, which will help them break down their goals, understand information, use concepts , theorem, etc., better; make judgments based on the consideration of the relationship of the common and the private and perform operations of similar thinking, generalization, specialization; make clear and insightful reasoning, and logical reasoning based on answering questions about cause and effect relationships They are trained in sorting and classification activities based on subdivision and finding logical relationships, structural relationships, etc Examining relationships in many aspects helps them see problems in many ways Method is directed at the requirement of "clarity and insight" in the analytical thinking manifestations of students It aims to influence students' awareness of thinking process, emphasizing the making of their own comments, attentions and experiences That helps them develop thinking in general and analytical thinking in particular Method helps students to think more methodically in the analysis process The "Whole-Part-Whole Strategy" helps them localize their knowledge to predict the appropriate analytical direction, with close attention to the whole during the analysis Answering a question with a question helps them navigate the thinking process and find the answer on their own The use of types of diagrams helps students identify relationships, and make clear and coherent inferences The use of SQR4 reading technique helps students identify main ideas, ask questions, think, memorize, etc Method focuses on the development of some other types of thinking concurrently with analytical thinking, and application of analytical thinking when solving problems It helps students become familiar with issues that requires the synthesis of a variety of knowledge Imagination before analysis helps limit the error of "analytical thinking that hinders creative thinking" Method emphasizes the development of analytical thinking activities for students during self-study at home, because there is not enough time to conduct these activities regularly in class, moreover, students need to practice independent thinking In addition, this method makes teachers pay more attention to the types of exercises that are only suitable for students to study at home, practical or specialized exercises, scientific research exercises The division of the above method is only relative to help understand and implement it easier, because the implementation of one method can simultaneously help to implement another For example, when students conduct activities to understand the problem clearly and deeply, it is also necessary to split, interpret and find any relationships, etc When students take steps to solve problems, they also analyze problem solving and think deeply The above methods help students have a method of reading, recognizing the main ideas, memorizing, so they can grasp knowledge better Thus, we think that the implementation of such methods can contribute to the development of analytical thinking for students while improving the effectiveness of Advanced Mathematics teaching In addition, the way to present illustrative examples has shown the use of some positive teaching methods and techniques, for example: Case studies, diagram techniques, "KWL" technique, etc These are results we have not found in previous studies of 20 Advanced Mathematics teaching Based on the above methods, we will design lectures, organize teaching, and evaluate the development of analytical students' thinking in order to test how difficult they are to implement and whether the use of such methods really develop analytical thinking for students or not CHAPTER 3: PEDAGOGICAL EXPERIMENT 3.1 Purpose, content and organization 3.1 Experimental purposes Pedagogical experiments are conducted for the purpose of: - Test scientific hypothesis of the thesis - Initially assess the feasibility and effectiveness of proposed methods 3.1 Experimental content 3.1.2.1 Subjects participating in the experiment The experiment was conducted at the Posts and Telecommunications Institute of Technology in two sessions Session (for university students majoring in engineering): Experimental period: From February 2018 to May 2018 We compared results between experimental class (TN1) and control class (DC1) Subject: Mathematical analysis - Class TN1: Group 14 (65 students) Lecturer: Nguyen Thi Dung - Class DC 1: Group 13 (62 students) Lecturer: Nguyen Thi Dung Groups 13 and 14 have similar knowledge and qualifications, entrance examination scores, and nearly equal number of students Session (For university students majoring in economics): Experimental period: From early September 2018 to mid-October 2018 We compared results between experimental class (TN2) and control class (DC2) Subject: Mathematical analysis - Class TN2: Class D18 QT 3, (90 students) Lecturer: Nguyen Kieu Linh - Class DC2: Classes D18QT1, (92 students) Lecturer: Nguyen Kieu Linh In addition, we selected students to monitor the development of analytical thinking during the experiment, that is, students in group 14 studying Mathematical analysis With the experimental classes, the lecturers followed methods proposed herein In the control classes, the lecturers taught normally as before, without the impact of experimentation In the course of experimental teaching, we attended the lessons as well After the lesson, we held discussions, assessments and lessons learned At the same time, we polled from the students to make timely adjustments to achieve the purpose of pedagogical experiment 3.1.2.2 Experimental teaching program A For university students majoring in economics - Advanced Mathematics 1: Basic concepts of functions, Limit of function, Continuous functions, Derivative of functions, Differentials of functions, Derivative and high-level differentials, Average value theorems, Some applications of derivatives 21 - Examination and evaluation: The examination content corresponds to the above lessons: (60 minutes) B For university students majoring in engineering: - Mathematical analysis 2: The lessons in Mathematical analysis are for students majoring in engineering at the Posts and Telecommunications Institute of Technology - Examination and evaluation: Contents: Multivariate function, multiple integral, line integral type two: (50 minutes) 3.1.2.3 Experimental lesson plans We present two illustrative lesson plans: "Line integral type two" for university students majoring in engineering "Derivative" for university students majoring in economics 3.2 Evaluation of experimental results 3.2 Qualitative aspect For the lecturers: By exchanging with the lecturers and observing and evaluating from the teaching hours in the experimental classes, it is found that: - Lecturers participating in the experimental teaching have understood and used pedagogical methods, strictly implementing ideas set out in the lesson - They believe that teaching by the methods mentioned above makes their lessons livelier, enable their students to understand the lesson and feel easier to memorize, while the lecturers know how and build a similar example system by themselves For the students and classroom environment In general, the atmosphere of the experimental class is quite exciting when students seem to love the subject and are excited about examples and exercises Students in the experimental classes also performed better than those in the control class (with more progress than themselves) in the ability to analyze errors and correct, explain in detail, only make connections, identify perspectives, see things in a number of ways, infer in forward and reverse aspects, understand the nature and make personal conclusions, etc For case studies, we randomly selected four students in the experimental class and supervise their progress Student Nguyen Xuan A, class D17CQCN12 During the teaching process, we followed A’s expressions in a number of ways: Learning and using information, making judgments, determining the purpose of thinking Student A showed certain his or her progress Student Nguyen Duc B, class D17CQCN7 We followed B’s expressions in a number of ways: Understanding the concepts, theorems, rules; Determining the perspective Student B showed a lot of progress Student Dong Thi Thu C, class D17CQCN8 We followed C’s expressions in a number of ways: Understanding the concepts and theorems; Asking analytical questions Student C showed her progress little by little Student Chu Quoc D, class D17CQCN5 During the teaching process, we followed D’s expressions in a number of ways: Making judgments, making inferences, determine the perspective Student D showed a lot of progress 22 3.2.2 Quantitative aspect The quantitative aspect is mainly evaluated based on the results of tests These tests were shared for the experimental and control classes 3.2.2.1 Evaluation of results of the first experimental session Before the experiment, we required the students in classes TN1 and DC1 to the test (25 minutes) It is shown in the results that the two groups have similar qualifications and many limitations in analytical thinking For example, they did not delve into concepts and theorems; failed to make arguments based on evidence or evidence; without thinking clearly step by step when encountering complex problems After the experiment, we required the students in classes TN1, DC1 to the test The answers and scores are presented in the Appendix The scores show that students generally scoring or above will have better analytical thinking than those with lower scores Therefore, comparing the analytical thinking of students in the two classes may be based on a comparison of percentage of students scoring at a score greater than or equal to *The test results indicate that: - The average score in class TN1 (6.0) is higher than that of class DC1 (5.2) The percentage of students with poor scores (3 and below) in the class TN is lower than that of the class DC Proportion of students achieving weak and moderate scores (4  6) in class TN1 is higher than that of class DC1 - The percentage of students with good scores (grades or higher) in class TN1 is higher than that of the class DC In addition, when referring to the final exam results, (General test based on the questions for all classes throughout the Institute), we get the following results (from the Registrar's Office): The average scores in the class TN1 class and class DC1 are 5.86 and 5.0, respectively 3.2.2.2 Evaluation of results of the second experimental session After the experiment, we required the students in classes DC2 and TN2 to the test The answers and scores are presented in the Appendix The scores show that students generally scoring or above will have better analytical thinking than those with lower scores Therefore, comparing the analytical thinking of students in the two classes may be based on a comparison of percentage of students scoring at a score greater than or equal to * Test results: The number of students scoring or higher in the class TN2 is higher than that in the class DC2 50 Lớp TN2 Lớp ĐC2 40 30 20 10 Frequency chart of scores for classes TN2, DC2 In addition, when summarizing questions students posed in the test, we found 23 that many students asked questions that were incorrect, irrelevant, unrelated to the content or unclear These questions show that they failed to understand the content of the lesson, show carefulness, and make grounded arguments Students in the class DC2 had more unsatisfactory question than those in the class TN2 CONCLUSIONS OF CHAPTER Through the process of conducting experiments above, we draw some conclusions as follows: It is possible to carry out solutions described in chapter The lecturers who participated in the experimental process thought that understanding of analytical thinking, pedagogies, and illustrative examples helped them better understand requirements of analytical thinking Thus, they can find it easier when developed examples and exercises, organizing situations in teaching Advanced Mathematics towards developing analytical thinking for students The solutions emphasize student self-study and the practice of analytical thinking exercises at home, teachers have enough time to teach in the classroom Although teaching in this direction requires research and more time for preparing lessons, it helps the lesson somewhat livelier Most of these opinions are similar to our comments when directly teaching and observing the activities of lecturers in the experimental classes The solutions have contributed to improve the efficiency of Advanced Mathematics learning and development of analytical thinking for students, expressed through a number of manifestations: Students pay more attention to learning in details of concepts and theorems; draw out the meaning of that concept and theorem; find out any relationship between the concepts and theorems; make predictions on the basis of considering the relationship and using thinking manipulations They can show better ability to infer in forward and reverse aspects Initially, they know to draw some comments, attention; limit some mistakes; offer some solutions and choose the direction briefly; ask better analytical questions (extensive questions, specific questions, reverse analysis questions, deep thought questions), etc They can use some commonly used methods of analytical thinking without the lecturer's suggestion The most difficult problem in teaching in this way is that students' self-study in general has not been well done, with little progress made, while some students not focus on listening to the lectures and not have the spirit of volunteering to their homework despite being reminded Therefore, the students’ manifestation of analytical thinking development is faint or even absent With some comments above, we initially assume that the scientific hypothesis of the dissertation has been tested Teaching according to the above-mentioned methods is possible, contributing to improving the effectiveness of teaching Advanced Mathematics and developing analytical thinking for students The development is shown more clearly for those who are conscious of studying hard 24 CONCLUSIONS AND RECOMMENDATIONS Conclusions Thank to conducting the research, we have gathered following main results: - Systematize the basic theories on thinking, analytical thinking - Propose concept on analytical thinking, teaching Advanced Mathematics towards developing analytical thinking for university students majoring in economics and engineering - Find out some manifestations of analytical thinking of university students majoring in economics and engineering in Advanced Mathematics teaching - Investigate, analyze and draw conclusions about the current situation of analytical thinking of students majoring in engineering and economics and Advanced Mathematics teaching towards developing analytical thinking for university students majoring in economics and engineering - Propose solutions for teaching Advanced Mathematics towards developing analytical thinking for university students majoring in economics and engineering - Conduct experiments at the Posts and Telecommunications Institute of Technology Qualitative and quantitative results justify the feasibility and effectiveness of the proposed methods, confirming the validity of proposed scientific hypothesis As a result, the research purpose and research tasks have been completed The contents herein can be used as a useful reference for the lecturers and students in teaching Advanced Mathematics in the course of realizing the goal of developing analytical thinking for students Recommendations The research process encourages us to give some following recommendations: - Because analytical thinking is essential for each student, university leaders need to encourage and facilitate (time, attention to teaching methods, etc.) to help teachers and students better understand analytical thinking, know how to choose content and teaching methods to be able to implement well the methods of developing analytical thinking for students - The problem of developing analytical thinking for students can continue to be further studied in more specific areas, such as in solving the real-world problems, or when teaching Probability theory and mathematical statistics-a subject with many applications in analysis, prediction and decision making LIST OF SCIENTIFIC WORKS BY AUTHORS RELATED TO THE DISSERTATION Nguyen Thi Dung (2014), To analyze Analysis exercise solving in order to enhance the Maths solving competence to students, Journal of Educational Sciences, No 105, June 2014, p 26-29,36 Nguyen Thi Dung, Do Phi Nga (2015), To develop reasoning skills for students in advanced mathematics teaching, Educational Equipment Magazine, No 122, October 2015, p 12-15 Nguyen Thi Dung (2016), Using concept map in teaching advanced mathematics to develop analytical thinking for students, Journal of Education, No 395, period 1-December 2016, p 36-39 Nguyen Thi Dung (2017), Several examples in teaching Calculus by situations researching method , Educational Equipment Magazine, No 158, period 1, December 2017, p 17-19.47 Nguyen Thi Dung (2018), Analyzing definitions of mathematic in university through practical math issue, Journal of Education, No 443, period 1-December 2018, p 42-46 Nguyen Thi Dung (2018), Applying the SQR4 reading strategy - An effective measure to develope student’s thinking analytical in teaching advanced mathematics in economic and technical schools, Vietnam Journal of Educational Science, No 9, September 2018, p.39-43 Nguyen Thi Dung (2019), Improve the ability to think clearly, thoroughly and deeply for students in advanced math teaching, Educational Equipment Magazine No 185, period 2-January-2019, p 14-16 ... Students pay more attention to learning in details of concepts and theorems; draw out the meaning of that concept and theorem; find out any relationship between the concepts and theorems; make predictions... better ability to infer in forward and reverse aspects Initially, they know to draw some comments, attention; limit some mistakes; offer some solutions and choose the direction briefly; ask better... and correctly use the concepts, clauses, theorems, logical laws; know how to combine various theorems to deduce in many steps It also helps students make better deductive inferences When applying