bồi dưỡng cho học sinh trung học cơ sở năng lực biến đổi thông tin toán học trong quá trình dạy học môn toán bản tóm tắt tiếng anh

1

PREAMBLE
1. Rationale
1.1. The need for educational reform at the present time
The scientific and technological revolution has continued to grow with the
great leap in the 21
st
century, bringing the world from the industrialized era to
the era of information and the development of knowledge-based economy.
Accordingly, all countries require their citizens to be competent, self-motivated,
creative and especially capable of receiving and processing information timely
and effectively in their learning and working activities and in life. These
qualities are needed to respond to the requirements of the integration process
and the rapid development of the world.
In Vietnam, the development of the country has been creating not only
many opportunities and great advantages but also challenges for the
development of education. Therefore, the cause of education and training needs
new strategies and solutions for its development, which need to be more
comprehensive and more robust. The changes need to be started from general
education. The implementation requires comprehensive measures in the various
fields, in which the innovation of the content of the textbooks and teaching
methods needs to: "be based on the assessment of the current programmes of
general education and on the reference to the advanced programmes of other
countries. The innovation of programs and textbooks after 2015 will be
implemented with the orientation of developing student competence, ensuring
the nationwide consistency and appropriateness to particular features of each
locality.
1.2. The goal of teaching mathematics in secondary schools
The goal of teaching mathematics in secondary schools is to provide
students with: the basic, and practical mathematical knowledge and methods;

shaping and training the students essential mathematical skills in order to
initially figure out the student capacity to apply knowledge of mathematics to
life and other subjects; training students’ capability of logical reasoning and
thinking, the ability to observe, the ability to use language accurately; fostering
students’ flexibility, independence, and creativity in learning mathematics.
After 2015, mathematics teaching and learning activities in schools will be
designed based on the goals of general education, the characteristics of the
mathematics, the trend of and experience in developing the mathematical
program from other countries, and the tradition of mathematics teaching and
2

learning in our own country. The proposed objective is to provide students with
the general knowledge and basic kills of mathematics of general education.
With these objectives, the innovation of teaching methods must be aimed
at fostering learner’s capacity; the process of teaching mathematics in
secondary schools should be designed with the aim of promoting students’
activeness and creativeness to mobilize students’ capacities in learning,
exploring and obtaining new knowledge. These will help to foster students’
mathematical capacity, in which SCTMI plays a crucial role.
1.3. Overview of research problem
Many international and Vietnamese researchers have been interested in the
study of human capability in general and the capability of teaching mathematics
in particular.
H. Poincaré, a French mathematician, was one of the researchers who
initiated this research problem in the early years of the twentieth century.
Researchers such as A. N. Kolmogorov, A. A. Stoliar, E. L. Thorndike have
studied students’ the mathematical capacity; An international assessment
organization of mathematics achievement (UNESCO) publicized 10
fundamental attributes of mathematical capacity.
In particular, V. A. Kruchetxki has a study on “Psychology of students’

mathematical capacity”, in which the primary and most important outcomes of
his study discuss the analysis of the structure of student’s mathematical
capacity based on the information theory.
In Vietnam, the researchers as Tam Dao, Ton Than, Tran Dinh Chau Tran
Luan, Nguyen Van Thuan Le Nhat, Nguyen Thi Huong Trang, Nguyen Anh
Tuan, Tran Duc Chien have studied different types of students’ mathematical
capacity. They also suggest a number of approaches for fostering the students’
mathematical capacity.
Recently , at the Vietnam - Denmark International Conference, as
discussed the target of mathematics teaching and learning at general schools in
Vietnam, Tran Kieu and his colleagues suggested a number of mathematical
capacities that need to be built and developed for students through mathematics
teaching process in general schools in Vietnam, including: the capacity of
thinking, the capacity of information acquisition and processing, the capacity of
problem solving, the capacity of mathematics modelizing, the capacity of
communication, the capacity of utilizing mathematical tools and means, and the
capacity of independence and collaboration in learning.
3

Although such studies have created “colorful paintings” of students’
learning capacity in general and mathematical capacity in particular, no single
study examined students’ capacity transformation of mathematical information
(SCTMI) in teaching mathematics in secondary schools.
1.4. Fostering the students’ capacity of transformation of
mathematical information.
Our surveys and learning and teaching observations in some schools show
that the teachers of mathematics demonstrated their interest in fostering SCTMI
in the explicit or hidden form, which contributed to enhancing the effectiveness
of their teaching and promoting students’ creativeness and activeness in their
learning process. Nevertheless, besides many of its advantages that should be

promoted and exploited, the organizing of teaching activities aimed at fostering
SCTMI also revealed certain limitations in some teachers and a number of their
lessons . The observation of pedagogical practices and the survey outcomes also
showed that SCTMI did not reach the desired level that it is expected to be
achieved. Hence, fostering secondary school students’ mathematical capacity in
general and SCTMI in particular in mathematics teaching process is necessary
and should be taken into consideration.
For those reasons, in order to meet the goals of mathematics teaching in
secondary schools and the goals of the education of students’ personality
development; in order to promote secondary school students’ activeness and
creativeness in their learning and improve the effectiveness of mathematics
teaching in secondary schools, I chose the research topic: Fostering secondary
school students’ capacity of transformation of mathematical information in
the process of mathematics teaching.
2. Aim of the study: On the basis of research and analysis of theoretical
perspectives for determining the elements of SCTMI and CTMI process, a
number of pedagogical measures will be proposed to foster CTMI for
secondary school students with the purpose of improving the teaching
effectiveness.
3. Research Tasks
3.1. Review, codify and clarify the theoretical perspectives and practical
basis of CTMI and the cultivation of CTMI.
3.2. Conceptualize the transformation of mathematical information,
SCTMI, and the procedures of the transformation of mathematical information.
4

3.3. Propose a number of elements and the expression levels of CTMI in
teaching mathematics.
3.4. Identify some basic orientations as a basis for the building up and
implementation of measures of fostering CTMI in mathematics teaching.

3.5. Propose measures of fostering SCTMI in teaching mathematics in
secondary schools.
3.6. Implement pedagogical experiments to examine the feasibility of the
proposed pedagogical measures.
4. Research Methods: Review literature, surveys, pedagogical
experiments, mathematical statistics in educational science.
5. Hypothesis: On the basis of theory and practice, given that the elements
of CTMI are identified and proper pedagogical measures of fostering SCTMI
are constructed and implemented, the effectiveness of mathematics teaching in
secondary schools will be improved.
6. The contributions of the thesis
6.1.1. Codify and clarify the fundamental problems of theoretical
perspectives and practical basis of CTMI and the cultivation of CTMI.
6.1.2. Conceptualize the transformation of mathematical information,
SCTMI, and the procedures of the transformation of mathematical information.
6.1.3. Propose a number of fundamental elements and the expression levels
of CTMI in teaching mathematics.
6.1.4. Identify some basic orientations as a basis for the building up and
implementation of measures of fostering CTMI in mathematics teaching.
6.1.5. The thesis can be used as the reference for the teachers of
mathematics and the pedagogical students of mathematics as contribution to
improving the effectiveness of mathematics teaching.
7. Outline of the thesis: In addition to the introduction, conclusion,
references, the thesis is presented in the three main chapters:
Chapter I: Literature review and practical perspectives.
Chapter II: A number of measures of fostering CMTI for secondary
school students in teaching mathematics.
Chapter III: Pedagogical experiments.
CHAPTER I. LITERATURE REVIEW AND PRACTICAL
PERSPECTIVES

1.1. Concepts of capacity and mathematical capacity
1.1.1. Basic concepts of capacity
5

Researchers from different countries have the same interest in the research
on human capacity. The concept of capacity is differently articulated and
interpreted by different researchers. However, their different definitions have
the same fundamentality and consistency in general.
Based on the literature review, a number of basic concepts of capacity are
summarized as follows:
Capacity can be divided into two major categories: general capacity and
specialized capacity. General capacity is needed for many different types of
activity. Specialized capacity is a unique quality of a single individual that
needs to be trained to meet the requirements of a specific type of activity.
Capacity is always defined and indentified with a particular activity within
which it is utilized to implement that activity. Capacity is exposed and can be
observed in a new situation in which it is needed for dealing with new
requirements; the capacity of an individual requires him to have personal
qualities that meet the requirements of a certain kind of activity and that ensure
such an activity to be accomplished with high efficiency. Hence, capacity is
associated with creativity but different in levels.
Capacity represents the differences of psychological and physiological
features of individuals. In terms of biology, capacity is influenced by genetic
innate factors. It is also developed or constrained by the conditions of the living
environment.
The innate factors of capacity need the favourable conditions of social and
living environment to develop or they will be corroded. Hence, capacity is
associated not only with innate factors but also with activities and it only exists
and represents itself in a particular activity.
Capacity itself needs to be attached to the knowledge background and a set of

correlative skills of an individual. Therefore, an individual’s knowledge
background and skills need to be cultivated in order to foster his capacity.
Capacity can be built and developed. It can also be observed and evaluated.
Building and developing students’ basic capacity in a learning and real situation
is one of the important tasks of schools.
1.1.2. Some fundamental concepts of math ability
There have been many research works on mathematical capacity from
different aspects and under different perspectives. The structure of students’
mathematical capacity is one of the research subjects that many research
scientists are interested in. In particular, according to a comprehensive study of
6

the structure of mathematical capacity conducted by V. NL A. Kruchetxki, the
structure of students’ mathematical capacity consists of the following
components: mathematical information receiving, mathematical information
processing, mathematical information storage; mathematical tendency of
intelligence.
1.1.3. Some remarks are drawn from the perspectives of the
previously-mentioned researchers.
1.1.3.1. It can be seen that:
The two structures from different perspectives of researchers that have the
same name may not be homogeneous in its inferred meaning and components.
In terms of the components of capacity, there is an interference among the
components of mathematical capacity. These components are closely linked
together. It is, therefore, the components of SCTMI discussed in section 1.3.2
of this thesis must have interference among them.
It is not easy to compare the rationality among different perspectives of
mathematical capacity or its components. This concept may be more logical
than the other one if it is considered from students of this level, but it may not
be rational if it is considered from the students of another level. Similarly, the

capacity may vary if it is considered from different subjects as: Arithmetic,
Algebra, Geometry, Calculus.
Due to the importance of the mathematical capacity and mathematical
structure, there is a growing in the discovery and fostering mathematical talents.
Many scholars propose different perspectives of the structure of mathematical
capacity. However, due to the research focus, the perspectives of the structure
of capacity proposed by V. A. Kruchetxki have been chosen as a theoretical
framework for this research.
1.1.3.2. Reviewing the perspectives of V. A. Kruchetxki on mathematical
capacity and the structure of mathematical capacity, some basic points can be
summarised as follows:
In a same learning condition, some students acquire knowledge and use it
better than the others. However, such a capacity is primarily built and
developed through the activity of doing mathematics. Thus researchers need to
understand the essence of capacity and find out the approach to building,
developing and perfecting capacity.
7

The meaning of capacity implies the differences among learners. When we
mention “capacity”, we assume that there is a difference among learners. One
can have capacity that is good for in this field but it may be poor in the other.
In the life of a person, there is a period of time that is more appropriate for
establishing and developing his mathematical capacity. A number of
psychological and educational research works also indicate that the age between
11 - 15 is best time for building and developing a learner’s mathematical
capacity.
Mathematical capacity is not an innate feature but it is constructed and
developed through daily life activities.
An individual’s effectiveness of performance in a particular field depends
on a set of capacities. The outcome of learning mathematics also keeps that

rule. Besides that, it depends on some other factors such as the learner’s
passion, diligence and the encouragement and support that the learner has from
his teachers, family and the society as a whole.
Training an individual to achieve a high level of performance for an
activity requires us to examine the individual’s capacity and seek out the best
approach to fostering his capacity.
To foster a learner’s mathematical capacity, in addition to understanding
his strengths to help him develop his capacity, we also need to find out the
weakness to help him overcome the difficulty. V. A. Kruchetxki confirmed that
fostering a learner’s mathematical capacity need to be associated with fostering
and developing his comprehensive capacity as fostering the learner’s
mathematical capacity will contribute to fostering the learner’s capacity as a
whole.
1.1.3.3. Thus, based on the review of theoretical and practical perspectives,
it can be found that:
Mathematical capacity is the psychological characteristics that reflect in
learners' intellectual activity. It helps them acquire mathematical knowledge
and apply it easily in learning mathematics.
A learner’s mathematical capacity is established and exists and develops in
the activity of learning mathematics. The expression of mathematical capacity
is only realised through the analysis of mathematical learning activity.
Therefore, when examining a learner’s mathematical capacity, the researcher
needs to pay attention to the mathematical operations and particularly
considering the activities of doing mathematics.
8

Mathematical capacity is established, expressed and developed through the
students’ activities of learning mathematics: building and applying the
concepts, demonstrating and applying theorem, solving a mathematical problem
est.

1.2. Mathematical information, the transformation of mathematical
information (TMI)
1.2.1 Mathematical information
1.2.2 The transformation of mathematical information
1.2.2.1. The rationale for conceptualising the transformation of
mathematical information
1.2.2.2. The concepts of the transformation of mathematical
information
The thesis proposes a number of concepts of TMI based on the
psychological and philosophical perspectives, theory of operation, and the
theoretical perspectives of teaching methods.
Based on my empirical examination, analysis and the concepts of TMI
from different perspectives, I define TMI as follows: TMI is an intellectual
activity of a subject (learner) with the purpose of changing the form of
information expression in order to collect the existing knowledge that helps to
understand the content and the hidden relationships within the given
information to acquire new knowledge effectively.
Thus, in teaching mathematics, TMI is characterized in a number of
aspects as follows:
- To transform mathematical information, learners must be able to observe,
read and understand the information; articulate it correctly; connect the
relationships between the information that the learner has already known and
the information that the learner wants to discover.
- The learner must use the knowledge reasonably and flexibly.
- The learner must know how to use thinking operations effectively as
comparison , analysis, synthesis, similarization, and to generalization to
facilitate the process of the transformation of information.
- TMI must be performed through a variety of activities to “get into the
object”; re-organize and restructure the cognitive diagram for the better
processing of new information;

- The appropriate TMI helps to acquire new knowledge effectively and
solve problems emerged in mathematics teaching process.
9

1.3. Capacity of the transformation of mathematical information
1.3.1. Capacity of the transformation of mathematical information
On the basis of examining and analysing the perspectives and concepts of
capacity, mathematical capacity, capacity of the transformation of mathematical
information, and teaching experience in secondary schools, I propose my own
concept as follows: SCTMI is a kind of mathematical capacities that includes a
combination of capacity components to perform TMI in the learning process.
1.3.2. The components of CTMI in mathematics teaching
To propose the capacity components, a number of the following
backgrounds are used to underpin:
- The perspectives of scholars who studied mathematical capacity and
categorized the mathematical capacity. Particularly, we used the perspectives of
mathematical capacity proposed by V. A. Kruchetxki , which underpin to
identify the components of SCTMI in secondary schools.
- A number of characteristics of secondary school students.
* The objectives, curriculum, mathematics textbooks in secondary schools.
* The teaching mathematics in secondary schools, especially the current
situation related to SCTMI.
The division of the components of SCTMI is considered based on the
following requirements:
- The components must be shown in real mathematics teaching situations
and activities in secondary schools.
- The components must play a meaningful role in improving the
effectiveness of teaching mathematics in secondary schools.
- In the teaching process, if such components can be fostered and
developed or not.

Based on the references and analysis of the perspectives of different
scholars and from the previously-discussed arguments, I come to conceptualise:
SCTMI is characterized by the following elements:
1.3.2.1. The capacity of observing, reading and understanding information
1.3.2.2. The capacity of using accurate language to articulate the
information
1.3.2.3. The capacity of idea association to connect the information
1.3.2.4. The capacity of knowledge mobilization to transform information
1.3.2.5. The capacity of mathematicalizing the information from real
situations
10

1.3.2.6. The capacity of testing and assessing the process of the
transformation of information
1.3.3. The expression levels of CTMI
1.3.3.1. Level 1: Students are able to implement basic requirements and
operations of TMI when mathematical problems are given out by clear
instructions, explanation and presentation from the teacher.
1.3.3.2. Level 2: Students are able to recognize the requirements of TMI
from the teacher and they are be able to complete TMI under hints and
instructions from the teachers when necessary. The teacher does not need to tell
students the transformation methods. He only explains and guides students how
to work out the difficult tasks or gives some more details to narrow scope or the
level of problem-solving so that students can mobilize knowledge and seek the
appropriate approach to transforming information.
1.3.3.3. Level 3: Students proactively detect problems and information that
need to be transformed and they actively implement the process of
transformation in order to obtain that new knowledge. The teacher does not
provide any guides or hints. He only assesses significant results that the
students achieve.

1.4. The process of TMI in teaching mathematics
Based on the theoretical basis reviewed and analyzed and the practical
traditions of mathematics teaching in secondary schools, we wish to propose the
process of performing students’ TMI in learning mathematics as follows:
Step 1: To receive initial information. Make observations, read and
understand the information.
Step 2: Base on the information received, recognizing problems to build
the mission for TMI, transforming the mathematical expressions.
Step 3: Perform TMI activities through the use of reflecting capacity,
mobilizing knowledge.
Step 4: Assessing and testing TMI activities (which activities are
performed smoothly and which ones are difficult to perform; which TMI
activities lead to the mathematical solution and which ones don’t; checking
outcomes) . Receiving the information and building up new knowledge.

11

Example: Performing the process of TMI to solve the following equation:
2 2 2 2
2 2 9 9 2 2 9 9 2013 2013
x x x x x x x x          

Stept 1: Information received: solve the equation containing several square
roots and absolute values.
Step 2: Observing the form of the equation, analyzing all the terms in the
equation to identify the properties and the relationship among the teams inside
the square roots.
- Identifying the problems and making the plan to transform the
information: undo the square roots in the equation and change the original
equation into an absolute-value equation or take the powers to undo the roots.
Step 3: Transforming the information.
- Looking at the terms inside the roots, to undo the roots, we use associated
property
   
2
( )
f x f x

and the identities




2 2
;

a b a b
 
to rewrite the terms as
the square of linear expressions.
- Change the equation into the following absolute-value equation:
2 2
9 9 2013 2013
x x x x x x        

Applying the properties of the absolute value
0
a b a b ab
    
in
both directions to transform the information, we have:




2 2
9 9 9 0
x x x x
     

Mathematical Information

1. Receiving initial
information

2. Setting up tasks of TMI

3. Performing TMI
activities

4. Assessing, checking,
receiving information.
Building up new knowledge


Components of capacity
Model: The pro
cess of TMI

12

It then follows that:
2 2 2 2
9 9 ( 9) ( 9) 2
x x x x x x x x x
           

And so the equation is turned to be:
2 2013 2013
   x x x

The transformation of information by taking the powers to undo the roots
is very complicated. Therefore we should not use this approach to solve the
equation.
Step 4: Checking, evaluating the transformation of information (the
transformation of information from square-root expressions in the equation to
the absolute-value expressions is more appropriate and easier while the

transformation of information by taking the powers to undo the roots is much
more difficult; and checking the results simultaneously). As a result, students
have another approach to solve radical equations.
1.5. The current status of teaching mathematics in secondary schools
towards fostering SCTMI.
To examine the current status of teaching mathematics towards fostering
SCTMI, we conducted a survey of teaching practices with the participation of
110 secondary school teachers of mathematics and 294 students in Quang Tri
province. For teacher participants: We used questionnaires with 25 questions
which were divided into 2 parts: secondary school mathematics teachers’
perceptions on the implementation of the current innovation of teaching
methods and their perceptions and understanding about TMI in teaching
mathematics and about fostering SCTMI. For student participants: using
questionnaire to survey SCTMI. The questionnaire consists of 20 questions
used to assess the components of SCTMI categorized into four groups: capacity
of receiving information and reflecting, capacity of mobilizing relevant
knowledge, capacity of transforming information, and capacity of checking.
Each question was rated on a scale of 1-5.
Outcomes of the survey:
In general, teachers understood the requirements and the content the
innovation of teaching method and they implemented it effectively. Besides that
teachers pointed out the difficulties and limitations in the process of
implementation of teaching method innovation, such as: infrequent
deployments and formalism, heavy curriculum, the irrational allocation of time,
inadequacy in facilities and teaching equipment, uneven level of student’s
capacity.
13

Most teachers were able to give their own perceptions on the TMI
activities in mathematics teaching process although their perceptions were not

fully explained. Many teachers were able to point out the important components
of CTMI. The results showed that all of the teachers agreed that SCTMI plays
an important role in the process of mathematics teaching. They also pointed out
the factors affecting the fostering of SCTMI that come from learners, teachers
and the learning environment. Many teachers pointed out the difficulties that
they faced in fostering SCTMI. The teachers also proposed a number of
measures aimed at fostering SCTMI.
However, the teachers’ understanding on TMI, SCTMI and the expressions
of SCTMI is not fully complete. Some teachers did not pay attention to
fostering SCTMI in mathematics teaching process.
The outcomes of survey on 294 secondary schools students and our
teaching observations show that: In general, the basic components of SCTMI
were taken into consideration but at a low level and incomplete (in average
scores rated from 2.1 to 3.5). The survey results also show that SCTMI did not
reach the desired level that it is expected. Some elements of SCTMI evaluation
were low and need to be fostered.
1.6. Conclusion of Chapter 1
Thesis systematizes the perspectives of scholars on capacity and
mathematical capacity. Based on the analysis of some theoretical basis,
concepts of TMI and SCTMI have been proposed, which were used to analyze
the expressions of TMI activities in teaching mathematics. The thesis has
identified six components of SCTMI and three expression levels of CTMI in
teaching mathematics, and developed the procedures to implement TMI
activities with four steps. Based on the research methods of questionnaires,
face-to-face interviews, teaching observations, the thesis has proposed the
evaluations and discussions on the status of teaching mathematics in secondary
schools towards fostering SCTMI and assessed SCTMI in secondary schools.
CHAPTER II. METHODS OF FOSTERING CTMI FOR
SECONDARY SCHOOL STUDENTS IN MATHEMATICS TEACHING
PROCESS

2.1. Orientations of building and implementing methods
2.1.1. Orientation 1. Building methods must be based on the targets of
teaching mathematics, the contents, the methods of teaching mathematics and
14

the methos must be considered with the status of teaching mathematics in
secondary schools, particularly SCTMI.
2.1.2. Orientation 2. Methods must show clear proposals, aimed at
fostering CTMI for secondary school students in mathematics teaching process.
2.1.3. Orientation 3. Methods must ensure the feasibility in the current
teaching conditions.
2.1.4. Orientation 4. The implementation of methods will contribute to
improve the quality of teaching mathematics in secondary schools, especially
maximizing the promotion of students’ activeness, sense of initiative and
creativeness in mathematics learning process.
2.1.5. Orientation 5. Methods must be carried out through the positive
teaching methods, the application of which has now been promoted in schools.
2.1.6. Orientation 6. Methods are not only applicable in teaching
mathematics in secondary schools but also applied in other levels.
2.2. Methods of fostering SCTMI in secondary schools
2.2.1. Methods of fostering students’ capacity of observation, reading
and understanding information.
2.2.1.1. Aims of methods
Through examining the different expression forms of information to guide
students how to read and understand information. In addition, helping students
use the information that they already received to analyze the expressions of
information, so that students can apply it effectively and offer solutions for
problems as required.
2.2.1.2. Methods of implementation
- Teachers provide students with different expressions of information in

teaching mathematics such as natural language, symbols, images, diagrams,
tables…
- Teachers guide students to read and understand mathematical information
from its different expressions.
- Students analyse each component of information, then synthesize them in
order to give out predictions and propose solutions.
2.2.1.3. Content of methods
Method 1. Training students how to read and understanding
information shown in different expression forms
In the process of learning mathematics, students approach mathematical
information by many different expression forms. Contents of current
15

Mathematics textbook for secondary school students are presented in varied
forms, using a variety of diagrams, figures, symbols, graphs, and summary
table. Aside from the presentation in the form of text, information is also
illustrated by images, which is believed to make the textbooks more attractive
to students in learning mathematics. Thus,
* First, it is necessary for teachers to guide students how to read
information expressed in the form of text with common language.
* Teachers guide students how to read information expressed in the form
of figures, diagrams and tables.
Method 2. Guiding students how to analyze and synthesize
information to acquire the orientation for the transformation of
information in order to solve the given problems
When solving a mathematical problem, after determining the given
information and the information of
mathematical problem required, students
need to analyze to associate to problems
related and find the logic chain

connecting the given information and the
information required to be solved, then
synthetize to change information.
For example: When students solve
the following problem:
With the triangle ABC, creating 2
equilateral triangles ABM and ANC outside the triangle ABC. Proving that the
angle between two lines MC and BN is
60
o
.
When guiding students to solve this problem, teachers need to require
students to:
- Read the given information: triangle ABC, triangles ABM and ANC are
equilateral; information needs to be proven: angle between two lines MC and
BN is
60
o
.
- Based on the information that angles and edges of two equilateral
triangles ABM and ANC are equal, it is possible to guide students to analyze
the information required to be proven of the problem on other same proving
ways and choose the result of using the given information most suitably:
To prove:

( , ) 60 ,
o
MC BN  what must be proved? (the desired answer is to
prove


60
o
NIC  ).
A
B
C
M
N
I
Fig.

2.1

16

Sum of 3 angles is
180
o
, considering in the triangle NIC to prove

60
o
NIC 
, what must be proved? (the desired answer is to prove


120
o
ICN I NC 
).

Analyzing information of angles in two triangles ANC, INC and the
relation of the angles in the triangles to guide students to implement next
change operations. We have


120
o
ACN ANC 
, thus, to prove


120
o
ICN I NC  , what must be proved? (the desired answer is to prove


ANB ACM
 ) and finally, comparing two triangles ABN and AMC based given
information.
Thus, based on analyzing the required information, it is possible to find the
way to solve the problem by using the diagam of changing information as
follows and give the suitable proving way:
AB = AM; AN = AC và


BAN MAC





ABN=

AMC




ANB ACM





120
o
ICN I NC 



60
o
NIC 


( , ) 60
o
MC BN 

2.2.2. Methods of fostering students’ capacity of using appropriate

language, technical terms and symbols to articulate accurate information
2.2.2.1. Aims of methods
Throught real teaching situations, teachers helps students understand
correct terms, use accurate languages and symbols to express information
effectively.
2.2.2.2. Methods of implementation
- Guiding students to use accurate languages such as common language,
mathematical language, figures, diagram, tables to articulate mathematical
information.
- Guiding students to understand and use correct terms such as
conjunctions: and; or; if; when and only when.
- Training students to how to express languages by using different
expression forms if possible so as to propose different appoaches to given
problems.
2.2.2.3. Content of methods
Method 1. Guiding students to understand correct terms and use
accurate language and symbols to express information
Understanding correct terms, using accurate languages and symbols to
express information plays an important role in teaching mathematics. When
students know how to articulate information of concepts, theorems which is
17

expressed in common language by using mathematical language and symbol, it
means that the students understand the key mathematical nature and specificity
of the information.
Thus, during teaching mathematics in secondary schools, in order to help
students to understand correct terms, using appropriate languages and symbols
to express information, it is necessary to note that:
* To help secondary school students express logically, first, it is necessary
to students to understand right, differenciate mathematical meanings to use

correct conjunctions such as “and”, “or”, “if”, “if and only if”, “necessary and
sufficient”; symbols
;
 
; …
* It is necessary to require students to understand and use rightly
mathematical terms and symbols.
* Training students how to change skillfully between common language
and mathematical symbols.
* With symbols with different meanings used in textbook due to teaching
reason, depending on each stage of developing concepts, teachers pay attention
to guide students to understant correctly in each case.
Method 2. Training students to use language to express definitions,
contents of mathematics in different ways
A concept can be defined in many similar ways. A mathematical content can
be expressed in different forms. In each case, it is necessary to select the most
suitable form for the aimed purpose. Then, with a given mathematical content, it is
possible to mobilize different knowledge to solve logically and effectively.
2.2.3. Group of methods to fostering students’ capacity of reflection to
connect information and mobilize knowledge to transform mathematical
information
2.2.3.1. Aims of methods
Helping students able to connect information, create relations between
mathematical objects and the given information, and mobilize relevant
knowledge and skills to solve problems in teaching mathematics.
2.2.3.2. Methods of implementation
- Training students how to find relations between the given information
and relevant information to make reflections.
- Guiding students to analyze the given information in different ways of
expression. With each way of analysis, helping students to mobilize relevant

knowledge and skills for solving problems.
18

- Creating some suitable teaching situations to help students to increase
their abilities of association and mobilize suitable their knowledge for changing
information.
2.2.3.3. Contents
Method 1. Training students to transform reflections to solve the given
problems and receive new knowledge
Transforming reflections can be implemented in languages of geometry and
algebra… Besides, during teaching, it is possible to create situations to help
students to transform to associate objects from a mathematical language to another
one.
Method 2. Regularly attaching special importance to systematize
knowledge and methods after each part, each chapter or each form of
mathematics so that students understand the relations in knowledge and
know how to connect information
When implementing systematization, it is necessary to make students
understand knowledge and basic skills of chapter, remember knowledge relating
to next the section through designing teaching activities. Using diagrams and
tables to help students to understand vertical, horizontal and continuous relations
of the contents, which helps students foster their capacity of reflections and
mobilize relevant knowledge to solve the given problems.
Aside from systematizing knowledge according to the theory line, it is also
necessary for teachers to help students to classify and imagine basic
mathematics forms and steps of implementation for solving the problem. It is
necessary to select each form of problem to help students to strengthen,
remember and systematize knowledge and raise awareness and basic skills.
Basing on teaching to solve exercises to transmit knowledge and methods, and
raise students’ abilities of changing mathematical information

In order to systematize knowledge and methods effectively, teachers must
play two roles: being learners to know contents that that need to be reviewed,
skills need to be practiced and way to express problems. Then, being teachers to
prepare optimal methods to answer questions from learners. Thus, students are
supported to understand deeply knowledge, to be skilled and to use it
effectively.
Method 3 Building a chain of problems from simple to complex related
in order to raise students’ abilities of associating and mobilizing
effectiveness of information
19

When teaching way to solve mathematics exercise, in order to training
students in abilities of association, teachers can guide students to know to build
and solve chain of problems to find the relation of the exercises, then when
solving the problem, it is possible to associate the way of solve other problems.
Method 4. Training students to use the relation of the general and the
particular to create advantages for students to associate and mobilize
knowledge then changing well mathematical information
Method 5: Building situations of training with difficulties and
contradictions then creating motivation and demand for students to know
to mobilize logically knowledge to change effectively mathematical
information
* First, teachers create teaching situations including contradiction and
difficulties to create motive of awareness of students.
* Guiding students to detect and determine contradictions and difficulties
to mobilize knowledge, skills and methods related to change mathematical
information.
* Guiding students to solve contradictions through changing information to
receive information and given problems as required of awareness.
2.2.4. Group of methods of fostering students’ capacity of

mathematicalizing information in reality.
2.2.4.1. Aims of methods
To train students in abilities of solving problems through transforming
information between mathematical language and the real one. Then, to raising
the ability of using mathematical knowledge in the reality, implementing
effectively the tast of teaching mathematics and developing students’ abilities
of changing information.
2.2.4.2. Methods
- Creating specific teaching situations to help students to know the relation
between real information and mathematical one.
- Guiding students how to change between the real information and
mathematical one.
- Training students’ abilities of using mathematics to solve problems with
real content.
2.2.4.3. Contents
Method 1: Guiding students to understand the relations of
mathematical information and the reality, then helping students build
20

capacity to use mathematics in the real situations to implement targets and
principles of training effectively
Method 2: Training students to know the way how to transform
between mathematical language and the reality to solve problems
To solve the problems built from practical situations, teachers must know
to help students implement in accordance with the procedure of changing
language as follows:
Step 1. Changing related information of the problem from the real language
to the mathematical one: Coding mathematical language from practical situation.
Step 2. Implementing operations of changing information in mathematical
language to give the conclusion of the problem under the mathematical

language.
Step 3. Changing the problem’s conclusion from mathematical language to
the real one: considering result or coding mathematical information of the real
language.
2.2.5. Group of methods to training students in abilities of testing and
evaluating the process of changing information
2.2.5.1. Purpose
Training students in abilities of testing and evaluating the process of
chaning information in learning mathematics. Helping students to evaluate the
advantageous information change, the difficult one and recognize their strength
and the weakness in changing information, then learn from experience and
adjust suitably.
2.2.5.2. Methods
- Guiding students to test and evaluate after completing changing
information by self-test and self-evaluation, mutual exchange and evaluation,
and discussions with teachers and friends.
- Training students to know to detect mistake forms and find solutions,
then to raise their abilities of evaluating rightly the process of chaning
information.
- Training students in the awareness of gathering an interesting way of
solving, a new method, an effective way of changing information or finding a
new way of solving the given problem.
2.2.5.3. Contents
Method 1: Training students to test and evaluate result of changing
mathematical information
21

Test and evaluation can be implemented at any time in learning
mathematics in general as well as in changing mathematical information in
particular. The abilities of evaluating the process of changing information will

help students to know their accuracy of reading, understanding and expressing
information, their abilities of association to connect information suitably, their
abilities of mobilizing knowledge and skills to changing information logically
and optimally then, it is possible to adjust the process of changing information
better and improve the learning effect. If students are able to test and evaluate
the result of changing information in particular and the result of study in
general, they can be more self-aware, self-reliany and self-confident in their
study.
* First, teacher must regulary require students to self-test and evaluate results
of changing information in learning mathematics.
* Teacher must create many opportunities and practices for students’
awareness of cooperating and exchanging for mutual test and evaluation of
changing mathematical information.
Method 2: Building suitable training situations for students to have
chance to detect and correct mistakes in solving mathematics in order to
training students in abilities of testing and evaluating rightly mathematical
information change.
From the teaching situations including mistakes, teachers help students to
change information more skilledly and exactly, at the same time, teachers form
students’ thought of criticization which is one of the important parts of
mathematical thought to raise abilities of testing and evaluating information.
* First, teachers design some teaching situation including normal mistakes
so that students have opportunities to face to those mistakes.
* Guiding students to detect mistakes in specific situations.
* Guiding students to analyze reasons of mistakes and solutions during
learning mathematics.
2.2.6. Group of methods to build question system in the typical
training situations for students to changing well mathematical information
2.2.6.1. Purpose
Through building the system of questions in the typical teaching situation,

to guide and orient students in learning and developing their activeness.
Especially, to training students in components of abilities of changing
mathematical information.
22

2.2.6.2. Methods
- Building system of questions to test information of students’ knowledge
and skills.
- Building system of questions to stimulate abilities of reading,
understanding and expressing information; abilities of connecting information
and promote students’ knowledge and guide students to implement the process
of changing mathematical information.
- Building system of questions to provide students information of
knowledge and skills…
2.2.6.3. Contents
Method 1: Building system of questions in training concepts
Method 2: Building system of questions in training theorems, rules
and methods
Method 3: Building system of questions in training exercise solving
2.3. Conclusion of chapter 2
The thesis shows clearly orientations in order to build teaching solutions to
foster SCTMI during learning mathematics in secondary schools linking to
targets, contents, programs and methods of teaching mathematics in secondary
school, ensure practicability in the current teaching condition, contribute to
develop students’ activeness and initiative and raise quality of teaching
mathematics. Then, to propose 6 groups of teaching methods including 16
specific methods to foster SCTMI in secondary schools in teaching
mathematics.
CHAPTER III. TEACHING EXPERIMENT
3.1. Purpose

Teaching experiment is carried out to test the scientic supposition of the
thesis through teaching practice; to test practicability and effectiveness of the
proposed teaching methods.
3.2. Contents
Contents of hours of experimental teaching in mathematics program of
secondary school from class 6 to class 9 include 18 hours.
3.3. Way of organization
3.3.1. Steps of implementation
Step 1: The thesis author compiles some lectures as required, summarizes
results of theoretic studies shown in the thesis in the chapters I and II, and some
requirements to experiment.
23

Step 2: Selecting location and school of experiment. Coordinating with the
school management board to select class and teacher of experiment.
Step 3: Teaching at the control class. Then, disseminating purposes and
requirements to teachers who participate in the experimental teaching.
Disseminating methods of applying the author’s studied theories into finding,
preparing lecture and organizing the teaching for teachers participating in the
experiment. Teachers in some secondary schools implement the teaching in
accordance with the prepared lectures. Implementing class observation to learn
from experience.
Step 4: Evaluating and learning from experience the experiment results.
Step 5: Testing the stage II and implementing a survey by using survey
note to evaluate the experiment results and control. Analyzing, evaluating
results, learning from experience, concluding problem, and adjusting the study
results to be suitable.
3.3.2. Objects
* Stage 1: was implemented from January 2012 to April 2012 at Nguyen
Trai Secondary School, Vinh Linh District and và Tran Quoc Toan Secondary

School, Dong Ha City, Quang Tri.
* Stage 2: was implemented from January 2013 to April 2013 at Tran
Hung Dao Secondary School, Dong Ha City, Quang Tri and và Tan Hop
Secondary School, Huong Hoa District, Quang Tri.
3.4. Evaluation of results
3.4.1. Qualitative analysis
- Through the experiment and professional exchanges, class observations
and organization for experience, we recognize that the teachers are enthusiastic,
confidential with stable professional knowledge and desire for knowledge.
- Preparation: Based on the experiment documents, the teachers prepared
considerately, meticulously and carefully in accordance with guidance with
special style and nuance of each teacher and suitable to their students. Before
and after each hour of teaching, they discussed their views with the experiment
guider. Although initially some teachers flinched and worried about receiving
and implementing experiment tasks, gradually they showed fully contents and
spirit of the experiment documents.
- Process of teaching was implemented proactively and effectively.
Teaching activities were designed clearly and shown fully ideals. The question
system was designed specifically, clearly and suitably to typical teaching
24

situations; students were guided and had advantageous conditions in changing
mathematical information in teaching hours.
- Atmosphere of learning in experiment class was exciting, enthusiastic
and seriously. In hours of learning, students showed their abilities of changing
mathematical information through components such as abilities of association
and promotion of related knowledge, abilities of changing, and abilities of
transforming language, abilities of detecting and correcting mistakes…
- After each hour of experimental teaching, the teachers implemented self-
commendation, evaluation, discussion for experience seriously in the profession

group and proposals to the experiment guider.
- Common evaluation of many teachers shows that this is an interesting
topic with active methods and important effect in teaching mathematics.
Teachers were interested in using the methods proposed by the experiment
guider, and students were active and positive. Applying the methods was
practical and many teachers suggested that it should be implemented widely in
the whole province. However, there were difficulties during experiment offered
by the teachers such as some suggested specific methods such as building chain
of similar problems, widening problems, transforming language in internal
mathematics… were used advantageously to good and fair students but
difficultly to average and weal students; some classes are crowned with
students with small class room, unequal learning abilities of students, limited
teaching and learning facilities… so building and organizing teaching activities
did not gain the given requirement.
3.4.2. Quantum analysis
Quantum analysis is based on the results of examination in the end of
chapter of experiment times with the selected form at schools and showed in the
tables 3.4; 3.5; 3.7 of the thesis.
The quantum results in the table 3.6 and 3.9 show that the average mark of
stage II examination, average mark of evaluating abilities of changing
information of experiment classes in each school correlative are 1% higher that
the average mark of examination, average mark of evaluating abilities of
changing information of control classes in that school.
3.5. Conclusion
Experiment and its results show that: the purposes of experiment were
completed, practicability and effectiveness of methods were affirmed, scientific
supposition of the thesis can be practically accepted.
25

CONCLUSION

Main results of the thesis are as follows:
1. Systematising attitudes of many scientists to abilities, mathematical
abilities, information, and mathematical information. Different attitudes of
changing mathematical information were given;
2. Proposing basises and ideals to determine connotation of the concept
changing mathematical information, abilities of changing mathematical
information of secondary school students expressed in teaching mathematics
and the process of TMI;
3. Clearing somewhat the status of CTMI of secondary school students in
teaching mathematics and the status of training the abilities through survey with
survey note;
4. Determining 6 components of CTMI from analyzing some theoretical
and practical bases:
5. On that basis, the thesis also shows 3 expression levels of CTMI in
teaching mathematics;
6. Building process of TMI in teaching mathematics.
7. Proposing basic orientation for direction in building groups of solutions
to ensure effectiveness and practicability.
8. Proposing 6 groups of teaching methods in which focusing on 16
specific methods in order to contribute to foster SCTMI in teaching
mathematics;
9. Organizing teaching experiment to illustrate effectiveness and
practicability of the proposed teaching methods.