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

MINISTRY OF EDUCATION AND TRAINING
VIETNAM NATIONAL INSTITUTE OF EDUCATIONAL SCIENCES

NGUYEN THI DUNG

TEACHING ADVANCED MATHEMATICS TOWARDS
DEVELOPING ANALYTICAL THINKING FOR UNIVERSITY
STUDENTS MAJORING IN ECONOMICS AND ENGINEERING

SUMMARY OF DOCTORAL THESIS IN EDUCATIONAL SCIENCE
Major:
Code:

Theory and methodology of Mathematics teaching
9 14 01 11

HANOI, 2020


The thesis is completed at
Vietnam National Institute of Educational Sciences

Supervisor:
1.
Dr. Tran Dinh Chau
2.
Ass Prof. Dr. Do Tien Dat

Reviewer 1: ………………………………………………………….
……………………………………………………………
Reviewer 2: …………………………………………………………

……………………………………………………………
Reviewer 3: …………………………………………………………..
…………………………………………………………….

This thesis will be reported in front of the Council of thesis evaluation level
Institute at the Vietnam National Institute of Educational Science, 101 Tran Hung
Dao, Hanoi.
On…Date…………………………………………………………

The thesis can found at
- The National Library
- The Library of Vietnam National Institute of Educational Sciences


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INTRODUCTION
1. Reason for choosing the topic
 The necessity of analytical thinking development
In recent years, learning and developing thinking has been getting more and more
attention from researchers, learners and working people who want effective thinking to
apply in all aspects of life.
By referring to the literature on teaching and learning methodologies, teaching and
learning theories at universities, etc., we can find certain parts related to analytical
thinking, highlighting the importance of analytical thinking for students.
In fact, analytical thinking requires a clear, detailed and insightful understanding
of the problem. This is necessary when you want to classify and use information, when
self-study to help remembering, judging, choosing to make decisions, and avoiding
making mistakes. Dividing complex problems into many small problems creates one of
the methods to solve the problem.

For those who work in the engineering industry, they need to be careful,
meticulous, pay attention to details, interactions between components in the system of
machinery or construction, network system, etc. For those who work in economics, they
need to collect and select the correct information, categorize, compare, evaluate and
explain causes, investigate thoroughly and make judgments, forecasts, etc. There have
been examples of applying analytical thinking in considering market needs, analyzing
financial statements, analyzing causes of failure of a company or a team, etc. Clear and
logical analytical skills can help each person be more successful in his or her career.
As pointed out by several recent studies around the world that analytical thinking
is necessary for students in the 21st century that will lead to the development of critical
thinking, creative thinking and problem solving competence. It helps improve thinking,
create a habit of thinking and asking questions in every aspect of life. Analytical thinking
and critical thinking contribute to reducing the unemployment rate of students after
graduation.
Today, there are many domestic and foreign universities that really appreciate the
practice of analytical thinking for students. In 2009, the Posts and Telecommunications
Institute of Technology announced output standards for students, in which students need
to have analytical thinking.
 The role of students’ analytical thinking in teaching Advanced
Mathematics
Advanced Mathematics consists of a number of subjects during the first academic
year of students at their universities. For freshmen, most of them are like fish out of
water to the new life and methods of learning and research at the university level. They
will find that the Advanced Mathematics has more abstract contents than those of
mathematics at the high schools. In order to handle the vast amount of knowledge, they
need to know how to self-study, and have a high level of creative and independent
thinking. Analytical thinking will help them learn the Advanced Mathematics more
easily. By asking analytical questions, for example, they can orient themselves on the
process of thinking and solving problems independently. Such orientation helps them to
see and state propositions in a number of different ways, making it easier for them to

understand the problem and present it more clearly. This also helps them analyze and
choose how to solve problems, avoiding just sticking in one direction which can lead to


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deadlock. Paying attention to relationships helps them create a habit of linking Advanced
Mathematics’s knowledge with the reality and their major, etc. Thus, analytical thinking
plays an important role for students in studying Advanced Mathematics. Advanced
Mathematics teaching towards developing student analytical thinking is considered
necessary.
In addition, we believe that in the Advanced Mathematics teaching process, it is
necessary to create many opportunities to develop analytical thinking for students,
thereby contributing to train them some skills according to the output standards.
At present, we have not found any comprehensive and systematic research on
analytical thinking and Advanced Mathematics teaching towards developing analytical
thinking for students.
Those are the reasons making us decide to choose the topic: “Teaching Advanced
Mathematics towards developing analytical thinking for university students majoring in
economics and engineering”. Within the scope of content and time frame of the subject,
we only provide some illustrative examples which have not covered such specific and
rich situations like the real life. However, from which students can find suggestions in
order to partially relate to the specific cases they will encounter later.
2. Research objectives
Based on theoretical and practical research on analytical thinking and Advanced
Mathematics teaching towards developing analytical thinking for university students
majoring in economics and engineering, there are a number of proposed measures for
teaching Calculs towards developing analytical thinking for university students majoring
in economics and engineering.
3. Research overview

So far, there have been studies on analytical thinking of many authors such as
Saccacop, Koliagin, Ayman Amer, Ben Johnson, etc. These studies have provided some
conceptions of analytical thinking; the importance of analytical thinking; the relationship
between analytical thinking and some other types of thinking; some tools and strategies
which are often used for analytical thinking.
As argued by some authors: The use of problems with parameters, graphs, min,
max functions, etc., in Calculs teaching can help develop analytical thinking for students.
4. Research object and subject
- Research object: Advanced Mathematics teaching process for university
students majoring in economics and engineering.
- Research subject: Advanced Mathematics teaching process towards developing
analytical thinking for university students majoring in economics and engineering
5. Scientific hypothesis
When manifestations of analytical thinking of university students majoring in
economics, engineering are identified in Advanced Mathematics teaching, the
development and implementation of a number of appropriate measures in teaching
Advanced Mathematics in the direction mentioned above can contribute to develop
analytical thinking for students while improving the Advanced Mathematics teaching
effectiveness.
6. Research tasks and scope
Research tasks:


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- Conduct research on theoretical and practical basis of analytical thinking and
Advanced Mathematics teaching towards developing analytical thinking for university
students majoring in economics and engineering
- Identify Advanced Mathematics teaching methods towards developing analytical
thinking for university students majoring in economics and engineering

- Conduct research on pedagogical experimentation in order to initially test the
feasibility and effectiveness of proposed measures.
Research scope:
- The scope of Advanced Mathematics in this research is only referredto Calculus.
- Experimental
will be conducted for freshmen at the Posts and
Telecommunications Institute of Technology.
7. Research methods
Theoretical research, survey, observation, expert method, case study, experience
summary, pedagogical experimentation, mathematical statistics are methods to be used.
8. Issues that are brought up for defense
- Some manifestations of analytical thinking of university students majoring in
economics and engineering in Advanced Mathematics teaching.
- Feasibility and effectiveness of some pedagogical methods in Advanced
Mathematics teaching methods towards developing analytical thinking for university
students majoring in economics and engineering
9. Contributions of the dissertation
- Concepts of analytical thinking and Advanced Mathematics teaching methods
towards developing analytical thinking for university students majoring in economics
and engineering
- Some manifestations of analytical thinking of university students majoring in
economics and engineering in Advanced Mathematics teaching.
- Some pedagogical methods in Advanced Mathematics teaching towards
developing analytical thinking for university students majoring in economics and
engineering
10. Dissertation structure
The dissertation consists of 3 chapters, in addition to the Introduction, Conclusions
and recommendations, List of scientific works by authors related to the dissertation,
Reference and Appendix.
CHAPTER 1: THEORETICAL AND PRACTICAL BACKGROUND

1.1. General information about thinking
In this section, some of the issues are presented: Concepts of thinking,
characteristics of thinking, thinking processes, basic elements of thinking processes,
thinking manipulations, and types of thinking.
Richard Paul and Linda Elder stated that every thinking include these elements:
Setting goals, raising questions, using information, using concepts, creating inferences,
making assumptions, raising implications and containing a perspective.
1.2. Analytical thinking
12.1. Concepts of analysis


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This section covers some concepts of analysis by Bloom, Sardakov, Chu Cam
Tho, etc.
Bloom gave some common errors in analysis: Failure to see the relationship
between the elements and the meaning that makes up the whole; Analysis without
quality, without insight, etc.
1.2.2. Concept of analytical thinking
From the analysis, summarization of some concepts of analytical thinking of the
authors Koliagin, Sardakov, etc., we boldly express our opinion of analytical thinking as
follows:
Analytical thinking is the type of thinking which clearly and thoroughly
understand the subject. The thinking process takes place based on a close examination of
elements of the subject and their relationship to each other, the whole and the external
elements. Then reflection, judgement and reasonable conclusion are made based on
logical background and reasoning.
1.2.3. Characteristics of analytical thinking
Analytical thinking has a number of following basic characteristics: Preferring
analysis when approaching the subject; Frequently raising and answering questions,

especially questions which break down problems, explain and reasoning; Examining the
subject clearly and thoroughly; Using thinking manipulations, judgments and reasoning;
Problem-based method basing on information, evidence and logics.
1.2.4. The relationship between analytical thinking and some other types of thinking
and problem-solving skill
Analytical thinking is related to many other types of thinking (synthesis thinking,
critical thinking, creative thinking, logical thinking, etc.) and to the problem-solving
competency.
1.3. Characteristics of Advanced Mathematics
Advanced Mathematics of universities majoring in economics and engineering
generally includes Mathematical analysis and Linear Algebra. Depending on each
discipline, each period, the content of this subject is adjusted by each university
management in detail as appropriate. The modules of Advanced Mathematics are usually
taught to the freshmen. For Mathematical analysis, the students majoring in engineering
at the Posts and Telecommunications Institute of Technology study Mathematical
analysis 1 and Mathematical analysis 2. Students in the field of Economics study
Advanced Mathematics 1 with the following main contents:
- Mathematical analysis 1 (45 periods): Chapter 1: Set of numbers, limit of a
sequence. Chapter 2: Differential calculus of single-variable function. Chapter 3: Integral
calculus. Chapter 4: Theory of series.
- Mathematical analysis 2 (45 periods): Chapter 1: Differential calculus of multiple
variable function. Chapter 2: Multiple Integrals. Chapter 3: Line integrals and surface
integrals. Chapter 4: Differential equation and systems of differential equations.
- Advanced Mathematics 1 (30 periods): Chapter 1: Functions and limits. Chapter
2: Derivative and differential. Chapter 3: Integral calculus. Chapter 4: Multiple variable
function. Chapter 5: Differential equation.
The Advanced Mathematics teaching process does not go into proving math
problems but focuses on creating foundation knowledge so that students can study
specialized subjects and apply a part of that knowledge in the future. Although there are

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a number of theorems not proven in the subject content, students are still required to
understand the concepts, theorems and their practical applications.
The Advanced Mathematics has more abstract contents than those of mathematics
at the high schools. The content of knowledge in the program is systematic, rich and
profound. The students must acquire a lot of knowledge during an hour of study,
understand the nature, think independently and learn more than they did when studying
maths in high schools.
There are many situations in Advanced Mathematics teaching suitable for
practicing logical reasoning and reasoning; helping to express clear, careful and
insightful thoughts; finding out any relationships to reality or specialized issues.
1.4. Some characteristics of university students majoring in economics and
engineering
In this section, we will present some characteristics of thinking development of
university students compared to that of high school students, especially independence
and creativity; as well as some difficulties for the freshmen. We also point out some
characteristics of people working in economics and engineering sectors (and many
students who love these fields may have some of the same); some characteristics of
thinking that university students majoring in economics and engineering need to practice
during their study process to contribute to the development of career skills.
1.5. Manifestations of analytical thinking of university students majoring in
economics and engineering in Advanced Mathematics teaching
For the purpose of identifying manifestations of analytical thinking of university
students majoring in economics and engineering in Advanced Mathematics teaching, we
rely on:
- Basic elements of thinking: Setting goals, raising questions, using information,
using concepts, creating inferences, making assumptions, raising implications and
containing a perspective. (As provided in the chapter 1.1.3, according to Richard Paul

and Linda Elder).
- Concepts and basic characteristics of analytical thinking: Thinking clearly and
deeply; dividing subjects, finding out any relationships, making judgments, deducing,
making logical arguments, etc.
- Characteristics of Advanced Mathematics; characteristics of university students
majoring in economics and engineering; activities of university students majoring in
economics and engineering when studying Advanced Mathematics. For examples: The
students need to determine the direction and choose the appropriate solution for the
problem; understand and use the meaning of mathematical concepts to model
mathematics in solving practical problems (or specialized problems); select, organize,
classify information during the course of scientific research or problem presentation, etc.
These manifestations are also considered through activities that show the
connection between analytical thinking and some other types of thinking and with the
problem-solving skill and activities towards contributing to training vocational skills for
students.
Since then, we believe that some typical manifestations of analytical thinking
those university students in economics and engineering in Advanced Mathematics
teaching should have are:
1.5. 1. Determining the purpose of thinking, breaking down the goal


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The students should identify goals when solving problems (when solving
problems, reading textbooks, etc.), then break that goal down into many meaningful
goals, and regularly adjust their thinking to that goal.
1.5.2. Determining the direction of view
The students need to determine the direction when solving problems. When
solving problems, they need to divide a number of solutions and prilimilarily envisage a
number of steps for each direction, as well as advantages and disadvantages of each

direction. When reading the material, they recognize which the direction and views the
author presents. They may explain a problem or re-state the proposition in a different
way, looking at some of the problems in a way that is relevant to the reality or their
major. When understanding information, they review from multiple sources. They realize
structure of a document, etc.
For
example,
when
studying
about
definite
integrals:
b


a

n

f ( x)dx 

lim
maxxi 0

( x1 , x2 ,..., xn

 f (i )xi
i 1

is

the

arbitrary

points

on

 a, b  ,  i

is

any

point

on

 xi 1, xi  , xi  xi  xi1, i  1, n).
The students majoring in engineering can relate to problems involving area and
volume calculations and they have to consider. xi (i  1, n) very small.
The students majoring in economics can relate to functions and variables that are
commonly used in economics and they consider xi to be at least 1.
1.5.3. Finding out information clearly and deeply. The information may be used to
solve the problem
The students should explore information (assumptions of problems, facts, etc.) in a
detailed, clear and insightful manner; with information interpretation; investigate any
connection between information with each other and with the problem to be solved;
organize, classify and synthesize information; recognize the correct information, wrong

information, important information, missing information; correct and supplement
information as well. They can learn information and materials related to Advanced
Mathematics and apply it in practice and specialized subjects. Use relevant information
to address the issue.
1.5.4. Understanding concepts (theorems, propositions, rules, methods) clearly and
deeply and using them to solve problems.
The students can interpret and present clear and detailed concepts (theorems, etc.);
identify any relationships between concepts (theorems, etc.) with each other and with the
problems to be solved, problems in the reality or their major. It is possible to use
concepts (theorems, etc.) to solve related problems, make some meaningful comments in
the process of in-depth study of concepts (theorems, etc.).
1.5.5. Making a grounded judgment, in relation to the question posed
The students need to make judgments related to the problem posed (predict a
solution, give a hypothesis, guess the author's ideas, etc.) based on following evidences:
consideration of relationships, use of information and thinking manipulations:
specialization, generalization, analogization, overturning, etc. Some judgments can be
made for solving practical or discipline-related issues. They often have to review


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judgments carefully.
1.5.6. Deducing clearly and thoroughly, step by step, based on grounds related to
the issue
Students should make clear inference step by step. Such inference is based on
grounds and evidence, make in-depth inference through a number of several steps and
directions. They often make logical, meaningful conclusions that relate to each other and
either practical or major-related issues. They often flip back and forth the problem,
scrutinize the whole process of inference and draw comments and lessons.
1.5.7. Raising analytical questions

The students often raise questions that require detailed explanations, identification
of relationships (the relationship between the elements with each other and with the
problem to be solved, with the reality or majors); pose questions about purpose,
questions about vision, information, bases, concepts, theorems, propositions,
experiences, related solved problems; questions about judgment and reasoning; questions
about clarity, deep thinking, etc. These questions are often broken down.
 Analytical thinking of students is manifested through the process of carrying out
activities. To give the manifestation degrees, we rely on consideration that the students
can perform in the manifestation, as well as the hierarchy of activities as introduced by
Nguyen Ba Kim. The hierarchy is applied based on the following grounds: complex
nature of the operating object; abstract and general nature of the object; contents of any
activity; activity complexity; activity quality; coordination in many aspects serve á the
basis for activity hierarchy.
We propose the following levels: Level 1: Students can partly perform
requirements of each above-mentioned manifestation with lecture suggestion; Level 2
Students can partly perform requirements of each above-mentioned manifestation
without lecture suggestion; Level 3: Students have above-mentioned manifestation
without lecture suggestion.
In addition, at each level, under appropriate circumstances, smaller levels may be
considered based on the complexity ofoperating object; abstraction and generalization of
the object, content of the activity or complexity of the activity.
For example: When applying the Maclaurin expansion of the function to find the
limit, the lecturer states that:

x3

x  x 
 o(x 3 )



3!
x  sin x
lim
 lim
(1)

2
x  0 x (1  cos x )
x 0 


x

x 1  1 
 o(x 2 )
2!
 

x3
 o(x 3 )
6
(2)
 lim
x 0 x 3
 o(x 3 )
2


8

x3
1
 lim 6  . (3)
x 0 x 3
3
2
and raises questions to the students: “Do you have any question? Whether you can
explain the solution above or not?”
- Student A: Why do you do that?
When that question is commented by the lecturer as too general, students should
ask more specific questions in the part that is not understood. For example, at step (1), it
is necessary to compare two sides, and ask: Have you replaced sin x with

x3
x
 o(x 3 ) ?”, Student A continues to ask: In step (1), have you replaced cos x with
3!
x2
1
 o(x 2 ) ?
2!
x3
 o(x 3 )
?
- Student B: At step (2), why don't we write lim 36
x 0 x
2
 x .o(x )
2
From (2) to (3) have we used higher order infinitesimals omission rule?

Student C: “I think that in order to understand this solution and work it out when
doing another task, we need to answer the following questions:

+ Whether it is x . o(x 2 )  o(x 3 ) or not? Is o(x 3 ) also is o(x 3 ) or not?
+ From (2) to (3) we have removed some terms, have we used higher order
infinitesimals omission rule?
x3
+ Why do we write sin x  x   o(x 3 ) in stead of sin x  x  o(x ) .
3!
2
x
x4
cos
x

1


 o(x 4 )? What happens if we write
Similarly, why don't we write
2! 4 !
like that?
In addition, to solve this problem on our own, we must first ask and answer the
questions: What knowledge does this exercise involve? It is related to Maclaurin
development and limit, but Maclaurin development is related to infinitesimal. So, are
there any properties and comments related to the infinitesimal and limit parts? For
examples: Replace equivalent infinitesimals, use the higher order infinitesimals omission
x3
 o(x 3 )
rule, etc., in addition, we can ask that: If we need to find out lim 36

, can the
x 0 x
2
 x .o(x )
2
computer do it, and do we need to pay attention to the typing of this limit search
command?
In the above case, student A shows a better ability to ask questions than student B


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and student C has a better ability to ask questions than student B. Student A asks general
questions, which need additional suggestions from the lecturer. Student B raises himself
more detailed, situation-fitted questions, without the lecturer’s suggestion, but that
question only explores the causes, without paying attention to the concepts and related
definitions and different perspective. Student C asks specific questions, helping him or
her to solve the problem, and also show a sense of connection with the major.
1.6. Teaching Advanced Mathematics towards developing analytical thinking for
university students majoring in economics and engineering
1.6.1. The concept of teaching Advanced Mathematics towards developing
analytical thinking for university students majoring in economics and engineering
We believe that teaching Advanced Mathematics towards developing analytical
thinking for university students majoring in economics and engineering is understood as
the type of teaching that the lectures help students to improve their analytical thinking to
higher levels through providing the students with a combination of solid foundations of
knowledge and activities that are appropriate for them.
Thus, the above-mentioned teaching method should be based on the analytical
manifestations of students and at the same time application of methods, organizational
forms and teaching techniques to design and organize teaching situations, create an

appropriate working conditions, and design lessons in a series of activities so that the
product produced by students performing those activities represents their level of
analytical thinking. Teaching towards developing analytical thinking should be combined
with the development of a number of other types of thinking, developing their problemsolving skills.
In order to assess students 'thinking development, it is necessary to use both
qualitative and quantitative methods, monitor student behavior throughout the process
(ways of thinking, asking questions, results of answering questions, exercises, acquiring
knowledge, sense of self-thinking, proficiency in performing activities) and results of
tests. Tests must include questions, exercises at different levels of requirements,
associated with the manifestations of analytical thinking.
1.6.2. Chance of developing analytical thinking of university students majoring in
economics and engineering in Advanced Mathematics teaching
Developing analytical thinking for students can be done through typical teaching
situations such as: Teaching mathematical concepts, mathematical theorems, rules,
methods, and solutions of mathematical exercises. For examples: An opportunity to
develop analytical thinking for students in teaching some concepts is expressed in many
situations:
- Upon approaching the concept:
+ Analytical operations (using a subdivision technique) can be used when solving
practical problems leading to concepts.
+ Students need to break down the attributes of the concept and study each part
carefully: interpret words and phrases (answer the question: what does it mean), can
explain in some ways to clarify the meaning, or infer to relate and understand based on
old knowledge base.
+ They must synthesize, link details to understand the whole concept.
- About strengthening the concept:
+ They need to analyze, organize, classify when dividing and systematize the


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concepts
+ Diagrams can be used to show the relationship with the learned concepts.
In addition, during the self-study process, they may have to gather information,
read some concepts from different references to understand the problem. They then have
to classify documents into groups, analyze representative concepts in each group,
compare and contrast them in order to realize the similarity in meaning in the different
expressions for that concept. Thus, they can evaluate, select concepts that are easy to
understand, and point out deficiencies (if any) in those references.
1.7. The current situation of teaching Advanced Mathematics towards developing
analytical thinking for university students majoring in economics and engineering
1.7. 1. Investigation and survey process
- Purposes: To investigte the current situation of students’ analytical thinking and
Advanced Mathematics teaching towards developing analytical thinking for university
students majoring in economics and engineering
- Subject: Lecturer of Advanced Mathematics and students majoring in economics
and engineering
- Contents: Manifestation of analytical thinking of university students majoring in
economics and engineering when studying Advanced Mathematics; The awareness of
lecturers and students about the manifestation of analytical thinking of university
students majoring in economics and engineering when studying Advanced Mathematics;
A number of activities, organizational forms and teaching techniques of lecturers in
teaching Advanced Mathematics towards developing analytical thinking of university
students in economics and engineering.
- Methods: Questionnaire; interview; tests; observation and comment during the
teaching process.
- Investigation and survey time: For the lecturer: in November 2016 and 2019; For
students: in 2018 and 2019
1.7. 2. Investigation and survey results
a) Survey results by questionnaire

The questionnaire is used to consult with lecturers in terms of analytical thinking
of students, developing analytical thinking for students in teaching Advanced
Mathematics (The questionnaire is presented in the Appendix). The questionnaires are
collected 64 lecturers teaching Advanced Mathematics at a number of universities.
b) Interview results
We interviewed a number of lecturers of Advanced Mathematics at the Posts and
Telecommunications Institute of Technology and Hanoi University of Industry. The
interview process is aimed at examining the lecturers’ views on analytical thinking,
analytical thinking manifestations of students when studying Advanced Mathematics and
the lecturers’ teaching method for the purpose of developing analytical thinking for
students.
c) Results of tests and comments during the direct Advanced Mathematics
teaching process.
We analyzed some tests of students in experimental class 1 (TN1), control 1
(DC1) and control class 2 (DC2) in oder to get more assessments of students' analytical
thinking. In addition, we relied on our observations on the teaching process.
The investigation and survey results revealed that in Advanced Mathematics,


11

generally, the university students in economics, engineering currently show little
analytical thinking. Many of them are not active in thinking; are not willing to find out
the details of details about concepts and theorems; fail to state the meaning of the
concept to solve practical problems or specialized problems. They do not pay much
attention to finding out and correcting any false information; as well as raising incorrect
questions, or general questions. Some students make judgments or conclusions without
any solid basis. In general, they rarely use comparative, similar and special
manipulations in making any judgments. Many of them fail to know how to use diagrams
to make the deduction process become clearer and more coherent. They do not create a

habit of reviewing, anticipating a few solutions to the problem and analyzing any
advantages and disadvantages of those solutions. Few students make a habit of drawing
any comments or lessons for themselves, etc.
Although each lecturer has a true awareness of the importance of analytical
thinking and has taught Advanced Mathematics towards developing analytical thinking
for students, he or she still fails to take sufficient measures and specific methods in the
teaching process. As the lecturers have not yet fully envisioned necessary manifestations
of analytical thinking, they fail to select many suitable examples, design activities
corresponding to the manifestations, and provide their students with methods often used
for analytical thinking.
CONCLUSION OF CHAPTER 1
The above theoretical and practical studies help us draw some conclusions:
Analytical thinking has general characteristics of thinking, which requires
analytical manifestations and division into small parts as well as finding out any
relationship. The analysis process requires insight. From the basic ideas above, it is
possible to conceive that: Analytical thinking is the type of clear and deep thinking about
understanding the subject. The thinking process takes place on the basis of full
consideration of parts of the subject and any relationship between them, with the whole
and the external elements. From this point, it is possible to think, judge, draw logical
conclusions based on logical bases and reasoning.
The concept of analytical thinking mentioned above shares something in common
with some conceptions given by other authors and is consistent with the students’
thinking in the course of studying Advanced Mathematics and analyzing some problems
in the life, as well as studying some specialized subjects.
Analytical thinking is closely associated with the general thinking, creative
thinking and problem solving.
Based on basic elements of thinking (Setting goals, raising questions, using
information, using concepts, creating inferences, making assumptions, giving rise to
implications and containing a perspective) combined with the characteristics of analytical
thinking (breaking down, finding out any relationships, clarity, insight, etc.) and the

characteristics of Advanced Mathematics, students’ activities majoring in economics,
engineering when studying Advanced Mathematics, we are able to determine
manifestations of analytical thinking of university students majoring in economics and
engineering when studying as follows: Determining the purpose of thinking, breaking
down the goal; Determining the direction of view; Finding out information clearly and


12

deeply to solve problems; Understanding concepts (theorems, propositions, rules,
methods) clearly and deeply and using them to solve problems; Making a sound
judgment, in relation to the question posed; Deducing clearly and deeply, in steps, based
on grounds related to the issue; Raising analytical questions.
Analytical thinking of students is manifested through the process of
implementing activities. Based on the hierarchy of activities and any parts shown in the
requirements of the stated manifestation, we provide a number of levels of analytical
manifestation with illustrative examples in teaching Advanced Mathematics. Level 1:
Students can show a part of requirements of each of the above manifestation but need the
lecturer's suggestion. Level 2: Students can show a part of requirements of each of the
above manifestation without the lecturer's suggestion. Level 3: Students can show the
above manifestation without the lecturer's suggestion In addition, for each of the
aforementioned levels, under appropriate circumstances, smaller levels may be
considered based on the complexity of the operating object; the abstraction and
generalization of the object, the content of the activity or the complexity of the activity.
By referring to some concepts related to the development of thinking , students’
thinking in teaching, analytical thinking for students, we do argue that teaching
Advanced Mathematics towards developing analytical thinking for university students
majoring in economics and engineering is understood as the type of teaching that help
students to improve their analytical thinking to higher levels, on the basis of providing
the students with a combination of solid foundations of knowledge and activities that are

appropriate for them. In term of teaching in this direction, the lecturers are recommended
relying on the characteristic manifestations of analytical thinking of students in
Advanced Mathematics teaching to select any contents, teaching methods, lesson design
in a sequence of activities so that the product produced by students performing those
activities represents their level of analytical thinking.
From results obtained in the course of surveying the current situation, comparing
with necessary manifestations of analytical thinking of students in teaching Advanced
Mathematics and concepts of teaching Advanced Mathematics towards developing
analytical thinking for students, it is found that analytical thinking of students is limited.
Many lecturers fail to identify the manifestations of analytical thinking of students in
Advanced Mathematics, making them unable to select the appropriate contents and
methods in order to develop the students’ analytical thinking.
Research results on theoretical and practical basis above are considered the basis
for proposing a number of Advanced Mathematics teaching methods towards developing
the analytical thinking for students.
CHAPTER 2: ADVANCED MATHEMATICS TEACHING METHODS
TOWARDS DEVELOPING ANALYTICAL THINKING FOR UNIVERSITY
STUDENTS MAJORING IN ECONOMICS AND ENGINEERING
2.1. Orientations to determine the methods
There are some orientations to determine the methods: The methods given should
be designed to ensure the principle of university teaching, affect the development of
analytical thinking, and must be feasible.
2.2. Advanced Mathematics teaching methods towards developing analytical


13

thinking for university students majoring in economics and engineering
2.2. 1. Method 1: To enhance activities about dividing, understanding each part and
finding out relationship to be implemented by students

a) Purpose
This method is aimed at helping students form habits and be able to successfully
perform activities of dividing, understanding each part and finding out relationship
during the course of using information, raising questions, identifying directions, judging,
making inferences, solving problem; relating Advanced Mathematics knowledge to
reality or specialized issues.
b) Basis of method
Dividing and finding out any relationships are characteristic of analytical thinking
and included in the analytical manifestations of students but currently they have not well
performed these activities.
c) How to apply
- The lecturer shows students that division depends on the characteristics of the
object, purpose of the activity and each specific case. The students must look at the
whole at first to choose how to divide and interpret each part. They are required to
understand the meaning of words, phrases, explain each part, express in many ways,
illustrate or give examples, and make any comments.
- The lecturer may give the following questions: Why? How is it understood?
What does it mean? What's your opinion? Please analyze ...
- The lecturer should design situations that require students to go deeply into
details.
- Exercises which are changed in terms of some properties of concepts and
properties may be used.
- Exercises containing parameters may be given, because this type of exercise
requires students to divide the solution into cases and provide appropriate arguments for
each case.
In terms of pointing out the relationship, according to the Textbook of Basic
Principles of Marxism-Leninism, the relationship in dialectics is used to refer to the
regulation, the interaction and the mutual transformation between things, phenomena, or
between sides, elements of each thing, phenomena in the world.
- Pointing out the relationship between elements is a large-scale problem,

requiring students to pay attention to the situations and contents they have learned.
When solving problems, they are required to relate to many relationships, then
screen and select any relationship that are appropriate for the situation.
The relationship between A and B can be manifested in many forms, for example:
A  B (2)
+ A  B (1)
+ A is the condition to have B (3)
+ A combines with C to obtain B (4) + A combines with B to obtain X (5)
+ A, B combine with D to obtain Y (6) + B is property of A (7).
+ A and B have the same properties C (8)
+ For finding (proving) A you have to find (prove) B (9)
+ A and B are bound by a certain expression or rule (10)
+ A, B and C are bound by a certain expression or rule (11).
...
(A, B can be clauses, formulas, etc.)


14

The relationships (4), (5), (6) often appear in the course of solving problems, at
which considering these relationships is the use of synthesized operations. Identifying the
relationship (3) facilitates the process of ordering when you want to present, understand
the structure and make logical arguments.
-Students need to perform activities related to classification, systematization,
document arrangement (finding logical relationships, structural relationships).
When analyzing and teaching mathematics, people often pay attention to some
relationships such as "To find (prove) A, we must find (prove) B", causal relationship,
relationship about the common - the private,
- For the purpose of clearly seeing the relationship such as "To find (prove) A, we
must find (prove) B", students should use the forward and reverse diagrams.

-In order to easily identify the causal relationships, in Advanced Mathematics,
students need to pay attention to the relationship.



A1 
A1




A  B ; A  B1  ...  Bn  B; A  B1  ...  Bn  B;  
  B; A  
 ,...




An 


An


Thus, they are required to understand, memorize and correctly use the concepts, clauses,
theorems, logical laws; know how to combine various theorems to deduce in many steps.
It also helps students make better deductive inferences.
When applying the common-the private relationship, students are recommended
asking and answering the following questions: What do A and B share in common? What
attributes (characteristics, etc.) are common in these objects? Which objects have the

same properties (characteristics, etc.) with object A? What is the particular (special) case
of this? Is this content a particular (special) case of what? What would be more general
results? ...
If we want to know if the predictions about the common is true or not, we can try
specialization. If the specialization operation produces any false results, the hypothesis
(prediction) is eliminated. If the specialization operation produces any true results, we
find a proof. When provable, it is possible to state in the general case. After that, another
common thing can be predicted based on the analogization operation.
Finding the relationship between the common and the private leads to similar
inferences and inductive inferences.
Because the common appears in many private things, but it may not be in every
individual one, it is necessary to predict which objects have the same in common. This
involves the use of comparative and similar operations. (If the object X has properties A,
B, C, D and the object Y has properties A, B, C then Y also has property D). In addition, it
is also necessary to use the abstract manipulation when considering certain common
properties in the private without regard to other properties.
In some cases, when predicting the common, it is possible to analyze the private
into elements, view each element in many directions, or make any general statement
about it.
Summary:
* Breaking down, understanding each part and finding out relationships help
students understand information, concepts, theorems; consider in a number of directions;
break down the goal; make clear, insightful inferences; make sound judgments; provide


15

logical arguments; and help them remember and solve problems.
* In order to help students identify any relationship, the lecturers should suggest
and create opportunities for them to regularly engage in activities:

- Pointing out some common relationships, selecting any appropriate ones in
specific situations.
- Sorting and classifying (finding out the relationship between ideas, the structural
relationships).
- Remembering and correctly using concepts, clauses, theorems, logical laws;
combining theorems to find out any transitive relationships.
- Answering the questions: "Why ...?", "What will lead to this?", "What will this
entail?", "What do we look for to find A?", etc.
- Paying much attention to find the common, similar things, differences between
the knowledge contents. Combining comparative, analogy, specialization, generalization,
abstraction manipulations (to detect and manipulate the common-private relationship,
leading to similar inference, inductive inference, knowledge creation).
2.2.2. Method 2: To design, organize activities that demonstrate a clear and
insightful way of thinking for students
a) Purpose
This is aimed at helping students think clearly and deeply when defining goals and
objectives; determining the perspective; using information; raising questions; using
concepts (theorems, propositions, rules, methods), judgments and reasoning; relating to
the reality or the major.
b) Basis of method
- Thinking clearly and deeply is a characteristic of analytical thinking, included in
requirements of the manifestations.
- This affects the awareness of students' thinking process, thereby developing
thinking in general and analytical thinking in particular.
c) How to apply
The lecturers should make suggestions and create opportunities for students to
carry out the following activities:
+ Write summary contents, write detailed contents, detect errors and correct,
compare, classify.
+ Present, explain the problem clearly, answer questions "why?, use illustrative

examples and counter-examples.
+ Look back to see the knowledge which is still unknown, record and think more
or ask others.
+ Constantly supplement newly discovered aspects to the existing knowledge
system and adjust the thoughts.
+ Pay attention to the gaps of knowledge not clarified or mentioned in the
materials to think and find out any comments and conclusions.
+ Search for many elements that are related to a problem.
+ Take full consideration a problem, see it in different ways; make transitive
inference in many steps; Complete, supplement, expand or relate other issues; Dissatisfy
with the general answers but seek any deeper ones. It is possible to find out many
problems, relationships related to an element under consideration; make comments, pay
full attention, put in additional clauses, and think deeply about the problem.


16

(It should be suggested by the lecturers that these comments are often derived
from the course of generalizing, specializing, similarizing, comparing, viewing counterpropositions, paying attention to mistakes, etc.).
2.2.3. Method 3: To provide students with some common methods for analytical
thinking
Some commonly used methods for analytical thinking include:
- Understanding the subject with full consideration: Whole-Part-Whole.
- Using the question. Answering a question with a question.
- Using types of diagrams, graphs, outline.
- Using SQR4 reading strategy".
a) Purpose
This method is intended to help students to use analytical thinking methodically;
limit common mistakes in the analysis process; orientation, stimulating thinking; make
the thinking process clear, coherent, deeper and more effective.

b) Basis of method
Several studies have shown that the above methods are aimed at providing
students with the strategies, techniques, and tools often used for analytical thinking, thus
helping them to develop analytical thinking. However, many students still do not know
how to use these methods.
c) How to apply
- Understanding the subject with full consideration of Whole-Part-Whole
+ The lecturers should suggest to students the "Whole-Part-Whole strategy", and
take specific examples. In the process of solving math problems, at the first step when
considering the whole, students need to understand all the factors and relationships in a
rough way, visualize the relevant knowledge area to determine the analytical direction.
(However, the effectiveness at this step is highly dependent on whether students often
solve math problems or not, because this will help them gain experience and have a
better feeling when determining the analytical direction). The lecturers should also use
this strategy in presenting their lessons.
+ They may build forms of exercises that include many ideas, in which the ideas
have little change but require students to solve the problem in a different direction.
- Using the question, answering a question with a question.
+ The lecturers should ask students to learn about the importance of asking
questions, answering a question with a question.
+ The lecturers may show students the importance of using questions and
answering a question with a question: guide the thinking process, stimulate thinking, etc.
It is possible to pose any split questions, explain and find out any relationships, raise
questions about purpose, information, concepts (theorems, rules, methods), perspective,
logical reasoning, judgment, deduction, clarity, depth, etc.
+ The lecturers may suggest several types of questions, such as: Questions when
reading theory; Question when solving problems; Questions after solving problems.
+ The lecturers often pose questions, answer students' questions with a question
and suggest them to do the same.
- Using types of diagrams, graphs, outline:

+ The students are suggested by their lecturers studying about some types of
diagrams themselves: mind map; conceptual map; forward diagram, reverse diagram.


17

The lecturer then gives students additional hints about the above diagrams (usage,
advantages, etc.) and provides illustrative examples during the teaching process (or gives
documents to students for reference when studying at home).
+ They may design exercises and situations for students to perform activities that
use diagrams and graphs: Read the diagram, comment on a diagram; draw a conceptual
map showing the relationship between concepts; draw mind maps when summarizing
lessons, indicating problem solutions; organizing, classifying and using information and
planning; draw any shape; use information technology when drawing any shapes.
- Using SQR4 reading strategy:
+ The students are suggested by their lecturer studying about SQR4 reading
strategy by themselves.
+ The lecturers give students additional suggestions for SQR4 reading strategy and
suggest using it when reading materials at home, which can be in the form of exercises,
for example: "Show SQR4 reading strategy when you read the surface integral type two".
+ During the class teaching, it is possible for the lecturers to check some steps in
this reading strategy, for example: "When you read the surface integral type two and use
the SQR4 strategy, in step 1 (survey), what key points do you need to cover?”
2.2.4. Method 4: To design, organize activities that combine analysis with synthesis,
creativity and problem-solving skills
a) Purpose
This is aimed at helping students often use analytical thinking combined with
other types of thinking, in addition to using thinking types in a more flexible and
effective manner.
b) Basis of method

- In the teaching process, in addition to developing analytical thinking, it is
necessary to develop other types of thinking, as well as the problem-solving skills for
students.
- Analytical thinking is closely associated with the general thinking, creative
thinking and problem solving.
- The integrated power may be created from the combination of training many
types of thinking.
c) How to apply
- Combination of analysis and synthesis: Students should be asked to
summarize, prepare an outline, point out basic steps in a solution, and briefly such steps
after analysis. Exercises should be provided in a way that combines a variety of
knowledge.
- Combination with creative thinking: The lecturers should encourage students to
imagine what is about to read (or listen to) and analyze after reading (together with
comparison). The lecturers suggest to students that: In oder to develop analytical and
creative thinking, they should know how to use phrases such as: "I have a hunch on this
topic ...", "I can imagine it working in this way if ...", "This reminds me of when I ...", "I
still wonder about the question about ...”, “When looking at the whole thing, I think the
key point here is…”; Using exercises with many solutions, silent exercises, open
exercises, other exercises, etc., is also recommended.
- Combination of analysis and problem solving skills: The lecturers can
introduce and let students solve problems in four steps of Polya, with close attention to


18

requirements on the manifestation of analytical thinking in each solution step.
2.2. 5. Method 5: To strengthening the use of analytical thinking during the selfstudy process
a) Purpose
This method helps students train their ability to self-study and develop analytical

thinking by regularly conducting thinking activities on their own.
b) Implementation basis
Self-study plays an important role in the students’ study process. Self-study,
especially when reading and solving problems, often involves elements of analytical
thinking. However, it is shown by the reality that students' ability to self-study is very
limited. In addition, as the detailed analysis takes time, it cannot only be done in the
classroom, but need to focus on the student's self-study process. Moreover, it is necessary
for students to have a high level of independent thinking, so there is no need for the
lecturers to analyze problems like those at high school level. Therefore, the students must
analyze themselves during the self-study process in order to understand knowledge.
c) How to apply
In teaching Advanced Mathematics in this direction, the lecturer may:
- Provide brief introduction for students to understand more about self-study
methods at university.
- Develop a system of questions and exercises using analytical thinking and
assigning students to complete by self-study. The assignments must be appropriate for
their qualifications, with higher education and consistence to the above methods.
Attention should be paid to some forms of exercises suitable for being done at home. For
examples: Drawing specific comments and comments (after reading, completing
assignments, etc.); Drawing maps; Making classification, summary, etc.; Presenting how
to solve the problem in four steps of Polya; Large, thematic exercises; computer-based
exercises; practical or specialized application exercises, case studies; Using SQR4
reading technique, etc.
- Asking students to self-study to raise their own questions and exercises (suggest
them how to ask these types of questions).
- It is possible for students to study in groups at home and present, discuss in their
class.
- Examining more frequently (including checking new lessons before class and old
ones after studying) is required. The test is not exactly the same as the exercise already in
the learning materials, with more or less changes to require students to have inference.

- It is possible to use KWL teaching techniques.
CONCLUSIONS OF CHAPTER 2
In this chapter, we have elaborated five methods based on the principles of
university teaching and examining manifestations of students' analytical thinking when
studying Advanced Mathematics, combined with findings in theoretical and practical
basis, including: To design, organize activities that combine analysis with synthesis,
creativity and problem-solving skills; To design, organize activities that demonstrate a
clear and insightful way of thinking for students; To provide students with some common
methods for analytical thinking; To design, organize activities that combine analysis with


19

synthesis, creativity and problem-solving skills; To strengthening the use of analytical
thinking during the self-study process.
For method 1, the students carry out activities of division, interpretation,
understanding each detail, finding out any relationships, which will help them break
down their goals, understand information, use concepts. , theorem, etc., better; make
judgments based on the consideration of the relationship of the common and the private
and perform operations of similar thinking, generalization, specialization; make clear and
insightful reasoning, and logical reasoning based on answering questions about cause and
effect relationships. They are trained in sorting and classification activities based on
subdivision and finding logical relationships, structural relationships, etc. Examining
relationships in many aspects helps them see problems in many ways.
Method 2 is directed at the requirement of "clarity and insight" in the analytical
thinking manifestations of students. It aims to influence students' awareness of thinking
process, emphasizing the making of their own comments, attentions and experiences.
That helps them develop thinking in general and analytical thinking in particular.
Method 3 helps students to think more methodically in the analysis process. The
"Whole-Part-Whole Strategy" helps them localize their knowledge to predict the

appropriate analytical direction, with close attention to the whole during the analysis.
Answering a question with a question helps them navigate the thinking process and find
the answer on their own. The use of types of diagrams helps students identify
relationships, and make clear and coherent inferences. The use of SQR4 reading
technique helps students identify main ideas, ask questions, think, memorize, etc.
Method 4 focuses on the development of some other types of thinking
concurrently with analytical thinking, and application of analytical thinking when solving
problems. It helps students become familiar with issues that requires the synthesis of a
variety of knowledge. Imagination before analysis helps limit the error of "analytical
thinking that hinders creative thinking".
Method 5 emphasizes the development of analytical thinking activities for
students during self-study at home, because there is not enough time to conduct these
activities regularly in class, moreover, students need to practice independent thinking. In
addition, this method makes teachers pay more attention to the types of exercises that are
only suitable for students to study at home, practical or specialized exercises, scientific
research exercises.
The division of the above method is only relative to help understand and
implement it easier, because the implementation of one method can simultaneously help
to implement another. For example, when students conduct activities to understand the
problem clearly and deeply, it is also necessary to split, interpret and find any
relationships, etc. When students take steps to solve problems, they also analyze problem
solving and think deeply.
The above methods help students have a method of reading, recognizing the main
ideas, memorizing, so they can grasp knowledge better. Thus, we think that the
implementation of such methods can contribute to the development of analytical thinking
for students while improving the effectiveness of Advanced Mathematics teaching. In
addition, the way to present illustrative examples has shown the use of some positive
teaching methods and techniques, for example: Case studies, diagram techniques,
"KWL" technique, etc. These are results we have not found in previous studies of

20

Advanced Mathematics teaching. Based on the above methods, we will design lectures,
organize teaching, and evaluate the development of analytical students' thinking in order
to test how difficult they are to implement and whether the use of such methods really
develop analytical thinking for students or not.
CHAPTER 3: PEDAGOGICAL EXPERIMENT
3.1. Purpose, content and organization
3.1. 1. Experimental purposes
Pedagogical experiments are conducted for the purpose of:
- Test scientific hypothesis of the thesis.
- Initially assess the feasibility and effectiveness of proposed methods.
3.1. 2. Experimental content
3.1.2.1. Subjects participating in the experiment
The experiment was conducted at the Posts and Telecommunications Institute of
Technology in two sessions.
Session 1 (for university students majoring in engineering):
Experimental period: From February 2018 to May 2018. We compared results
between experimental class 1 (TN1) and control class 1 (DC1).
Subject: Mathematical analysis 2
- Class TN1: Group 14 (65 students). Lecturer: Nguyen Thi Dung.
- Class DC 1: Group 13 (62 students). Lecturer: Nguyen Thi Dung.
Groups 13 and 14 have similar knowledge and qualifications, entrance
examination scores, and nearly equal number of students.
Session 2 (For university students majoring in economics):
Experimental period: From early September 2018 to mid-October 2018.
We compared results between experimental class 2 (TN2) and control class 2
(DC2).
Subject: Mathematical analysis 1

- Class TN2: Class D18 QT 3, 4 (90 students). Lecturer: Nguyen Kieu Linh.
- Class DC2: Classes D18QT1, 2 (92 students). Lecturer: Nguyen Kieu Linh.
In addition, we selected 4 students to monitor the development of analytical
thinking during the experiment, that is, students in group 14 studying Mathematical
analysis 2.
With the experimental classes, the lecturers followed methods proposed herein. In
the control classes, the lecturers taught normally as before, without the impact of
experimentation.
In the course of experimental teaching, we attended the lessons as well. After the
lesson, we held discussions, assessments and lessons learned. At the same time, we
polled from the students to make timely adjustments to achieve the purpose of
pedagogical experiment.
3.1.2.2. Experimental teaching program
A. For university students majoring in economics
- Advanced Mathematics 1: Basic concepts of functions, Limit of function,
Continuous functions, Derivative of functions, Differentials of functions, Derivative and
high-level differentials, Average value theorems, Some applications of derivatives.


21

- Examination and evaluation: The examination content corresponds to the
above lessons: (60 minutes).
B. For university students majoring in engineering:
- Mathematical analysis 2: The lessons in Mathematical analysis 2 are for
students majoring in engineering at the Posts and Telecommunications Institute of
Technology.
- Examination and evaluation:
Contents: Multivariate function, multiple integral, line integral type two: (50
minutes).

3.1.2.3. Experimental lesson plans
We present two illustrative lesson plans: "Line integral type two" for university
students majoring in engineering. "Derivative" for university students majoring in
economics.
3.2. Evaluation of experimental results
3.2. 1. Qualitative aspect
1. For the lecturers:
By exchanging with the lecturers and observing and evaluating from the teaching
hours in the experimental classes, it is found that:
- Lecturers participating in the experimental teaching have understood and used
pedagogical methods, strictly implementing ideas set out in the lesson.
- They believe that teaching by the methods mentioned above makes their lessons
livelier, enable their students to understand the lesson and feel easier to memorize, while
the lecturers know how and build a similar example system by themselves.
2. For the students and classroom environment
In general, the atmosphere of the experimental class is quite exciting when
students seem to love the subject and are excited about examples and exercises. Students
in the experimental classes also performed better than those in the control class (with
more progress than themselves) in the ability to analyze errors and correct, explain in
detail, only make connections, identify perspectives, see things in a number of ways,
infer in forward and reverse aspects, understand the nature and make personal
conclusions, etc.
For case studies, we randomly selected four students in the experimental class and
supervise their progress.
1. Student Nguyen Xuan A, class D17CQCN12. During the teaching process, we
followed A’s expressions in a number of ways: Learning and using information, making
judgments, determining the purpose of thinking. Student A showed certain his or her
progress.
2. Student Nguyen Duc B, class D17CQCN7. We followed B’s expressions in a
number of ways: Understanding the concepts, theorems, rules; Determining the

perspective Student B showed a lot of progress.
3. Student Dong Thi Thu C, class D17CQCN8. We followed C’s expressions in a
number of ways: Understanding the concepts and theorems; Asking analytical questions.
Student C showed her progress little by little.
4. Student Chu Quoc D, class D17CQCN5. During the teaching process, we
followed D’s expressions in a number of ways: Making judgments, making inferences,
determine the perspective. Student D showed a lot of progress.


22

3.2.2. Quantitative aspect
The quantitative aspect is mainly evaluated based on the results of tests. These
tests were shared for the experimental and control classes.
3.2.2.1. Evaluation of results of the first experimental session
Before the experiment, we required the students in classes TN1 and DC1 to do the
test (25 minutes). It is shown in the results that the two groups have similar qualifications
and many limitations in analytical thinking. For example, they did not delve into
concepts and theorems; failed to make arguments based on evidence or evidence; without
thinking clearly step by step when encountering complex problems.
After the experiment, we required the students in classes TN1, DC1 to do the test.
The answers and scores are presented in the Appendix. The scores show that students
generally scoring 7 or above will have better analytical thinking than those with lower
scores. Therefore, comparing the analytical thinking of students in the two classes may
be based on a comparison of percentage of students scoring at a score greater than or
equal to 7.
*The test results indicate that:
- The average score in class TN1 (6.0) is higher than that of class DC1 (5.2). The
percentage of students with poor scores (3 and below) in the class TN 1 is lower than that
of the class DC 1. Proportion of students achieving weak and moderate scores (4  6) in

class TN1 is higher than that of class DC1.
- The percentage of students with good scores (grades 7 or higher) in class
TN1 is higher than that of the class DC 1.
In addition, when referring to the final exam results, (General test based on the
questions for all classes throughout the Institute), we get the following results (from the
Registrar's Office): The average scores in the class TN1 class and class DC1 are 5.86 and
5.0, respectively.
3.2.2.2. Evaluation of results of the second experimental session
After the experiment, we required the students in classes DC2 and TN2 to do the
test. The answers and scores are presented in the Appendix. The scores show that
students generally scoring 6 or above will have better analytical thinking than those with
lower scores. Therefore, comparing the analytical thinking of students in the two classes
may be based on a comparison of percentage of students scoring at a score greater than
or equal to 6.
* Test results: The number of students scoring 6 or higher in the class TN2 is
higher than that in the class DC2.
50

Lớp TN2

Lớp ĐC2

40
30
20
10
0
2

3

4

5

6

7

8

Frequency chart of scores for classes TN2, DC2
In addition, when summarizing questions students posed in the test, we found


23

that many students asked questions that were incorrect, irrelevant, unrelated to the
content or unclear. These questions show that they failed to understand the content of the
lesson, show carefulness, and make grounded arguments. Students in the class DC2 had
more unsatisfactory question than those in the class TN2.
CONCLUSIONS OF CHAPTER 3
Through the process of conducting experiments above, we draw some
conclusions as follows:
It is possible to carry out solutions described in chapter 2. The lecturers who
participated in the experimental process thought that understanding of analytical
thinking, pedagogies, and illustrative examples helped them better understand
requirements of analytical thinking. Thus, they can find it easier when developed
examples and exercises, organizing situations in teaching Advanced Mathematics
towards developing analytical thinking for students. The solutions emphasize student

self-study and the practice of analytical thinking exercises at home, teachers have enough
time to teach in the classroom. Although teaching in this direction requires research and
more time for preparing lessons, it helps the lesson somewhat livelier. Most of these
opinions are similar to our comments when directly teaching and observing the activities
of lecturers in the experimental classes.
The solutions have contributed to improve the efficiency of Advanced
Mathematics learning and development of analytical thinking for students, expressed
through a number of manifestations: Students pay more attention to learning in details of
concepts and theorems; draw out the meaning of that concept and theorem; find out any
relationship between the concepts and theorems; make predictions on the basis of
considering the relationship and using thinking manipulations. They can show better
ability to infer in forward and reverse aspects. Initially, they know to draw some
comments, attention; limit some mistakes; offer some solutions and choose the direction
briefly; ask better analytical questions (extensive questions, specific questions, reverse
analysis questions, deep thought questions), etc. They can use some commonly used
methods of analytical thinking without the lecturer's suggestion.
The most difficult problem in teaching in this way is that students' self-study in
general has not been well done, with little progress made, while some students do not
focus on listening to the lectures and do not have the spirit of volunteering to do their
homework despite being reminded. Therefore, the students’ manifestation of analytical
thinking development is faint or even absent.
With some comments above, we initially assume that the scientific hypothesis of
the dissertation has been tested. Teaching according to the above-mentioned methods is
possible, contributing to improving the effectiveness of teaching Advanced Mathematics
and developing analytical thinking for students. The development is shown more clearly
for those who are conscious of studying hard.