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MINISTRY OF EDUCATION AND TRAINING VIETNAM INSTITUTE OF EDUCATIONAL SCIENCES NGUYEN VAN THA TEACHING MATHEMATICS AT HIGH SCHOOLS IN THE DIRECTION OF EXPLOITATION OF MATHEMATICS BEAUTY IN ORDER TO CONTRIBUTE TO ACTIVATE STUDENTS' LEARNING ACTIVITIES Major: Theory and Methods of teaching Mathematics Code: 9.14.01.11 DOCTORAL THESIS Hanoi- 2019 The thesis is completed at: VIETNAM INSTITUTE OF EDUCATIONAL SCIENCES Scientific supervisors: DSc DANG THI THU THUY Assoc Prof., DSc NGUYEN THANH QUANG Opponent 1: ………………………………………… Opponent 2: ………………………………………… Opponent 3: ………………………………………… The thesis is defended at the Council of Thesis Examiners, held at Vietnam Institute of Educational Sciences At on month Year 2019 The thesis may be searched at: - National Library of Vietnam, Hanoi City - Library of Vietnam Institute of Educational Sciences PREAMBLE REASON FOR CHOOSING THE THESIS The Law on Education, as amended and supplemented in 2010, defines: “The goals of education are to educate Vietnamese into comprehensively developed persons who possess ethics, knowledge, physical health, aesthetic sense and profession, loyal to the ideology of national independence and socialism; to shape and cultivate one’s dignity, civil qualifications and competence, satisfying the demands of building and defending the Fatherland.” The goals of the renovation of general education as defined by the Resolution No 88/2014/QH13 by the National Assembly are as follows: “The renovation of general education textbooks and programs is aimed to create a fundamental and comprehensive change in the quality and efficiency of general education; combining the education of literacy and personality and career orientation; contribute to transforming the education which is bias in favour of transmitting the knowledge to the education of comprehensive development both in quality and competences, harmonizing the virtue, mind, body, beauty and maximizing the potential of each student” In Vietnam, the social investment in the mathematics in schools is increasingly focused on and cared for, with the main reason that the education of mathematics will help students develop the comprehensive thinking, become smart, confident and active, thereby know the way of solving problems in the life Learning math is one of the best ways to develop the thinking and enlarge the knowledge of creative workers According to Nguyen Ba Kim “Mathematics contributes to the development of creative capacity and ideological thinking, so the mathematics has the effect of aesthetic education” Therefore, the aesthetic education in general and mathematical aesthetic education in particular are the important parts of mathematical education They are penetrated into teaching mathematics in many different ways It not only helps students establish a correct perspective on learning and life, but also promotes the comprehensive development and improves students’ thinking and passion for mathematics The limitations that are quite common in teaching mathematics in high schools result in the existence of the following situations: Teachers read and students write down what teachers read in teaching and learning; then students write down and teachers mark in evaluation and examination; students complain that they have to study math too much and difficult; teachers reflect students’ laziness in learning the mathematics; managers have not appreciated the effectiveness and quality of teaching mathematics; Parents are worried about their children’s mathematics learning with much pressure The limitations may be resulted from: Pressure on marks, examinations and virtual achievements from teachers, learners and their parents; bias in favour of mathematics without focusing in the development of qualities and competencies that are formed through the mathematics learning; not arousing the desire to study and create, the students’ joy and passion for discovery In order to foster the learning of each person and the society, we need set the goals of discovery and investigation of the true beauty of science and not to consider the knowledge as a tool or a means Mathematics has a very unique beauty which is featured by its logic and accuracy Along with the knowledge of science, Mathematics is potential to offering the aesthetic education to students However, in the viewpoint of many people, mathematics in the school is now considered a dry subject which is limited to the extent of exercises, formulas and numbers In order to contribute to overcoming this situation, exploiting the mathematical beauty in teaching mathematics is a topic that has attracted many scientists and educators interested in studying in various points of view During the collection of research materials, we have found that there has been no systematic, systematic study of exploiting the beauty of mathematics and exploring their applications in teaching mathematics, in order to activate the learning activities of students at high schools So we decide to select and implement the research topic: “Teaching mathematics at high schools in the direction of exploitation of mathematics beauty in order to contribute to activate students’ learning activities” PURPOSE OF RESEARCH Design methods of teaching mathematics in the direction of exploitation of mathematics beauty in order to contribute to activate students’ learning activities, thereby contributing to improving the effectiveness of teaching mathematics at high schools SCOPE OF RESEARCH - Mathematics beauty and its expression in mathematics curricular at high schools - Pedagogical methods in teaching mathematics at high schools in the direction of exploitation of mathematics beauty in order to activate students’ learning activities THEORY OF SCIENCE If the mathematics teaching at high schools in the direction of exploitation of mathematics beauty, it will positively promote the learning activities of students, thereby contributing to improving the quality of teaching mathematics DUTIES OF RESEARCH 5.1 Overview of the theoretical and practical basis of mathematical beauty and study of new foundations and achievements in teaching mathematics in the direction of exploitation of mathematics beauty 5.2 Give the concept of mathematics beauty (content, characteristics and expression), teaching mathematics in the direction of exploitation of mathematics beauty 5.3 Propose some methods of teaching mathematics in the direction of exploitation of mathematics beauty 5.4 Organize the pedagogical experiments to check the feasibility and effectiveness of proposed teaching methods METHODS OF RESEARCH 6.1 Methods of theoretical research - Survey the philosophical and psychological basis of the activities of teachers and students in teaching and learning mathematics at high schools in the direction of exploitation of mathematics beauty - Analyze and select some contents which can exploit mathematical beauty in Vietnam's current mathematics textbook - Survey ideas and contents of exploiting mathematics beauty shown in mathematic textbooks in some countries in the world - Analyze, evaluate and select the contents which are suitable to the topics of the thesis in any instruction documents of teaching mathematics for high school teachers 6.2 Methods of survey and observation - Combine both types of scientific observations which are direct and indirect ones - Collect questionnaires, observation notes, feedback on relevant issues of the thesis from teachers, students, scientists, educators 6.3 Method of experts Using the wisdom of expert team to consider and review the nature of the research object, then find the optimal solution 6.4 Method of analysis and summarization of experiences Research and review the practical achievements in the past to draw any useful conclusions for practice and science 6.5 Method of pedagogical experiments - Organize the pedagogical experiments according to the proposals of the topic - Organize the pedagogical experiments to assess the feasibility and effectiveness of the proposed pedagogical methods 6.6 Statistical method Calculate and process data collected in practice and theoretical materials by using mathematical statistical tools and methods 6.7 Method of case study Used in the first pedagogical experiment to adjust and improve the proposed pedagogical methods NEW CONTRIBUTION OF THE THESIS a) Theoretically Give the concept of mathematics beauty and teaching mathematics in the direction of exploitation of mathematics beauty - Clarify requirements and opportunities of exploiting the mathematics beauty in the process of teaching and learning mathematics at high schools of teachers and students b) Practically - Methods of teaching mathematics in the direction of exploitation of mathematics beauty in order to activate students’ learning activities - Suggest the use of methods and illustrative examples in teaching concepts, theorems, exercises, revision in the topic of Algebra, Calculus and Geometry at high schools - Research results can be used as a reference for teachers and students THEORETICAL POINTS TO BE DEFENDED - Concept of mathematics beauty and teaching mathematics in the direction of exploitation of mathematics beauty at high schools - The proposed methods of teaching mathematics at high schools in the direction of exploitation of mathematics beauty are feasible and effective, contributing to the education of mathematics beauty to students and activating students’ learning of mathematics STRUCTURE OF THE THESIS In addition to the introduction, conclusions, published works, references and annexes The thesis includes chapters: Chapter Theoretical and practical basis; Chapter Some methods of teaching mathematics at high schools in the direction of exploitation of mathematics beauty; Chapter Pedagogical experiments Chapter THEORETICAL AND PRACTICAL BASIS In this chapter, we focus on clarifying some of the following main contents 1.1 OVERVIEW OF RESEARCHES RELATED TO THE TOPIC 1.1.1 Overview of some foreign researches Since the beginning of the eighteenth century, there have been numerous publications on the approaches to aesthetic theory in full of beauty in science and mathematics In 1735, A G Baumgarten introduced the aesthetic term to describe the issues related to the beauty that we now consider as a philosophical approach to beauty and art This event has marked the birth of modern aesthetics Since then, there have been many researches on this issue and most of the authors in the world have diversified and rich researches with many approaches to and on many different aspects of mathematics beauty: Aesthetics in mathematics; beauty of mathematics solution; various applications of mathematics into practice; the relationship between creativity and mathematics beauty; mathematics and life; the connection between mathematics beauty and human spirit and psychology; mathematics and art; joy and mathematics excitement; Mathematics entertainment 1.1.2 Overview of some domestic researches In Vietnam, there have also been a number of diverse studies on the beauty of mathematics and mentioned the current issue of the meaning of discovering, exploiting and applying mathematics beauty to mathematics teaching at schools by such authors as: Ngo Bao Chau, Nguyen Tien Dung, Nguyen Thi My Loc, Tran Dinh Chau, Dang Thi Thu Thuy However, neither specific researches on methods of mathematics beauty education for high school students in teaching mathematics have been carried out 1.2 CONCEPT OF MATHEMATICS BEAUTY 1.2.1 Mathematics beauty and concept of the author on the mathematics beauty The local and foreign authors have studied and examined the relationship between mathematics and aesthetics in different points of view and offer quite explicit idea of beauty in mathematics, it is: mathematics and art have many similarities Both are towards the beauty and are creative Mathematics is a tool to describe the surrounding world and the foundation of creative thinking It is creativity that creates beauty of mathematics The success in mathematics or art requires the ability of abstract visualization, rich imagination and bold thinking, as well as inspiration and passion On the other hand, mathematics always inspires people to learn, discover and create because they crave to find new beauty, but not where there is such perfect beauty, that is the aesthetic beauty of mathematics Within the scope of this thesis, on the basis of acknowledgment and acceptance of the research results of local and foreign authors on mathematics beauty, we have determined: The ideas of mathematics beauty given by such authors as Bertrand Russell, Alfred S Posamentier, Ha Huy Khoai, Ngo Bao Chau, Vu Ha Van, and Nguyen Cang have been approached by us in some viewpoints and serve as a basis for giving the concept of mathematics beauty as follows: Mathematics beauty is a scientific category used for referring to the beauty brought about by mathematics for humans, which is the clever combination of natural beauty with elements featured by mathematical science through contact, cognitive and mathematical applications 1.2.2 Components of mathematics beauty It can be seen that mathematics beauty is also in the process of our direct contact, finding and application of mathematical techniques Because people find it more difficult to learn and research mathematics, but when it comes to overcome these difficulties, the more excited and happy they are, the results of the bright, intuitive are more and more interesting; the more and more beautiful the mathematics is, and so the mathematics beauty is formed by many components, which are: Component 1: Generalization and abstraction Component 2: High creativity of cognitive process Component 3: Forms of expression Component 4: Mathematics thinking Component 5: Rich benefits that mathematics brings to people Component 6: People experience the beauty through intellectual, excitement, habits and thinking ability, 1.2.3 Characteristics of mathematics beauty From such concept of mathematics beauty, according to the author of the thesis, mathematics beauty has the following characteristics: Harmonious, unique, creative and universal 1.2.4 Mathematics beauty is shown in high school mathematics curricular Based on the contents and practice of teaching mathematics at high schools today and in the future, we select some components of mathematics beauty in the content of mathematics curricular at high schools which are potentially exploited in teaching, that is: The closeness, accuracy, simplicity, surprise, consistency, harmony and symmetry as well as profound applications of mathematical contents, objects and tools including: - Symbols, languages, expressions, reasoning; - Axioms, definitions, theorems, consequences, properties; - Mathematics, formulas, drawings, images, graphs, diagrams; - Mathematics problems, solutions, games; - Approaches and ways to demonstrate creativity With the above concept of the thesis, we can identify mathematics beauty mathematics curricular at high schools through the following six expressions: 1) Expression by harmony in drawings, structures and mathematical expressions 2) Expression by language, symbols and logic of mathematics 11 and without any classification according to the capacity of each student He thinks that: “With the way of study for exams and tests as it is in schools today, the positive side is that students are diligent to exercises but it will not be able to make them passionate for exploitation Of course the exam is also important, but the more important thing is to give them the skills, the passion for exploration, the discovery that is the need for them later” According to Tran Nam Dung: “There are many shortcomings in mathematics education in Vietnam in particular Teachers give clauses and students apply them mechanically without understanding the nature The more you go to higher levels, the drier and farer away from the reality the mathematics is” 1.4.2 Learn the situation of teaching and learning mathematics at high schools today 1.4.2.1 Purposes of survey - Learn the situation of learning mathematics in the direction of promoting the active learning of students at high schools, specifically: + Regarding students: Learn the positivity, proactivity, interest in learning Mathematics; students’ views and perceptions of creativity in solution of mathematical problems, the role of mathematics in practice, historical values of mathematics and mathematics beauty in mathematics curricular at high schools + Regarding teachers: Learn the renovation of teaching methods, apply teaching techniques in the direction of activating students’ learning activities; understanding, recognition of the roles of mathematics beauty and application of mathematics beauty to create the excitement and improvement of positivity of students in teaching mathematics at high schools 1.4.2.2 Methods and objects of survey - Method of survey: Use questionnaires which are designed in the form of closed questions, open questions and combined questions for students and teachers 12 - Objects of survey: + Regarding students: We randomly surveyed 1,600 students who are studying mathematics at high schools across the country + Regarding teachers: We randomly surveyed 300 teachers 1.4.2.3 Contents of survey - Active teaching activities of teachers and active learning of students - Positive changes of students through the teaching process of teachers - The level of awareness of the beauty by students - The level of exploiting the beauty in teaching mathematics of teachers and the level of interest of students when teachers apply 1.4.2.4 Results of survey A) Regarding teachers Only 20% teachers at high schools: - Desire to inspire students to arouse a passion for learning - Recognize the role of the exploitation of mathematics beauty in teaching mathematics - Find measures to overcome the problem of students who are afraid to study mathematics, afraid of learning math - Read some documents (books, translations, articles) about mathematics beauty in recent times in Vietnamese by Nguyen Cang, Nguyen Tien Dung, Ngo Bao Chau or other translators - Feel some elements of mathematics beauty in general and the expression of mathematics beauty in the mathematics curriculum Most of the remaining surveyed teachers face with obstacles and difficulties in exploiting the mathematics beauty in teaching mathematics, expressed by the following problems: - Not invest time in guiding students to find good solutions for the mathematical problems - Not invest time in guiding students to experience the application of mathematics 13 - Not understand carefully the cause resulting in students’ hesitation of learning mathematics and being afraid of learning mathematics - Not interested in exploiting mathematics beauty, presenting various difficult problems by simple, clear and easy to understand words - The perception is superficial about mathematics beauty, even teachers have never read any materials or books about mathematics beauty - Not yet realized the components of mathematics beauty shown in the mathematics curriculum at high schools - Less use of methods to exploit mathematics beauty in teaching mathematics; - Less search for specific mathematical content to help students feel beauty - Organize less activities to encourage students to seek and feel beauty in solution of mathematical problems - At different levels, teachers have not really focused on sensing the beauty and interesting of mathematics, but also favoring teaching mathematics because of math, not focusing on exploiting aesthetic factors and the beauty of mathematics in teaching b) Regarding students: - Most students not realize the importance of mathematics - The majority of students not grasp the meaning of mathematical concepts and results - Many students still exercises and tests in the very formal way They only know how to solve mathematics problems according to a model lesson, but they not really understand the problem as well as finding various many solutions for a problem is very difficult for them - Most students not realize the beauty of mathematics? Not know that the mathematics knowledge at schools will be applied to life 14 - Most surveyed students have the idea that teachers’ teaching methods are not really interested in learning for them Particularly, during the teaching: + Teacher only focus on teaching the theory and instructing or correcting the model exercises for students according to the curricular but not focusing on analyzing any good solutions, concise and concise solutions of the mathematical problem + Teacher have not really focused on the relationship between mathematics and practice and vivid representations of mathematics in nature + Teacher have not expressed expressively the mathematical formulas which contain great ideas of science along with practical activities that people need to express + Teachers have not highlighted the original, core techniques and basic methods that go into the nature of mathematical concepts and results + Teachers have not focused much on training students about analogy, similarization, generalization or regularization of each strange exercise on familiar forms; It has not been noticed to combine the individual exercises into a series of exercises which is closely related Such restrictions, in our opinion, are resulted from: - Some teachers have not been trained carefully about active teaching methods, so they cannot apply new methods to their teaching tasks - A part of high school teachers has not focused on selflearning ability, lack of creative thinking, not approached to new methods, less use of IT in teaching, weak logical reasoning methods; heavy presentation in conveying the knowledge, light on training skills of detecting and solving problems; There are not many practical examples illustrating mathematical models - Some teachers find it unnecessary to use new teaching methods, the use of traditional teaching methods is more convenient with their teaching methods 15 - The preparation of a positive, elaborate and expensive lesson depends on many factors: time, equipment, support facilities, infrastructure - The distribution of mathematics curricular at high schools is still inadequate, especially in terms of contents, heavy on theory, lack of practicality, not updated with new and more practical knowledge From the above results, we realize that most students are passive in learning mathematics and not yet aware of mathematics beauty; Teachers are not really interested in inspiring mathematics for students through recognizing mathematics beauty, besides some teachers are aware of the role of exploiting mathematics beauty in teaching mathematics at high schools However, there are still many teachers who are not really interested in, this problem needs to be overcome CONCLUSION OF CHAPTER Through theoretical and practical researches, we have obtained the following results: System and clarify theoretical basis related to the thesis Point out six components of mathematics beauty, propose a concept of mathematics beauty, outline four characteristics of mathematics beauty and six explicit expressions of mathematics beauty Clarify the concept of teaching mathematics at high schools in the direction of exploitation of mathematics beauty and identify opportunities and orientations to teach mathematics in the direction of exploitation of mathematics beauty at high schools Through the survey of reality of teaching mathematics at high schools from the viewpoint and requirements of the exploitation of mathematics beauty The author has pointed out that teaching mathematics in the direction of exploitation of mathematics beauty in some contents of mathematics at high schools is feasible 16 Chapter SOME METHODS OF TEACHING MATHEMATICS AT HIGH SCHOOLS IN THE DIRECTION OF EXPLOITATION OF MATHEMATICS BEAUTY IN ORDER TO ACTIVATE LEARNING ACTIVITIES OF STUDENTS Originating from the theoretical and practical basis presented in Chapter 1, we propose three pedagogical methods to help teachers teach mathematics in the direction of exploitation of mathematics beauty to contribute to activate the learning activities of students at high schools 2.1 ORIENTATION OF FORMULATION OF METHODS 2.1.1 Orientation In accordance with characteristics and principles of teaching Mathematics 2.1.2 Orientation In accordance with the orientation of renovation of teaching methods of mathematics, especially requirements of activating learning activities 2.1.3 Orientation Consistent with the physiology of students at high schools 2.1.4 Orientation Ensure the feasibility in actual conditions of teaching mathematics at high schools 2.2 SOME METHODS OF TEACHING MATHEMATICS IN THE DIRECTION OF EXPLOITATION OF MATHEMATICS BEAUTY AT HIGH SCHOOLS 2.2.1 Method Focus on exploitation of many good and creative solutions for each mathematical problems, synthesize and develop into series of exercises Purpose of the method: This method directly affects the second component - The beauty lies in the creativity of mathematics and the sixth component - the beauty of mathematics lies in the process of overcoming difficulties to find solutions of mathematical problems Thereby it arouses the excitement, passion, aspiration, discovery and promotion of students’ activation of learning activities 17 Ways of implementation + Use method in periods of exercises + Use method in periods of review and summarization 2.2.2 Method Enhance the exploitation of the practicality of mathematics through mathematical models of mathematical problems with actual contents Purpose of the method: This method directly affects the component (generalization, abstraction) and component (universal practicality both in terms of origin and application of mathematics), while also practicing component (creative mathematics, mathematical language), motivating students to think, explore, discover, actively exploit the beauty of mathematics to contribute to fostering the capacity of discovering and creating new techniques, improving further skills of problem solving in mathematics learning as well as life for students Ways of implementation + Use method in teaching concepts + Use method in teaching through experience, practice + Use method in periods of review and summarization 2.2.3 Method Enhance students to learn the history of mathematical knowledge in textbooks Purpose of the method: This method directly affects the fifth component, the origin and the rich practical application of mathematics Ways of implementation + Use method in teaching concepts + Use method in teaching theorems + Use method in teaching through experience, practice 2.3 SOME PEDAGOGICAL SUGGESTIONS HELP TEACHERS USE THE SYSTEM OF METHODS Based on the goals of teaching, contents, programs and plans for teaching each lesson, teachers choose the content to set up the lesson plan so that it is appropriate to the circumstances and specific 18 techniques that teachers choose to present or coordinate reasonable methods which shall make the teaching livelier, more attractive, Suggestion 1: Teaching mathematics at high school in the direction of exploitation of mathematics beauty is not a new teaching method, but just a clearer approach, emphasizing the goal of educating mathematics beauty for students Suggestion 2: Not any content, any student can fully apply the methods It is required to select and coordinate the methods in accordance with the contents, objects, conditions of time and means of teaching mathematics practically at high schools Suggestion 3: Depending on the type of mathematics teaching contents (theory includes concept, theorem, rules of mathematical methods, exercises, practice and revision), integrate the content and methods of educating mathematics beauty accordingly Suggestion 4: Combine harmoniously and synchronously the requirements and methods of teaching mathematics associated with the reality and the goals of educating mathematics beauty for students Including activities of learning the historical origin and practical applications of mathematical knowledge at high schools CONCLUSION OF CHAPTER Based on the results of theoretical and practical research on mathematics beauty in mathematics at high schools, we have formed three pedagogical methods, affecting the components of mathematics beauty in mathematics at high schools The pedagogical methods along with specific examples of application in teaching mathematics at high schools have clarified the scientific basis, meaning and method of exploiting mathematics beauty in teaching mathematics at high schools, contributing to activate the learning activities of students 19 Chapter PEDAGOGICAL EXPERIMENTS In this chapter, we focus on clarifying some of the following main contents 3.1 PURPOSES, REQUIREMENTS, TASKS, PRINCIPLES AND CONTENTS OF EXPERIMENTS 3.1.1 Purposes of experiments Through teaching practice on some high school students, pedagogical experiment records are conducted for the purposes of testing: The correctness of the scientific theories stated in the thesis; The feasibility and effectiveness of pedagogical methods have been proposed in Chapter 2, through answering the following key questions: (1) Can the pedagogical methods proposed by the thesis be implemented in the process of teaching mathematics at high school? (2) Do these pedagogical methods have a positive impact on the acquisition of techniques and experiences that need to be equipped for students? (3) Are these pedagogical methods feasible and effective in the current conditions of teaching mathematics at high schools? (4) Do these pedagogical really inspire and maximize students’ learning activities through teaching mathematics in the direction of exploitation of mathematics beauty? 3.1.2 Requirements of experiments - Pedagogical experiment must ensure honesty and objectivity - Pedagogical experiment must be suitable for students, close to teaching situation 3.1.3 Duties of experiments - Compile pedagogical experiment materials and conduct the teaching according to some pedagogical methods as proposed in Chapter 2; 20 - Collect, analyze and process pedagogical experimental results to test the feasibility and effectiveness of proposed pedagogical methods 3.1.4 Principles of experiment organization - Only carry out the experiment upon sufficient arguments; - Provide a meticulous and reasonable plan to facilitate implementation; - Choose experimental samples which are randomized classes of students; - Experimental objects include the full educational levels of high school students; - Ensure that students have enough time to think and implement learning activities in a small content; 3.1.5 Contents of experiments a) In the first pedagogical experiment, we mainly apply the method of case study as a basis to adjust and perfect pedagogical methods in the direction of exploitation of mathematics beauty in Chapter b) In the second pedagogical experiment, we mainly conduct some lessons designed in the direction stated in Chapter of the thesis and tested and reaffirmed the feasibility of the proposed pedagogical methods 3.2 EVALUATION OF PEDAGOGICAL EXPERIMENT RESULTS 3.2.1 Results of the first pedagogical experiment Through the observation of all pedagogical experimental lessons, observation and exchange to check the interest, students’ ability of acquiring lessons with the methods as proposed in the thesis, we have realized that: - The atmosphere of the experimental class is lively and the students are more excited when not using pedagogical experimental methods 21 - The majority of students in the experimental class have acquired the basic knowledge with certainty, can orient and find many solutions to a mathematical problem, and have the ability to solve the practical mathematical problems in the faster manner Especially some good and excellent students have proposed short, unique and creative solutions Through the results of pedagogical experiment in phase 1, we have also realized that it is necessary to adjust some of the following issues for the implementation of pedagogical experiment in phase to get better results Firstly, regarding any preliminary mathematical problems, teachers need to assign students to at home in the previous lesson so that the organization of group discussions in the next class session is effective and ensures the class time Secondly, teachers should have more detailed instructions on homework, topics for students, ask students to submit the report to the teacher before the class presentation Thirdly, guide students to solve practical mathematical problems more carefully, more detailed and assign more exercises to at home, help students apply knowledge more proficient Fourthly, it is necessary to compile some separate documents, help students have reference sources and learn how to use some mathematics software, encourage students to look for similar exercises, search for various mathematical problems which are solved in many ways, history of formation and development of mathematics in reference books and on the Internet Based on the above adjustments, we conduct pedagogical experiment in phase 3.2.2 Results of the second pedagogical experiment a) Qualitative evaluation: For students who participate in experiments, most of them are excited about learning, which is reflected in the fact that they actively participate in building lessons In each lesson, the role of students is 22 highly appreciated because each of their ideas becomes a small part of the lesson so they feel confident, excited, boldly make suggestions to build lessons Most students of the group participating in pedagogical experiment think that: Teaching Mathematics in the direction of exploitation of mathematics beauty, not only helps students to acquire basic and core skills of the subject but also offers the aesthetic education for them, helping them feel the hidden beauty of mathematics, create aesthetic emotion for students, create excitement, passion, love and be more active in learning Mathematics b) Qualitative evaluation Students of both experimental and control classes of the pedagogical experiment seriously took the test and cooperated with the author in the process of processing observation, inspection and evaluation votes + With the control class, the process of teaching mathematics in class according to the usual lesson plan is still focused on academic, the exploitation and application of subject knowledge in practice is still limited Therefore, students perform the test exercises slowly, facing with many difficulties in finding various solutions of a mathematical problem and the number of students who are able to model and solve the mathematical problems with the actual content is still modest, the percentage of 7, 8, 9.10 marks is still modest + With the experimental class, the majority of students in the experimental class mastered the academic knowledge, besides, the students were more proficient in mobilizing knowledge in the implementation of finding various solutions of mathematical problem as well as good skills of modeling problems with practical content, especially some good and excellent students have solved exercises given in the form of thinking exercises, higher practical application than the control group, the percentage of 7, 8, 9.10 marks is higher than those of the control class 23 This proves that the implementation of pedagogical methods proposed in the thesis has created students’ interest, initiative, creativity, habit formation, ability to apply mathematical knowledge to solve the practical issues, thereby forming and developing students a number of necessary skills, serving as a foundation for creative thinking in learning, thus contributing to the active learning of mathematical activities of students at high schools CONCLUSION OF CHAPTER The pedagogical methods proposed by the thesis are feasible in the process of teaching Mathematics for students at high schools and are initially effective; had the effect of activating students’ learning activities and contributing to improving the quality of teaching mathematics for students The proposed methods not only create conditions for students to acquire knowledge, improve skills, have positive attitude and interest in the subject but also help students form and develop aesthetic capacity, feel the mathematics beauty, exploit the beauty in mathematics, and serve as the initial foundation for creative thinking of students Through the implementation of teaching in the direction of exploitation of mathematics beauty, students have gradually approached and realized mathematics beauty, formed and developed some basic skills in finding and creating the beauty in mathematics CONCLUSION The thesis has made the following contributions: On the basis of approaching scientific analysis of mathematics, aesthetics, philosophy, we have clarified the role of mathematics beauty as an important component in cultural capital in general and mathematical cultural capital in particular which needs to equip for mathematics learners Clarify the concept of mathematics beauty and teaching mathematics in the direction of exploitation of mathematics beauty 24 Learn the situation of teaching and learning mathematics from the viewpoint and requirement of the exploitation of mathematics beauty in some high schools in Vietnam Determine the directions of exploiting mathematics beauty in teaching mathematics at high schools Develop three methods of teaching mathematics and exploiting the mathematics beauty in mathematics teaching at high schools to contribute to educating mathematics beauty for students and learning mathematics for them (with four pedagogical suggestions and 32 examples in teaching Algebra, Calculus and Geometry) Organize pedagogical experiments to check the feasibility and effectiveness of proposed solutions and obtain results: - In practical conditions at high schools in Vietnam, the implementation of teaching mathematics in the direction of exploitation of mathematics beauty is feasible, allowing us to exploit some new achievements of mathematics and technology in teaching mathematics - The lesson plans are designed according to the orientations proposed by the thesis, which have exploited many strengths in teaching Mathematics, in accordance with the objectives, technical standards, and skills towards the development for learners’ competences With the results and contributions mentioned above, it can be affirmed that: The scientific hypothesis of the thesis is reasonable, the research purpose of the thesis has been achieved and the research task of the thesis has been completed 25 SOME SCIENTIFIC WORKS OF GRADUATE STUDENT RELATED TO PUBLISHED TOPIC [1] [2] Thanh Quang Nguyen Van Tha Nguyen, Anh Tuyen Phan (2015), Exploiting the beauty of inequality in teaching mathematics, Education Journal, May 2015, p 175-177 Thanh Quang Nguyen Van Tha Nguyen (2016), Exploiting the [3] beauty of the unity of mathematics through solving some geometry problems by algebraic methods, Journal of Educational Science, Vietnam Institute of Educational Sciences, 8/2016, p 32-35 Van Tha Nguyen (2017), Exploiting the mathematics beauty [4] through solving some practical mathematics problems, Journal of Educational Science, Vietnam Institute of Educational Sciences, No 142 - 7/2017, p 68-72 Van Tha Nguyen, Viet Duong Nguyen (2017), Teaching 11th- [5] [6] grade Integrated Mathematics - Probability from G Polya’s viewpoint contributing to promoting students’ positivity and creativity, Education Journal, / 2017, p 176-180 Van Tha Nguyen, Ngoc Giang Nguyen (2017), The Mathematical Beauty, International Journal of Learning, Teaching and Educational Research, Vol 16, No 5, pp.14-30 Hai Tam Dinh, Van Tha Nguyen (2018), Analyzing and correcting common mistakes of students when solving exercises of applying derivatives to survey and draw function graphs (Analysis- grade 12), Journal Education, No 427 (Period - 4/2018), p 23-26 ... TOPIC [1] [2] Thanh Quang Nguyen Van Tha Nguyen, Anh Tuyen Phan (2015), Exploiting the beauty of inequality in teaching mathematics, Education Journal, May 2015, p 175-177 Thanh Quang Nguyen... Theoretical and practical basis; Chapter Some methods of teaching mathematics at high schools in the direction of exploitation of mathematics beauty; Chapter Pedagogical experiments Chapter THEORETICAL... needs to be overcome CONCLUSION OF CHAPTER Through theoretical and practical researches, we have obtained the following results: System and clarify theoretical basis related to the thesis Point out