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
Trang 1MINISTRY 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
Trang 2VIETNAM INSTITUTE OF EDUCATIONAL SCIENCES
Scientific supervisors:
1 DSc DANG THI THU THUY
2 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
Trang 3PREAMBLE
1 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
Trang 4not 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
Trang 5mathematics, 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”
2 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
4 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
5 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
Trang 66 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
Trang 76.7 Method of case study
Used in the first pedagogical experiment to adjust and improve
the proposed pedagogical methods
7 NEW CONTRIBUTION OF THE THESIS
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
8 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
9 STRUCTURE OF THE THESIS
In addition to the introduction, conclusions, published works, references and annexes The thesis includes 3 chapters:
Chapter 1 Theoretical and practical basis;
Chapter 2 Some methods of teaching mathematics at high
schools in the direction of exploitation of mathematics beauty;
Chapter 3 Pedagogical experiments
Trang 8Chapter 1 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
Trang 9The 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
Trang 10beautiful 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
Trang 113) Expression by arguments, proofs, short and concise expressions
in a number of solutions with the creativity, originality and surprise of the problems
4) Expression by the internal and interdisciplinary relationship of mathematics and other sciences
5) Expressed by the rich applicability of mathematical knowledge
to life and production
6) Expression by the history of formation and development of mathematics
1.3 CONCEPT OF TEACHING MATHEMATICS AT HIGH SCHOOLS IN THE DIRECTION OF EXPLOITATION OF MATHEMATICS BEAUTY
1.3.1 What is teaching mathematics in the direction of exploitation of mathematics beauty?
We consistently think that: Exploitation of mathematics beauty
in teaching and learning mathematics is the process of organizing a series of activities “Discovering beauty - Feeling beauty - Creating beauty - Exciting learning - Actively learning - Effective learning” Based on the current process of teaching mathematics, teachers shall teach mathematics in the direction of exploitation of mathematics beauty by adding requirements, clarifying intentions and implementing techniques to integrate into the lecturers with necessary activities to educate the mathematics beauty for students It
is detailed as follows: In the process of teaching mathematics, teachers focus on clarifying some expressions of mathematics beauty hidden inside the lessons aiming to make students see the beauty and effects of mathematics, thereby creating inspiration of learning, improving the aesthetic sense of mathematics, active learning and achieving mathematical results in a more solid and profound way
1.3.2 Opportunities and orientations of teaching mathematics in the direction of exploitation of beauty
In our opinion, teaching mathematics in the direction of exploiting mathematics beauty at high schools should be focused on exploiting some of the following orientations:
Trang 121- Focus on finding various solutions for the same mathematical problem, creating beautiful and interesting solutions in
a short, clear, creative, unique and unexpected way
2- Exploiting various applications of mathematics in the real life and interdisciplinary mathematics as well as interdisciplinarity between mathematics and other subjects (physics, chemistry, biology, informatics, foreign languages, fitness and sports, ) 3- Exploiting the history of formation and development of mathematics related to mathematical techniques at high schools Besides, neither all contents included in the mathematical curricular at high schools can also facilitate the exploitation of mathematics beauty while implementing the lesson Therefore, in order to bring the best teaching efficiency, we need to select the contents related to teaching to formulate the concept, the content of the exercises, revision outside the class on the basis of following closely to the curricular included in the current textbook
1.4 SITUATION OF TEACHING MATHEMATICS IN THE DIRECTION OF EXPLOITATION OF MATHEMATICS BEAUTY AT HIGH SCHOOLS
1.4.1 The situation of teaching mathematics at high schools in general
strongly and achieved some remarkable achievements, on some aspects, it has approached the science of education in the world Some positive teaching trends have been studied and applied
in teaching mathematics at all levels in high schools However, the situation of teaching and learning mathematics at high schools has still faced with many limited issues
Within the framework of the open mathematics festival held
by Vietnam Institute for Advanced Studies in Mathematics on August 13, 2017 at the C2 Hall of Hanoi University of Science and Technology Some mathematics experts "analyze" inadequate teaching mathematics in high school as follows: According to Ngo Bao Chau, the curricular is harden the education method is imposed
Trang 13and 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 do 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