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. [...]... 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 22 Step 2: Selecting location and school of experiment Coordinating with the school management board... 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... 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... 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,... 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... 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... 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... 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... 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... 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 . secondary schools. - A number of characteristics of secondary school students. * The objectives, curriculum, mathematics textbooks in secondary schools. * The teaching mathematics in secondary schools,. 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. 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