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

- Research subject: Advanced Mathematics teaching process towards developing analytical thinking for university students majoring in economics and engineering 5.. Scientific hypothesis

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: Theory and methodology of Mathematics teaching

Code: 9 14 01 11

HANOI, 2020

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

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

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:

- 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

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

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

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:

0 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

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:

( )6

lim

( )2

x

x

o x x

3 0

16

32

x

x x



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

( )6

( )2

x

x

o x x

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?

rule, etc., in addition, we can ask that: If we need to find out

3

3 3

( )6

lim

( )2

x

x

o x x

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 problem-solving 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

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,

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