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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;

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.
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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

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

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
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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

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.
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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.

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

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.

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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

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

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
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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

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


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





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
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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

knowledge and skills for solving problems.
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- 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

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
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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

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,

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.
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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

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.
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CONCLUSION