Góp phần phát triển ngôn ngữ toán học cho học sinh dự bị đại học ở vùng tây nguyên tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY

KIEU MANH HUNG

CONTRIBUTING TO THE DEVELOPMENT
OF THE MATHEMATICAL LANGUAGE
FOR PRE-UNIVERSITY STUDENTS
IN THE CENTRAL HIGHLANDS
Major: Reasoning and methodology of teaching mathematics
Code: 9140111

SUMMARY OF DOCTORAL THESIS IN SCIENCE EDUCATION

NGHE AN - 2020


The thesis was completed at Vinh University

Supervisors:
1. Dr. Nguyen Van Thuan
2. Assoc. Prof. Dr. Nguyen Thanh Hung

Reviewer 1: Assoc. Prof. Dr. Vu Duong Thuy
Reviewer 2: Assoc. Prof. Dr. Dao Thai Lai
Reviewer 3: Dr. Nguyen Huu Hau

The thesis will be defended at the Boards
of Examiners of Univeristy level, at Vinh University, No. 182, Le Duan Street,
Vinh City, Nghe An Province,
Time: At , date month year 2020

This thesis can be found at:
1. National Library of Vietnam
2. The Information Center - Library of Nguyen Thuc Hao, Vinh University

1


INTRODUCTION
1. Rationale
1.1. By studying theoretical and practical teaching, we find that pre-university
students master the knowledge and skills of mathematics. Once they master the
mathematics language system, they will be able to use this language system in the
thought process, reasoning in solving math problems and putting them into practice. In
addition, language difficulties are a significant barrier to the acquisition and application
of scientific and technical knowledge, especially for highly abstract scientific fields
such as mathematics. The results of the grassroots research topics that the author had
done in 2009, 2013, 2015, 2016 and 2018 showed that pre-university students still have
many limitations of expression when solving the problem.
1.2. Flexibly understanding and applying concepts, theorems, consequences,
properties,... to solve math issues successfully is not an easy task. But how to present
the theoretical content briefly, concisely and highlight content to facilitate the use of
them in mathematical reasoning is much more difficult.
1.3. The Math curriculum for pre-university students do not have content
specifically to introduce and teach knowledge related to mathematical language. The
knowledge is introduced in an implicit manner in teaching process, in accordance with
the students' level of knowledge in order to serve mathematical reasoning as well as
apply it to other science subjects. This shows that teachers have to pay attention to
fostering self-studying skills for pre-university students so that these skills can be used
as a means to serve the thinking and reasoning process.

1.4. The mountainous areas in our country in general and the Central Highlands
in particular are places where socio-economic conditions are still facing many
difficulties. There are still many children of ethnic minorities with low educational
levels and uneven knowledge. Through the teaching process at the pre-university
Department of Tay Nguyen University, we found that the pre-university students in
the Central Highlands are mainly ethnic minority people with many different
languages, rituals and customs. In general, they have many difficulties in learning
subjects in general, and Maths in particular.
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The lecturers teaching mathematics always try to let students know how to
interpret the definitions, theorems and problems,... from ordinary language to
mathematics language and vice versa for the purpose of consolidation and applying
knowledge. However, in fact many pre-university students in the Central Highlands
are still confused and encountered many mistakes when performing the above tasks.
This greatly influences the acquisition of knowledge, mathematical reasoning and the
development of logical thinking.
1.5. The task of the pre-university training system is to help students
consolidate, systematize and better understand the basic knowledge of the high
school program, build learning methods and self-study methods. In order to help preuniversity training students confidently study Maths at the University and College
levels later, it is necessary to practice and develop mathematical language during the
studying period at pre-university. From that awareness, the proposal of pedagogical
measures in teaching to develop the language of mathematics in learning Maths is
meaningful and practical work. The study of this issue contributes to improving the
learning results of Mathematics for pre-university students in the Central Highlands
in particular and pre-university students in general.
Start from the above reasons we study the thesis “Contributing to the developing

mathematical language for pre-university students in the Central Highlands”.

2. Research purposes
Based on theoretical and practical research related to issues of language,
mathematical language, thinking, the relationship between language and thinking,
mathematical language and mathematical thinking, we propose some measures for
developing mathematical language to contribute to improving the quality of teaching
for pre-university students in the Central Highlands.
3. Subjects and scope of the research
3.1. Research subjects: Measures to develop mathematical language for
pre-university students in the Central Highlands.
3.2. Scope of the research
Scope of time: The Doctoral thesis collects data on students in two courses
K2016 and K2017 of pre-university Department of Tay Nguyen University.
3


Experiments on pre-university classes of blocks A and B, two courses of K2017 and
K2018 of pre-university Department of Tay Nguyen University.
Scope of space: The Central Highlands.
Scope of content: Mathematical language in the curriculum of Mathematics
which is used for pre-university students.
4. Research topic
- Theoretical study of language, mathematical language, thinking, mathematical
thinking, the relationship between language and thinking, the relationship between
mathematical language and mathematical thinking, development mathematical language.
- Research content and curriculum of Mathematics using for pre-university
students.
- Researching the development of thinking and language of pre-university
students in the Central Highlands.
- Studying on the situation of the use of mathematical language in teaching
mathematics at the pre-university.

- Proposing a number of pedagogical measures to develop mathematical
language for pre-university students in the Central Highlands in teaching Maths.
- Pedagogical experiment to test the effectiveness and feasibility of the proposed
pedagogical measures.
5. Research method
5.1. Methods of theoretical research
We use a combination of research methods: collecting information, documents,
analyzing, synthesizing, etc. to study the theories of: language, mathematical
language, thinking, mathematical thinking of Pre-university students of blocks A, B.
At the same time researching Math content, programs and subjects of Maths which is
used for pre-university students.
5.2. Practical research methods
Coordinating the practical research methods to clarify the situation and test the
effectiveness and feasibility of the Doctoral thesis.
5.3. Information processing method
4


Using statistical methods to process data after investigating the situation, data of
pedagogical experiment process.
6. Scientific hypothesis
In teaching mathematics for pre-university students, if building and
implementing a number of teaching methods such as: Fostering knowledge of syntax
and semantics (namely consolidating vocabulary, semantics, syntax, internal
conversion capacity training in one language, converting from one Math language to
another); Practice using mathematical language in typical teaching situations
(specifically in conceptual teaching - theorem, in teaching rules - methods and in
teaching math solving); Practice mathematical communication skills (listening,
speaking, reading and writing skills); Developing mathematical language through
active teaching methods (problem-solving method, role-playing method, game

method and group work method) will contribute to university pre-university students'
development of mathematical language, through then improve the quality of teaching
- learning Math for pre-university students in the Central Highlands.
7. The contributions of the Doctoral thesis
Systematizing some theoretical issues about language, mathematical language,
thinking, mathematical thinking, development of mathematical language.
Analyzing the problem of mathematical language in the content of the
mathematical program for pre-university students.
Find out the situation of using the mathematical language of the pre-university
students in the Central Highlands.
Proposing 4 groups of measures to contribute to the development of
mathematical language for pre-university students in the Central Highlands.
8. Supporting contents for Thesis
- Concepts of language, mathematical language, thinking, mathematical
thinking, development of mathematical language of pre-university students in the
Central Highlands.
- Pedagogical measures to contribute to the development of mathematical
language for pre-university students in the Central Highlands.
- The results of the pedagogical experiment.
5


9. The structure of the Thesis
In addition to the Introduction, Conclusions and References, the Doctoral thesis
is presented in three chapters:
Chapter 1. The theoretical and practical basis
Chapter 2. Developing mathematical language for pre-university students in the
Central Highlands.
Chapter 3. Pedagogical experiments.

6


CHAPTER 1
THE THEORETICAL AND PRACTICAL BASIS
1.1. Theoretical basis
1.1.1. Overview of research issues
1.1.1.1. International studies
In 1952, Hickerson studied the meaning of the arithmetic symbols formed
during a student's math class.
In the 70s of the twentieth century, Jesse Douglas (1897 - 1965) focused on
studying the relationship between the capacity of using mathematical language and
students' thinking ability.
In 1986, Andrew Waywood studied the influence of mathematical language on
junior high school students.
In 1986, Martin Hughes in the book "Children and Numbers" proposed a
perspective on the early efforts of children to understand mathematics. He describes
the incredible knowledge of numbers that children know before they start class.
Understanding of pre-school numbers is an obstacle to learning mathematical
knowledge in the classroom.
In 1988, in the works "Second international handbook of mathematics
education", two mathematicians Stigler and Baranes mentioned the use of
mathematical language of elementary students in Japan, Taiwan, South Korea and the
United States.
Pimm (1987), Laborde (1990), Ervynck (1982) confirmed that the use of
mathematical language of students in maths is a barrier because the mathematical
language is much different from the daily use language.
In 1993, Diane L. Miller concluded that developing mathematical language has
a profound influence on the development of mathematical concepts [78].
In 1995, Eula Ewing Monroe and Robert Panchyshyn studied the lexical

problem of mathematical language, the need for vocabulary in the development of
mathematical concepts.
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In 2007, Chard Larson emphasized the role of mathematical vocabulary in the
understanding and learning of junior high school students. He believed that
mathematics is a language and students who want to master it must be able to use and
understand vocabulary. By using vocabulary quizzes and vocabulary-related activities
taken from math, students will better acquire an understanding of mathematical
concepts [75].
In 2008, Charlene Leaderhouse studied the mathematical language in the subject
of Geometry. He studied the mathematical language of 6th grade students in learning
geometry and concluded that the ability to understand and correctly use mathematical
terms will help them master the mathematical concept. To study geometry well,
children need to have many opportunities to discuss ideas and practice in teaching
which uses mathematical language [80].
In 2008, Bill Barton [74] concluded that everyday mathematical ideas were
expressed differently in different languages. Diversity occurs in the way language
expresses numbers, the language that describes the position of numbers and the
grammar of mathematical content expressions.
In 2009, Rheta N. Rubenstein researched the issues of how to help teachers
teach mathematics in high school to recognize the challenges that students often
encounter with mathematical symbols to propose teaching strategies that can alleviate
those difficulties. The study proposed solutions to help teachers know how to use
different symbols and identify common difficulties that students often encounter
when they speak, read and write symbols; At the same time, he also provided
teaching methods to avoid or overcome these difficulties [93].
1.1.1.2. Domestic studies
In 1981, Pham Van Hoan, Nguyen Gia Coc and Tran Thuc Trinh affirmed that

the correct expression of the relationship between “mathematical ideology content”
and “mathematical language form” wais the basis of important methodology of
mathematical education [30, p. 93].
In 1990, Ha Si Ho presented some concepts and characteristics of mathematical
language. Accordingly, the language of mathematics is primarily the language of
8


using signs, not the language of "speech" as in the language of mathematics. The
major mathematical language is the "written" language, which is both tight and
flexible [31, p. 45].
In 1992, Hoang Chung studied mathematics language and teaching
mathematical notation in high school.
In 1998, authors Ha Si Ho, Do Dinh Hoan and Do Trung Hieu mentioned many
aspects of mathematical language. Accordingly, it is necessary to have a language
suitable for expressing mathematical content, and at the same time overcome the
disadvantages of mathematical language [32].
In 2004, in the Thesis "Contributing to developing the capacity of logical
thinking and correct use of mathematical language for high school students in algebra
teaching", the author Nguyen Van Thuan proposed the pedagogical measures: Set
students to express some definitions and theorems in different ways; Train students to
use correct transformations; Practice using terms and symbols of mathematical logic
to express mathematical propositions [57, p. 82-135].
Recently, there are many direct and indirect studies on languages in teaching
high school mathematics, such as Tran Ngoc Bich [4], Vu Thi Binh [5], Thai Huy
Vinh [63],...
pre-university students in the Central Highlands are mainly ethnic minority
people in Ede. In recent years, there have been many studies on Ede language from a
linguistic perspective, such as: Malyo - Polynesian languages in Vietnam of Romal
Del and Truong Van Sinh [21]; Doctoral thesis of Doan Van Phuc (2009) with the

topic Phonetic of Ede language [46]; The doctoral thesis of Linguistics by Truong
Thong Tuan on the subject of Comparative method in the customary language of Ede
[61]; Doctoral thesis of Linguistics by Nguyen Minh Hoat (2012) with the subject of
type nouns in the Ede language.
The doctoral thesis of Linguistics by Doan Thi Tam (2012) with the topic of the
system of human words in Ede language. However, these works are only studied
from the perspective of the Ede language - the mother language of most of the preuniversity students in the Central Highlands.
9


1.1.2. Language
1.1.2.1. Concept of language
In this thesis, we agree with the concept of language of author Nguyen Thien
Giap in [22, p. 28]: "Language is a special audio signal system, a means of
mechanical communication and the most important of the members in a community
of people; language is also a means of developing thinking, conveying culturalhistorical traditions from generation to generation”.
1.1.2.2. Function of language
Communicative function
Language is the most important means of human communication, helping people
understand each other in the life and labor process; is a production tool, a class
struggle tool.
Reflective function
Language is a means of thinking. Human language is born and developed
because people feel the need to say something to each other, to be informed to others
in the community, that is the results of the reflection of the world objects (the
thinking) of human beings.
Function to show thinking
Language is the actual expression of thought, directly involved in the process of
thought formation. Human language exists in the form of speech (sound symbols in the
brain) and writing. Therefore, the reflection of the language does not only indicate when

the language is spoken into the word, but when silent thinking or writing the paper.
1.1.2.3. Nature of language
1.1.2.4. Characteristics of language
1.1.3. Mathematical language
1.1.3.1. A brief history of the development of mathematical languages
1.1.3.2. Concept of mathematical language
a. Concepts
In this thesis, we agree with the viewpoint on mathematical language of author
Nguyen Duc Dan in [20]: "Mathematical language includes symbols, terms (words,
10


phrases), symbols. and the rules that combine them as a means to express
mathematical content in a logical, accurate, and clear manner. Symbols include
numbers, letters, alphabetic characters, mathematical signs, relational signs, and
parentheses are used in maths. Symbols include images, drawings, diagrams or
models of specific objects ”.
b. Semantics and syntax
1.1.3.3. Function of mathematical language
a. The language of mathematics is a direct realization of thinking
b. The language of mathematics reflects thinking
c. Mathematical language is a means of communication in mathematical
activities
1.1.3.4. The role of mathematical language in the Math curriculum for college
preparatory students
1.1.4. Thinking and Mathematical thinking
1.1.4.1. Thinking
1.1.4.2. Mathematical thinking
1.1.4.3. Some manipulations of mathematical thinking
a. Analytical and general operations

These are two contradictory operations, analysis to find the solution of
problems, synthesis is a process of discovering relationships that unify parts that
seem to be separate. This is a method of putting assumptions together to find
solutions.
b. The similarity
According to G. Polya: "The two systems are similar if they fit together in well
defined relationships between the corresponding parts" [26, p. 23]. According to
Hoang Chung: The similarity often means the same. People often consider similar
problems in mathematics in the following aspects: Two proofs are similar if the way
and method of proof are the same; The two figures are similar if they have many
similar properties, if their roles are the same in certain problems or if their respective
elements are the same [16, p. 8-9].
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C. Generalization, specialty
Generalization and specialty are two different thinking actions that are
contrary. Specialization is the presentation of specific cases, particular cases of the
problem. Generalization is also generalizing the problem from specific cases and
particular cases.
d. Comparison
Comparing mathematical objects helps students identify similarities and
differences of two or more objects. Regularly comparing students will have a more
comprehensive view of the problem. We often perform comparisons such as:
compare two concepts, compare two definitions, compare two issues,...
1.1.5. The relationship between language and thinking, mathematical
language and mathematical thinking
1.1.5.1 The relationships between language and thinking
Language and thinking are a unified but not a consistent relationship. The
language exists in the form of material while thinking is the form of spirit. Language

is perceived by humans by senses such as pitch, field, tone, etc. and thinking is the
inner awareness of the human brain in a certain logical order. The language is
nationalistic (the product of the nation) while thinking is humankind (all countries
have the same products of thinking about the problem: sovereignty, peace, education,
health,...) [22].
1.1.5.2. The relationship between mathematical language and mathematical
thinking
Mathematical language is both a tool and a material shell of mathematical
thinking. Mathematical language and mathematical thinking are a unified but not
consistent. This is reflected in the fact that mathematical language exists in physical
form, mathematical thinking exists in mental form. Units of mathematical language
are perceived by the senses and have physical properties such as pitch, intensity, etc.
And mathematical thinking is not perceived by such senses, there are no properties.
of matter such as mass, weight, taste, etc. The activity of mathematical thinking
requires rational and logical while the mathematical language operates according to
12


habit. Units of mathematical thinking are not identical with the units of mathematical
language. The function of mathematical language for mathematical thinking is to
express ideas and directly participate in the formation of ideas [28].
1.1.6. Communication skill and mathematical communication skills
1.1.6.1. Communication skills
Communication skills are a set of rules, the art of how behavior, responses are
molded through practical experiences, making effective communication and
achieving the purpose of specific circumstances.
1.1.6.2. Mathematical Communication Skills
Math communication skill is the ability to understand mathematical problems
through communication, speaking, reading and writing. It is the ability to effectively
use mathematical language in a close relationship with the mathematical language for

exchanging, presenting, explaining, arguing and proving mathematicians accurately,
logically, clarifying mathematical ideas in particular contexts.
1.1.7. Development and development of mathematical language
1.1.7.1. Development concept
According to the Vietnam Encyclopedia "Development is a philosophical
category that indicates the nature of the changes taking place in the world.
Development is an attribute of matter. All things and phenomena of reality do not
exist in a different state from appearance to death,... the source of development is
unity and struggle between opposites ”[62].
1.1.7.2. Developing mathematical language
a. Develop competency in using mathematical language
Competence in using mathematical language includes:
* The ability to receive and understand knowledge and skills about
mathematical language
* The ability to create and apply effectively mathematical language in
communication as well as thinking.
* The ability to select and convert languages in learning and in practice.
b. Capacity development of mathematical performances
13


* Conventional performances and non-conventional performances
* Internal performance and external performance
c. Develop mathematical communication competence
1.1.7.3. The scale of assessing development level of mathematical language for
pre-university students in the Central Highlands
Table 1.4: A scale for assessing Boleslaw's levels of thinking
Thinking level
Identification

Level
Students remember the basic concepts, can state or recognize
them when required
Students remember basic concepts and can apply them when they

Understanding

are presented in ways similar to the way that teachers teach or as
typical examples of them in the classroom.
Students can understand the concept of a higher level of

Operation

"understanding", creating logical links between fundamental

(Low Level)

concepts and being able to use them to reorganize the information
that is presented as similar to a teacher's lecture or in Textbooks
Students can use the concepts of subject-topic to solve new

Operation

problems, unlike those learned or presented in textbooks but it is
appropriate to be solved with the skills and knowledge taught at

(High Level)

the this perception. These are the same problems that students
will encounter in society.

Based on the rating scale of Boleslaw's thinking levels, in this thesis, we propose
the following levels of mathematical language development for university
preparatory students in the Central Highlands as follows:

14


Table 1.5: Scale for evaluating mathematical language development levels
for pre-students in the Central Highlands
Development

Indicator

level

Students remember symbols, mathematical terms and grasp the
Level 1

syntax of mathematical language. Students read the correct names,
recognize symbols, mathematical terms and correctly use
mathematical symbols and terms in a single form.
Students use correctly and accurately mathematical symbols and

Level 2

terms; Correctly associates mathematical symbols in simple form.
Initial reading, understanding mathematical content through
drawings, diagrams, visual images.
Students use correctly and accurately mathematical symbols in

Level 3

complex forms; Initially, knowing the mathematical content
through drawings, diagrams, visual images.
Students know: Reading and understanding correctly the
mathematical contents presented in written language or diagrams,

Level 4

drawings; using mathematical language to present math problems
in written language in a coherent, logical and accurate manner.
Using mathematical language to listen and understand what others
have to say and present mathematical problems.

Thus, in order to achieve the above levels, it is necessary to have a system of
measures to help pre-university students develop the mathematical language in
teaching Maths.

15


1.2. Practical basis
1.2.1. Mathematical program for pre-university students [2]
1.2.1.1. Goal, request
a. Target
b. Knowledge
c. Skills
1.2.1.2. content
Table 1.6: Distribution table of Mathematical curriculum

for pre-university students
Algebra
Number of periods
SQN Chapter

Name of chapter

Total Theory

Exercises,
Review

1

1

Combinations and probability

25

12

13

2

2

Equations, systems of equations,
inequalities

45

22

23

3

3

Trigonometric

15

7

8

4

4

Derivative and application

30

16

14

5

5

Primitive and integrals

18

8

10

6

6

Complex numbers

7

4

3

Total

140

69

71

Geometry
Number of periods
SQN Chapter

Name of chapter

Total Theory

Exercises,
Review

1

1

Vector

8

3

5

2

2

Straight lines and planes in space

29

14

15

3

3

Polyhedron blocks - Spherical surface
- Cylindrical surface - Cone surface

11

5

6

4

4

Method of coordinates in the plane

15

8

7

5

5

Method of coordinates in space

21

12

9

84

44

40

Total
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1.2.2 Review of the mathematical program of pre-university rank
A. Advantages
B. Limitations
C. Proposals
1.2.3. Characteristics of pre-university students

1.2.4. Surveying the status of mathematical language development of preuniversity students
1.2.4.1. Survey purpose
1.2.4.2. Survey object
1.2.4.3. Survey content
1.2.4.5. Survey results
1.2.4.6. Cause of the situation
1.2.5. Conclusion of the situation of mathematical language development in
pre-university students

17


Conclusion of chapter 1
In Chapter 1, we investigated and clarified the following present, analyze and
clarify a number of issues related to language, mathematical language, thinking,
mathematical thinking, the relationship between language and thinking, between
mathematical language and mathematical thinking, communicative skills,
mathematical

communicative

skills,

development

and

development

of

mathematical language.
Presents an overview of the concept, function, nature, characteristics of
language as well as the relationships between language and thinking.
Introducing the Maths program for pre-university students. Thereby analyzing
the characteristics of the program as well as raising some notes when teaching Maths
for pre-university students.
Finding out the situation of developing the mathematics language for the preuniversity students, thereby knowing the situation and seeing the necessary to train
and develop the mathematics language for the pre-university students in the Central
Highlands.
The above-mentioned content is the basis for which we propose the following
issues and also new points of the thesis.
- Clarify and analyze mathematical language of pre-university students.
- Finding out

and surveying

the status of the mathematical language

development of the pre-university students in Central Highlands at the current.
- Proposing some teaching measures to develop mathematical language for preuniversity students in the Central Highlands.

18


Chapter 2
DEVELOPMENT OF MATHEMATICAL LANGUAGE FOR
PRE-UNIVERSITY STUDENTS IN THE CENTRAL HIGHLANDS

2.1. Some principles in formulating and implementing measures

In order to develop mathematical language for pre-university students, we set up
a number of teaching measures, which must ensure the principles:
2.1.1. Consistent with the characteristics of teaching Maths in the pre-university
program
2.1.2. Consistent with the principle of teaching Maths in the pre-university program
2.1.3. Consistent with the psychology of pre-university students and special
characteristics of ethnic minority students
2.1.4. Ensuring the feasibility in the current practical conditions of teaching
mathematics at pre-university schools
2.2. A number of orientations in developing and implementing measures
2.2.1. Organize learning activities to enable students to realize the role of Maths in
the pre-university Program
2.2.2. Exploiting thoroughly the knowledge, experiment and experience of
students as a basis for creating new knowledge
2.2.3. Build a positive collaborative learning environment, always encouraging
students to exchange, discuss, explore, discover and solve problems
2.2.4. Focus on helping students to create a connection between theoretical
content, applying theory with practice
2.3. Some measures for the development of mathematical languages for preuniversity students in the Central Highlands
2.3.1. Measure 1: Reinforce the mathematical language knowledge and
fostering language transformation capacity for students
2.3.1.1. Measure 1.1: Reinforce vocabulary and semantics of mathematical
language for students
2.3.1.2. Measure 1.2. Reinforce the syntax of mathematical language for students
19


2.3.1.3. Measure 1.3: Develop mathematical language through fostering the
internal transformation capacity of a language
2.3.1.4. Measure 1.4: Developing mathematical language through fostering the

ability to convert from one mathematical language to another
2.3.2. Measure 2: Develop mathematical language through practice used in
typical teaching situations
2.3.2.1. Measure 2.1: Developing mathematical language through practice used
in teaching concepts and theorems
2.3.2.2. Measure 2.2: Developing mathematical language through practice used
in teaching rules and methods
2.3.2.3. Measure 2.3: Developing mathematical language through practice used
in teaching maths
2.3.3. Group of measures 3: development of mathematical language through
training of mathematical communication skills (listening, speaking, reading and
writing)
2.3.3.1. Measure 3.1: Development of mathematical language through training
of listening skills in mathematics
2.3.3.2. Measure 3.2: Development of mathematical language through training
speaking skills in mathematics
2.3.3.3. Measure 3.3: Development of mathematical language through training
of reading skills in mathematics
2.3.3.4. Measure 3.4: Development of mathematical language through training
in writing skills in mathematics
2.3.4. Group Measure 4: Development of mathematical language through
aggressive teaching methods
2.3.4.1. Measure 4.1: Development of mathematical language through the
organization of problem solving methodology
2.3.4.2. Measure 4.2: Development of mathematical language through the roleplaying organization
2.3.4.3. Measure 4.3: Development of mathematical language through a gamerelated teaching organization
2.3.4.4. Measure 4.4: Development of mathematical language through teaching
organization by teamwork method
20

Conclusion of chapter 2
Based on the theoretical and practical studies presented in Chapter 1 and Chapter 2,
we focus on researching and proposing measures to develop mathematical language for
pre-university students in the Central Highlands. Based on the principles and
orientations, we have built measures to foster and develop mathematical language for
students, including 15 specific measures in 4 groups of measures:
1) Measure 1: Developing mathematical language through strengthening
knowledge of mathematical language (4 measures). Implementing this group of
measures helps students consolidate their knowledge of mathematical language,
understand and master mathematical symbols and terms, semantics and syntax of
mathematical language; Know how to convert internally a language (synthetic
geometry language, vector language,...); Know how to convert from one language to
another (synthetic geometry language into vector language, synthetic geometry
language to coordinate language,...).
2) Measure 2: Develop mathematical language through training to use in
teaching situations (3 measures). Implementing this group of measures will train
students to use mathematical language in teaching formation, reinforcing concepts,
teaching rules and methods, teaching solving math problems;
3)

Measure

3:

Develop

mathematical

language

through

practicing

communication skills (4 measures). Implementing this group of measures will
develop communication skills (listening, speaking, reading and writing) in
mathematical language for students;
4) Measure 4: Develop mathematical language through active teaching methods
(4 measures). Implementing this measure group will develop mathematical language
through organizing non- traditional teaching activities (problem-solving method, roleplaying method, game method, working method group).
In each measure, we propose suggestions and guide teachers to organize activities
for students in the process of teaching Mathematical content in the pre-university
program issued by the Ministry of Education and Training in the year 2012 [2].
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The proposed measures are also always considered to ensure the scientific,
practical, regional and specialized elements of the subjects of pre-university students.
Measures indicate the development according to the rating scale which are
proposed in Chapter 1.
To confirm the feasibility and effectiveness of the proposed measures, we
conduct pedagogical experiments.

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Chapter 3
PEDAGOGICAL EXPERIMENTS
3.1. Experimental purposes

Pedagogical experiments are conducted for the purpose of testing the feasibility
and effectiveness of the pedagogical measures which are proposed in Chapter 2.
3.2. Experimental organization and content
3.2.1. Experimental organization
The experiment was conducted in two phases:
Phase 1: Conducted from January 2018 to April 2018. Experiment at pre-university
classes K2017A, K2017B, Pre-university Department of Tay Nguyen University.
- Experimental class is pre-university K2017A; Teacher: Tran Quynh Mai.
- The control class is pre-university K2017B; Teacher: Tran Quynh Mai.
Phase 2: Conducted from January 2019 to April 2019. Experiment at pre-university
classes K2018A, K2018B, pre-university Department of Tay Nguyen University.
- Experimental class is pre-university K2018A; Teacher: Tran Quynh Mai.
- The control class is pre-university K2018B; Teacher: Tran Quynh Mai.
3.2.2. Experimental content
Phase 1: The experiment was conducted in 15 periods with the content of
Coordinate Method in the plane of Chapter IV [35].
Phase 2: The experiment was conducted in 20 periods with the contents of
Equations, Inequalities, and Equations system of Chapter II [34].
3.3. Evaluate experimental results
3.3.1. Qualitative evaluation
3.3.2. Quantitative evaluation
Conclusion Chapter 3
This chapter presents the purpose, content and key outcomes of the experimental
batches. The pedagogical experiment aims to test the scientific hypothesis of the
thesis through teaching practices and the feasibility test and the feasibility of the
proposed pedagogical measures. Pedagogical experiments have been conducted two
times at pre-university classes of K2017A, K2017B, K2018A and K2018B, preuniversity Department of Tay Nguyen University.
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CONCLUSIONS AND RECOMMENDATIONS
The thesis has completed the research contents and tasks, building the
pedagogical measures to develop the mathematics language for the pre-university
students in the Central Highlands. The thesis obtained the following main results:
1. Overview of language, mathematical language, thinking, mathematical
thinking, the relationship between language and thinking, between mathematical
language and mathematical thinking.
2. To give an overview of the communicative skills, mathematical skills,
development and development of mathematical language.
3. Researching the current situation of studying Maths, situation of
mathematical language development of pre-university students, clearly analyzing the
causes as a basis for proposing measures to develop mathematical language.
4. Identify the four principles for developing and implementing measures for
developing mathematical language. Specifically, the measures must be suitable to the
characteristics of teaching Maths; in accordance with the principle of teaching Maths;
suitable with the psychology of pre-university students and special characteristics of
ethnic minority students; ensure feasibility in the current practical conditions of
teaching mathematics in pre-university schools.
5. Identify the four orientation to construct and implement the mathematical
language development measures for the pre-university students. Specifically, the
measures must be built in the direction of: Organizing learning activities to create
conditions for students to realize the role of Maths in the pre-university program;
Fully exploit the knowledge, experiment, experience of students as the basis for
creating new knowledge; Building a positive collaborative learning environment,
which always encourages students to exchange, discuss, explore, discover and solve
problems; Focusing on helping students create the relationship between the
theoretical content, applying theory and practice.
6. On the basis of principles and orientations, we proposed 15 specific
measures (belonging to four groups) to develop mathematical language. For each
measure, there are five parts: objectives, content, steps of implementation, examples

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