Dạy học toán trung học phổ thông theo hướng khai thác vẻ đẹp toán học góp phần tích cực hóa hoạt động học tập của học sinh tt tiếng anh

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:
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


1

PREAMBLE
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


2

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


3

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”.
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.
3. 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.
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.

4

6. 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.


5

6.7. Method of case study
Used in the first pedagogical experiment to adjust and improve
the proposed pedagogical methods.
7. NEW CONTRIBUTION OF THE THESIS
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.
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


6

Chapter 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


7

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


8

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


9

3) 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:


10

1- 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


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 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.


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 do not realize the importance of mathematics.
- The majority of students do not grasp the meaning of

mathematical concepts and results.
- Many students still do exercises and tests in the very formal
way. They only know how to solve mathematics problems according
to a model lesson, but they do not really understand the problem as
well as finding various many solutions for a problem is very difficult
for them.
- Most students do 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 1
Through theoretical and practical researches, we have
obtained the following results:
1. System and clarify theoretical basis related to the thesis.
2. 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.
3. 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.
4. 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 2
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 1 In accordance with characteristics and
principles of teaching Mathematics
2.1.2. Orientation 2 In accordance with the orientation of renovation
of teaching methods of mathematics, especially requirements of
activating learning activities
2.1.3. Orientation 3 Consistent with the physiology of students at
high schools
2.1.4. Orientation 4 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 1 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 1 in periods of exercises.
+ Use method 1 in periods of review and summarization.
2.2.2. Method 2 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 1 (generalization, abstraction) and component 5
(universal practicality both in terms of origin and application of
mathematics), while also practicing component 3 (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 2 in teaching concepts.
+ Use method 2 in teaching through experience, practice
+ Use method 2 in periods of review and summarization.
2.2.3. Method 3 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 3 in teaching concepts.
+ Use method 3 in teaching theorems.

+ Use method 3 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 2

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 3
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 2.
b) In the second pedagogical experiment, we mainly conduct
some lessons designed in the direction stated in Chapter 2 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 2
to get better results.
Firstly, regarding any preliminary mathematical problems,
teachers need to assign students to do 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
do 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 2.
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 3
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:
1. 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.
2. Clarify the concept of mathematics beauty and teaching
mathematics in the direction of exploitation of mathematics beauty.