Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.Vận dụng lí thuyết siêu nhận thức vào dạy học môn Toán trung học cơ sở theo hướng phát triển năng lực toán học cho học sinh.ĐỀ CƯƠNG NGHIÊN CỨU MINISTRY OF EDUCATION AND TRAINING HA NOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI HUONG LAN APPLYING METACOGNITIVE THEORY IN TEACHING MATHEMATICS IN SECONDARY SCHOOLS IN THE DI.
MINISTRY OF EDUCATION AND TRAINING HA NOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI HUONG LAN APPLYING METACOGNITIVE THEORY IN TEACHING MATHEMATICS IN SECONDARY SCHOOLS IN THE DIRECTION OF DEVELOPING MATHEMATICAL COMPETENCE FOR STUDENTS Major: Theory and Methods of Teaching Mathematics Code: 14 01 11 SUMMARY OF DOCTORAL THESIS IN EDUCATIONAL SCIENCES Hanoi - 2022 The dissertation is completed AT HANOI NATIONAL UNIVERSITY OF EDUCATION Scientific supervisor: Prof Dr Bui Van Nghi Reviewer 1: Assoc.Prof Dr Nguyen Huu Hau Hong Duc University Reviewer 2: Assoc.Prof Dr Le Van Hien Hanoi National University of Education Reviewer 3: Assoc.Prof Dr Nguyen Tien Trung Vietnam Journal of Education The dissertation will be defended in front of the Dissertation Defense Jury at Hanoi National University of Education Time: ……………… Date: ………………………… The dissertation can be found at: - National Library, Hanoi - Library of Hanoi National University of Education INTRODUCTION Rationale for choosing the research topic In the context of the Fourth Industrial Revolution, education in Vietnam faces challenges and negative impacts To address these challenges, it is necessary to change educational policies, contents and methods of training to create human resources capable of following new technological production trends Around the world, policymakers are making efforts to reform the education system in general and Maths education in particular to create a fundamental transformation in the content, programs and methods of Maths learning of students Metacognition is increasingly attracting the research interest of psychologists and educators Studies on the role of metacognition in the development of students' competencies focus on two basic components: individual’s knowledge of their thinking processes and the monitoring and control of individual activities in the learning process Teaching with metacognition (serving as a tool – a method of higher-order thinking in cognitive processes) will contribute to the development of students' competencies and help promote a more positive and effective learning environment There have been several research works on metacognitive skills in the process of teaching Maths such as doctoral theses by Hoang Xuan Binh, 2019; Le Binh Duong, 2019; Hoang Thi Nga, 2020; Le Trung Tin, 2016; Phi Van Thuy, 2021 Research works on metacognition in Maths teaching in Vietnam mainly focus on training metacognitive skills through Maths The problem of understanding the influence of metacognition on the formation and development of mathematical competence of secondary school students through Maths has not been researched specifically Therefore, conducting research in this direction to find out how to apply metacognitive theory to contribute to the development of mathematical competence for students in teaching Maths in secondary schools is necessary For the above reasons, the researcher chose the research topic: “Applying metacognitive theory in teaching Mathematics in secondary schools in the direction of developing mathematical competence for students” Research aims On the basis of studying the impacts of metacognitive activities on the mathematical competence of secondary school students, we have proposed measures for applying metacognitive theory in teaching secondary school Mathematics in the direction of developing mathematical competence for students Research questions The research question is "How can the application of metacognitive theory in teaching secondary-school mathematics contribute to the development of students' mathematical competence?" To answer this research question, it is necessary to answer the following specific questions: What is the theoretical basis for the application of metacognitive theory in teaching secondary school Mathematics to contribute to the development of mathematical competence for students? What is the current situation and effectiveness of organizing metacognitive activities for students in the direction of fostering mathematical competence in Mathematics at secondary schools today? How metacognitive activities in math learning affect the formation and development of mathematical competence for secondary school students? How to apply metacognitive theory in teaching secondary school Mathematics to contribute to the development of mathematical competence for students? Research subjects and objects + Research subjects: Measures for applying metacognitive theory in teaching secondary-school mathematics to contribute to the development of mathematical competence for students + Research objects: The process of teaching and learning Mathematics of teachers and students in secondary schools Scientific hypothesis If the impacts of metacognitive activities on students' mathematical competence in secondary-school Maths are clarified and measures for applying metacognitive theory in teaching secondary school mathematics in the direction of developing mathematical competence for students are implemented, they will contribute to improving the effectiveness of teaching and the development of mathematical competence for students Research tasks + Review related research issues on metacognitive theory and mathematical competence in the field of mathematics education + Analyze the impacts of organizing metacognitive activities in teaching Mathematics on the development of mathematical competence for secondary school students + Investigate the current situation of applying metacognitive theory in teaching Mathematics in the direction of developing mathematical competence for secondary school students today + Propose measures to apply metacognitive theory in teaching secondary school mathematics in the direction of developing mathematical competence for students + Organize experiments to test the feasibility and effectiveness of the measures Research Methods Theoretical research method Survey method Pedagogical experiment method Case study method: Mathematical statistical method Research scope - Programs and contents of secondary school Mathematics - Methods of teaching secondary school mathematics in the direction of developing mathematical thinking and reasoning competence and mathematical problem-solving competence for students through typical situations in teaching secondary school mathematics New contributions of the thesis 9.1 Theoretical contributions The thesis has: + Reviewed studies on applying metacognitive theory in teaching towards developing mathematical competence for secondary school students + Identified and clarified the role of metacognitive activities in developing students' mathematical competence in the process of teaching Mathematics + Proposed measures to apply metacognitive theory in teaching secondary school mathematics in the direction of developing mathematical competence for students The above results contribute to the theory and teaching methods of Mathematics 9.2 Practical contributions + The thesis has reflected part of the current situation of teaching Mathematics in the direction of developing mathematical competence for students in secondary schools + The results of the thesis contribute to innovating teaching methods of Mathematics and improving the effectiveness of teaching Mathematics in secondary schools 10 Arguments that will be protected (1) Metacognition plays an important role in developing students' mathematical competence through secondary-school mathematics (2) The measures for applying metacognitive theory in teaching secondary school mathematics in the direction of developing mathematical competence for students proposed in the thesis are feasible and effective 11 Structure of the thesis In addition to the introduction, conclusion and appendices, the thesis consists of chapters: Chapter Theoretical basis Chapter Practical basis Chapter Measures for applying metacognitive theory in teaching secondaryschool mathematics in the direction of developing mathematical competence for students Chapter 4: Pedagogical experiment CHAPTER THEORETICAL BASIS 1.1 Literature Review 1.1.1 Research on metacognition 1.1.1.1 Research on metacognition in the world Metacognition has been interested in research since the 70s of the twentieth century in the work of Flavell (1979) Although there are many different definitions of the term "metacognition", in general, these definitions provide a unified meaning of this concept The large-scale research works on metacognition in education were conducted in the late twentieth century and early twenty-first century, more or less related to comprehensive assessment programs to understand the role of effective teaching methods in schools 1.1.1.2 Research on metacognition in Vietnam In Vietnam, in the field of Psychology and Education, there are some authors researching metacognition, for example, Hoang Thi Tuyet with the research "Strategies for teaching nature-society in primary school"; Ho Thi Huong (2013) with the research "A study on metacognitive theory and its applications in secondary school education", institute-level project, Vietnam Institute of Educational Sciences; Hoang Xuan Binh, Phi Van Thuy (2016) had research articles on "The role of metacognition in teaching Mathematics in high schools" and "Developing metacognitive skills for students through solving geometry problems in high school” In these articles, the authors have analyzed and clarified the role of metacognition in the process of teaching mathematics in high schools, and concretized the fostering of metacognitive skills in teaching spatial geometry In this thesis, with the aim of applying metacognitive skills in teaching secondaryschool Maths in the direction of developing mathematical competence for students, we chose component metacognitive skills corresponding to metacognitive activities, which are orientation and planning, monitoring and adjusting, and evaluating to focus on organizing for students to apply metacognition when learning Math, thereby affecting components of mathematical competence that need to be developed through Math 1.1.2 Research on Mathematical competence 1.1.2.1 Overview of research on mathematical competence around the world The concept of competence There have been many studies abroad on mathematical competence from different aspects, such as: V.A Krutecxki; A.N Kolmogorov (refer to Pham Van Hoan et al., pp.128-129); UNESCO (1973) announced 10 basic mathematical competence indicators; Morgen Niss (2003); 1.1.2.2 Overview of research on mathematical competence in Vietnam The concept of competence According to Nguyen Cong Khanh “Competence is the ability to master systems of knowledge, skills, attitudes and operate (connect) them logically into successfully performing tasks or effectively solving problems in life Competence is a dynamic (abstract) structure, open, multi-component, multi-level, containing not only knowledge and skills but also beliefs, values, and social responsibilities, etc., expressed in the willingness to act in real conditions and changing circumstances" Thus, it can be seen that “competence is seen as a combination of individual attributes that are suitable for the specific requirements of a certain activity, enabling the individual to successfully perform certain activities, achieve desired results under specific conditions” Research on Mathematical competence The Mathematical General Education Program in 2018 specifically identified the goal of developing students' mathematical competence, including five components: (1) Mathematical thinking and reasoning competence; (2) Mathematical modeling competence; (3) Mathematical problem-solving competence; (4) Mathematical communication competence; (5) Competence to use mathematical learning tools and means; and directly set requirements for teaching Maths, focusing on the formation and development of mathematical competencies for students, making an important contribution to the development of necessary competencies to continue studying and working in life 1.2 Metacognitive Theory 1.2.1 Cognition, cognitive activities, skills 1.2.2 Concepts and approaches of metacognition Based on reference to research results on metacognition in the world and in Vietnam (section 1.1.), in this thesis, with the aim of applying metacognition as a "support means" for teaching Maths to approach the goal of developing students' mathematical competence, we define that “Metacognition is a form of cognition, a highorder thinking process, relating to human cognitive processes, reflecting their understanding of the nature of the cognitive process and strategies for carrying out cognitive activities” 1.2.3 Features and functions of metacognition According to John Flawell (1979), some of the main features of metacognition that need to be mentioned are “Awareness of one's own thinking process; Active and proactive monitoring of cognitive processes related to learning tasks; Learners find ways to solve problems by themselves; Monitor and regulate one’s own cognition; Evaluate the process and results achieved against the set goals” The functions of metacognition are “Awareness of one’s own cognition; Planning, monitor, and adjust the problem-solving process.” 1.2.4 Metacognitive skills Along with the concept of metacognition, scholars around the world have done indepth research on metacognitive skills and activities According to Veenman et al (2014), metacognitive skills are associated with procedural metacognitive knowledge Metacognitive skills are understood as the ability to monitor, direct, and adjust one's learning or problem-solving behavior Metacognitive skills are demonstrated through orientation, goal setting, planning, monitoring, and evaluation skills Metacognitive skills directly shape learning behavior and thus influence learning outcomes 1.2.5 Components of metacognition Research on metacognition cannot fail to mention Flavell, who is considered one of the first to define "metacognition" The metacognitive components proposed by Flavell serve as the foundation for later metacognition-related research Flavell (1979, identified the components of metacognition and specified their characteristics, including Metacognitive knowledge; Metacognitive experiences; Cognitive goals; Activities and strategies He argued that the ability of an individual to regulate cognitive outcomes depends on the interaction between the components of cognitive strategy, cognitive experience, metacognitive knowledge and metacognitive experience 1.2.6 Metacognitive activities According to the research by Tobias and Everson (2002), metacognition is a combination of factors such as skills, knowledge (understanding of cognition) in the cognitive process of learners as well as the control of the learning process according to the following “pyramid” model: Figure 1.1 The “pyramid” model of the metacognitive process (Tobias and Everson, 2002) 1.2.7 Metacognition in problem-solving 1.2.7.1 Problem-solving diagram There have been many scholars outlining different models of problem-solving Fernandez, Hadaway and Wilson (1994) gave a problem-solving model (Figure 1.4) in the form of a circle representing "the management overseeing the entire problem-solving process", which Schoenfeld and Flavell (2000) have recognized as metacognition Figure 1.4 The problem-solving model outlined by Fernandez, Hadaway and Wilson (1994) 1.2.7.2 Metacognitive activity during the Mathematical problem-solving process To describe the metacognitive aspects of learners in solving mathematical problems, Garofalo and Lester (1985) have proposed four metacognitive activities related to solving any mathematical task, including orientation, organization, implementation, and evaluation 1.2.7.3 Theoretical framework of metacognition in problem-solving processes Artzt and Armor-Thomas (1992) provided a framework for the interrelationship between metacognitive and cognitive processes in solving mathematical problems Stages: 1- Read the problem (cognition); 2- Understanding the problem (metacognition); 3- Problem analysis (metacognition); 4- Devising a plan (metacognition); 5a-Exploration (cognition); 5b- Modification (metacognition); 6aImplementation (cognition); 6b- Evaluation (metacognition); 7a- Revision (cognition); 7b-Confirmation (metacognition); 8- Watch and listen (uncategorized) 1.2.7.4 The five metacognitive components of problem-solving Howard, McGee, Shia and Hong (2000) have identified five learning strategies that self-regulated learners use in a problem-solving context and considered them as metacognitive components to guide learners' cognitive activities in the problem-solving process: 1.3 Cognitive and metacognitive activities in the process of developing mathematical competence for secondary school students` 1.3.1 Cognitive and metacognitive activities 1.3.2 Cognitive and metacognitive manifestations of secondary school students in components of mathematical competence Table 1.3 Cognitive and metacognitive manifestations of secondary school students in components of mathematical competence Cognitive and metacognitive Cognitive and metacognitive manifestations in manifestations in mathematical mathematical competence of secondary school competence students Mathematical thinking and reasoning competencies are demonstrated - Performing thinking operations, especially through: observing (cognition), explaining similarities - Performing thinking operations such and differences in many situations as: compare, analyze, synthesize, (metacognition) and expressing the results of specialize, generalize, analogize; observation (cognition) induce, and interpret (cognition) Reasoning (cognition) logically - Pointing out evidence and reasons (metacognition) when solving problems (cognition) and making reasonable - Asking and answering questions when arguments before concluding reasoning and solving problems; proving that (metacognition) the mathematical proposition is not too - Explaining or modifying how to complicated (cognition) solve the problem mathematically (metacognition) Mathematical modeling competence is demonstrated by: - Identifying mathematical models (including formulas, equations, tables, graphs, etc.) for situations appearing in practical problems (metacognition) - Solving mathematical problems in the established model (cognition) - Demonstrating (cognition) and evaluating (metacognition) the solution in the actual context and improving the model if the solution is not appropriate (metacognition) - Using mathematical models (including mathematical formulas, diagrams, tables, drawings, equations, representations, etc.) to describe situations appearing in some practical problems which are not too complicated (cognition) - Solving mathematical problems in the established model (cognition) - Demonstrating mathematical solutions in practical contexts (cognitive) and getting used to verifying the correctness of the solutions (metacognition) 11 relationship between cognitive and metacognitive activities, the role of cognitive skills in learning mathematics - especially in mathematical thinking and reasoning activities, mathematical communication and mathematical modeling, in this thesis, applying mathematical reasoning is understood as: On the basis of clarifying the effect of metacognition on the development of components of mathematical competence, build methods of teaching mathematics to exploit strengths and effects of metacognitive activities to affect students' self-study competences 1.3.4.2 Orientations for applying metacognitive theory in teaching secondary school Maths in the direction of developing mathematical competence for students - Evaluate the actual situation of organizing cognitive and metacognitive activities in the direction of developing mathematical competence for secondary school students in teaching mathematics - Study methods and techniques of teaching Mathematics in order to exploit the advantages of metacognition for the goal of developing mathematical competence for students through Mathematics - Develop measures to apply metacognition in teaching Secondary School Mathematics in order to develop mathematical competence for students 1.4 Summary of Chapter In this thesis, applying metacognitive theory is understood as: On the basis of clarifying the effects of cognitive thinking on the development of components of mathematical competences, build measures of teaching mathematics to exploit the strengths and effects of metacognitive activities to affect students' self-study competence Many studies have shown that when students are taught how to learn and think, they can achieve higher levels of education; effective teachers often integrate teaching contents with teaching strategies and metacognition information about choosing effective strategies in their daily teaching activities The comparison between cognitive activities and metacognitive activities can be seen in Table 1.2; The manifestions of cognition and metacognition in the components of mathematical competence for secondary school students can be seen in Table 1.3 The results of research and analysis show that there are many opportunities and ways of applying metacognitive theory to develop mathematical competence for students In order to perform well this activity, it is necessary to add an assessment of the actual situation of applying metacognitive theory to the process of developing students' mathematical competence, as presented in the next chapter 12 CHAPTER PRACTICAL BASIS 2.1 Objectives, tasks, participants and time of the survey on the current situation 2.1.1 Objectives and tasks of the survey This study was designed towards the following objectives: To clearly investigate the actual situation of applying metacognitive theory in teaching secondary-school Maths in the direction of developing mathematical competence for students of math teachers and the manifestations of metacognitive activities and mathematical competence of students at some secondary schools in Tuyen Quang province This serves as a practical basis for proposing measures in the next chapter To achieve the above objectives, the survey tasks are to answer the following scientific questions: + How are the perceptions of secondary school math teachers about metacognitive theory and their activities related to metacognition in the current math teaching process? + What is the current situation of teaching secondary-school Maths in the direction of developing mathematical competence for students today? + What are the opinions of some educational experts on the development of mathematical competence for students in teaching secondary-school Maths? + How are the students' metacognitive activities in the process of learning mathematics at secondary school? 2.1.2 Participants and time of the survey Survey locations: secondary schools in Tuyen Quang province (Ỷ La Secondary School; Phan Thiet Secondary School; Hung Thanh Secondary School, Le Quy Don Secondary School, Nong Tien Secondary School) Survey participants include: + 30 teachers of Mathematics at some secondary schools in Tuyen Quang province; + 100 students in grade at some secondary schools in Tuyen Quang province; + experts in mathematical education (including 01 professor, 01 associate professor and 05 doctors) Due to the situation of the Covid-19 epidemic, the survey participants and the sample size were considered to be minimal, but still ensure reliability and ensure epidemic prevention 2.2 Survey content, tools and methods 2.2.1 Survey contents + The perceptions of secondary school math teachers on metacognition and the requirements for developing mathematical competence for students 13 + The current practice of metacognitive activities of teachers and students in teaching and learning Mathematics at secondary school today +The current practice of teaching Mathematics in the direction of developing mathematical competence + The advantages and difficulties of teachers in applying metacognitive theory in teaching Mathematics in the direction of developing mathematical competence for students today 2.2.2 Survey tools and methods The tools used in this study are questionnaires with descriptions and rating levels (This questionnaire is shown in Appendix 1,3 and of the thesis) The data processing tool is the algorithms of the mathematical statistical method to calculate the percentage for each level 2.2.3 The procedure of the survey Step 1: Set up a research group of people (thesis author and teachers representing surveyed schools) Step 2: Study and find out the contents related to secondary-school Maths and requirements to develop mathematical competence for students today; apply metacognition in the process of teaching and learning mathematics in secondary schools Step 3: Discuss the contents of the questionnaire to be used in the survey; discuss issues of concern when discussing and consulting with teachers and educational experts; unify the contents for the survey Step 4: Conduct the survey on the current status of secondary-school math, the requirements for developing mathematical competence for students today, and the reality of applying metacognition in the process of teaching and learning mathematics in secondary schools Step 5: Consult with experts in mathematical education and teachers on related issues during the survey 2.3 Survey results 2.3.1 Teachers' perception of metacognition 2.3.2 Consult educational experts + Consult educational experts on the possibility to develop students' mathematical competence through Mathematics + Consult experts about the students' metacognitive activities in Math and the influence of metacognition on the development of students' mathematical competence + Consult experts about the benefits of students with metacognitive activities compared with students without metacognitive activities in the process of developing mathematical competence 14 + Consult experts on measures to apply metacognitive theory in teaching Mathematics in the direction of developing students' mathematical competence 2.3.3 Student survey results 2.3.3.1 Find out the students' metacognitive activities in the process of solving math problems 2.3.3.2 Survey students on the manifestations of metacognitive activities 2.3.3.3 Results of surveying students on the manifestations of metacognitive activity in the process of discovering open-ended Math problems 2.4 Analysis of the contents of secondary-school mathematics and requirements for the development of mathematical competence for students 2.4.1 Secondary-school Maths 2.4.2 Objectives of teaching mathematical competence and skills 2.4.2.1 For the contents of Arithmetic and Algebra 2.4.2.2 For the contents of Geometry and Metrology 2.4.2.3 For the contents of Statistics and Probability 2.4.2.4 For practice and experiential activities 2.4.3 Analysis of the characteristics of secondary-school students in learning activities 2.4.3.1 Learning activities of secondary-school students 2.4.3.2 Intellectual development, cognitive ability and metacognitive ability of secondary school students Secondary school students have begun to have the ability to analyze, synthesize and especially reason logically and effectively when perceiving objects and phenomena 2.4.3.3 The current situation of training metacognitive skills in teaching and learning Mathematics as reflected in published works Although there have been many research works on improving and renovating the teaching method of Mathematics in Vietnam for many purposes, with different tools and methods at different levels, it can be seen that in teaching Mathematics, teachers often still focus on conveying content knowledge and practicing basic skills to solve math problems Regarding developing mathematical competence for students, including thinking competence and other competences in the whole process of learning and applying Mathematics, there are limitations in both approaches and implementation methods, which need to be removed or overcome 2.5 Summary of Chapter Although not explicitly talking about metacognition, the survey results have shown that many teachers have applied metacognition in their teaching process; students have also had metacognitive activities However, the activities of applying metacognition by 15 both teachers and students are not regular; there is no habit of using metacognition as a thinking tool Therefore, the students have not made significant progress in mathematical competence and their learning results in Mathematics are low From the results of the survey on the actual situation of applying metacognition in teaching and learning Mathematics for secondary school students, we continue to consult with experts and teachers to orient and identify effective pedagog ical measures which we can apply to overcome current limitations in developing students' mathematical competence The results presented in this chapter will be the important practical basis for proposing measures to teach secondary school mathematics in the direction of developing mathematical competence for students in the next chapter 16 CHAPTER MEASURES FOR APPLYING METACOGNITIVE THEORY IN TEACHING MATHS AT SECONDARY SCHOOL IN THE DIRECTION OF DEVELOPING MATHEMATICAL COMPETENCE FOR STUDENTS 3.1 Orientation for building measures 3.1.1 Stick to the goal of developing students' mathematical competence as specified in the 2018 Maths general education program 3.1.2 Coordinate and flexibly apply metacognitive models suitable to the practice of teaching Maths in Vietnamese secondary schools 3.1.3 Exploit the relationship and mutual influence between cognitive and metacognitive activities 3.1.4 The pedagogical measures are synchronous with the methods, forms, and techniques of teaching Maths in the direction of developing students' competence 3.2 Methods of applying metacognitive theory to develop mathematical competence through secondary school Maths 3.2.1 Measure 1: Apply metacognition in teaching typical situations of secondary school mathematics according to the process of steps towards the goal of developing students' mathematical competence 3.2.1.1 Purpose of the measure This measure aims to make the best use of opportunities to apply metacognitive theory in teaching typical situations of secondary school mathematics towards the goal of developing students' mathematical competence 3.2.1.2 The basis of the measure For the purpose of developing mathematical competence, in which the ability to solve mathematical problems and the ability to think and reason mathematically are considered two core competencies, the center of the close connection between the components of mathematical competence, referring to the stages of cognition and metacognition when solving mathematical problems by Artzt and Armor-Thomas (1992) and the research results of Annemie Desoete (2007), we have proposed the process organizing cognitive and metacognitive activities in learning Mathematics This process has specified the orientation and application of metacognitive theory to teaching Mathematics through steps of organizing cognitive and metacognitive activities for students 3.2.1.3 How to implement the measure Teachers need to combine teaching theory of each typical situation with cognitive and metacognitive activities included in the teaching steps mentioned above, eliciting metacognitive activities for students The 5-step process of organizing cognitive and 17 metacognitive activities for students when teaching typical situations in Mathematics is shown in Table 3.1 3.2.2 Measure Apply the metacognitive theoretical framework to the process of solving mathematical problems in the direction of developing mathematical competence for students 3.2.2.1 Purpose of the measure This measure helps teachers to apply the metacognitive theoretical framework to the problem-solving process in teaching Mathematics in the direction of developing mathematical competence for students 3.2.2.2 The basis of the measure The basis of this measure is the metacognitive theoretical framework in the process of solving mathematical problems of Artzt and Armor - Thomas (1992) presented in Chapter 1, including stages with cognitive and metacognitive activities 3.2.2.3 How to implement the measure Based on the metacognitive theoretical framework including stages in the process of solving mathematical problems of Artzt and Armor - Thomas (1992), teachers can apply to each specific mathematical problem Teachers need to pay attention to exploiting metacognitive components in the problem-solving process that have been described by Howard, B.C., McGee, S., Shia, R., & Hong, N.S (2000) mentioned in section 1.2.4.4 in Chapter It is “Represent the problem; Knowledge of Cognition; Monitor the implementation process; Evaluate tasks; Ensure objectivity.” 3.2.3 Measure Apply the process of steps of metacognitive theory in teaching and applying mathematics into practice in order to develop mathematical competence for students 3.2.3.1 Purpose of the measure This measure aims to develop mathematical competence for students through organizing activities that combine the process of mathematical modeling with the process of stages of metacognitive theory in teaching and applying mathematics into practice 3.2.3.2 The basis of the measure Problem-solving competence in learning Math is not only solving math problems, but more broadly it is applying Math to solving practical problems (internally in Math; in other subjects; in real life) while metacognition has a great relationship, role and influence in detecting and solving practical problems The reason is that the "problems" in the internal subject of Mathematics are often in the form of questions and exercises 18 that are "purely mathematical" - quite close to knowledge - mathematical methods that students have learned, rarely encountered Difficulties require adjusting thinking direction, or re-evaluating the problem-solving process, etc Therefore, they can use their previous experiences to solve them without having to carry out "metacognitive" activities It can be said that the situation of applying mathematics in practice is a very good opportunity for students to practice using metacognition in solving various real-life problems, thereby developing mathematical competence - especially mathematical modeling ability and mathematical problem solving ability 3.2.3.3 How to implement the measure Based on the content of Mathematics in the textbook program, teachers select and collect relevant materials to build "problems with practical content" and use exercises for students to practice applying mathematical knowledge and methods known to solve Approaching from a problem-solving perspective, in the teaching situation of "applying mathematics to practice", it can be seen that students use cognitive and metacognitive activities to detect and solve the necessary "problems" This comprehensively affects mathematical competence, especially mathematical modeling ability and mathematical problem solving ability 3.3 Summary of Chapter The three proposed measures have exploited the advantages of metacognitive theory in combination with methods and techniques of teaching Mathematics to influence the components of mathematical competence that need to be developed for secondary school students through Mathematics In each measure, the author has clarified the scientific basis and method of implementation, shown through illustrative examples which are typical situations in mathematics learning at the secondary school level, including: systematization of knowledge; Teaching math exercises; Teaching the application of Mathematics in practice In the learning process, students have applied metacognition in math learning activities, and the initial research results show that students have completed tasks faster and have a deeper understanding of the problem The study found that there is a close relationship among the measures; therefore, it should be applied flexibly in the process of teaching Mathematics in secondary schools; synchronize with teaching methods and ensure the implementation of current Math objectives and content and the General Education Program in Mathematics 2018 19 CHAPTER PEDAGOGICAL EXPERIMENT In Chapter 4, the research is carried out in two phases In phase 1, the author first assessed whether the proposed measures in chapter are necessary and feasible with the current practice of teaching secondary-school Maths For this content, the author conducted a survey through questionnaires, consulted with educational experts and teachers of Maths at secondary schools In phase - pedagogical experiment, after having results from phase (then the measure can be modified), the study conducted a pedagogical experiment and used case study method to demonstrate most clearly the impact of the proposed measures on students' progress in mathematical competence 4.1 Assess the necessity and feasibility of the proposed measures 4.1.1 Evaluation methods The purpose is to collect information to evaluate the necessity and feasibility of the proposed measures to develop the mathematical competence of secondary school students; on that basis, adjust the inappropriate measures and confirm the reliability of the measures being evaluated Subjects of assessment: 74 participants responded to the questionnaire (including 12 educational experts, 62 secondary school Maths teachers) 4.1.2 Result of evaluation The measures proposed by the author are considered necessary and feasible In order to develop mathematical competence of secondary school students, according to the author, it is necessary to pay attention to the organization and implementation of measures; with each measure, there will be difficulties in implementation After surveying opinions from experts in mathematical education and teachers of secondary school Maths, the author received a consensus on the three proposed measures; There is no opinion on the need to adjust the measure as before the survey In the next step, the author conducted an experiment to continue to confirm the effectiveness of the measures in developing mathematical competence of secondary school students 4.2 Pedagogical experiment 4.2.1 Methods of pedagogical experiment 4.2.1.1 Purposes of pedagogical experiment The pedagogical experiment aims to assess the level of development of students' mathematical competence when fostered by the measures proposed in the thesis, hence testing the rationality of the scientific hypothesis and the ability to apply the research results of the thesis into practice 20 4.2.1.2 Participants and time of the pedagogical experiment We conducted a pedagogical experiment at Y La Lower-Secondary School To select teachers to participate in the pedagogical experiment, we base on the actual teaching and learning situation of teachers and students in some 9th-grade classes at Y La Lower-Secondary School, Tuyen Quang city The teachers involved in the experimental group and control group all have good and fair professional qualifications They also have good pedagogical skills and experience to meet the requirements of the pedagogical experiment tasks Based on the students' cognitive and metacognitive competencies, we chose students at the end of secondary schools in grades with the same number, cognitive level and starting knowledge Teachers in those classes have relatively similar competencies and qualifications To facilitate data collection and analysis, the author assigned labels to the experimental and control classes as follows: Table 4.3 Schools, classes, teachers and students participating in the experiment and the control groups Experimental class Control class School Number of Number of Class/Teacher Class/Teacher students students Y La Lower9B 9A Secondary 42 42 Luong Thi Kim Quyet Nong Thi Nga School 4.2.1.3 Organization of pedagogical experiment 4.2.1.4 Pedagogical experimental contents 4.2.1.5 Data collection and interpretation 4.2.2 Results of pedagogical experiment 4.2.2.1 Qualitative analysis Through the experiment process, we found that, in the pedagogical experimental class, there was a change in students' perception of metacognition, leading to a positive change in learning activities The students understood the problem better because they always tended to rethink and verify, so their doubts and questions were always looking for a satisfactory answer 4.2.2.2 Quantitative analysis Through the obtained data, it was found that, in the experimental class, the scores changed markedly across the assessments The research team once again affirmed that the teaching and learning methods being conducted in the experimental class are appropriate and effective for the development of students' Mathematical self-study competence 21 4.2.2.3 Analysis of the Pearson correlation coefficient (r) of the results of applying metacognition and the results of mathematical competence Based on the value of the Pearson Correlation coefficient (r) and the Sig coefficient, it is confirmed that the use of metacognition and mathematical competence have a close positive linear correlation If the use of metacognition is increased in the process of teaching and learning Maths, then, students' mathematical competence will be increased and vice versa, if the level of metacognition is reduced, it will lead to mathematical competence decreasing 4.3 Case study 4.3.1 Research Methods 4.3.1.1 Aims Using the case study method aims to assess the development of mathematical competence in students when teachers apply metacognitive theory to identify and organize cognitive and metacognitive activities for students in Maths classroom The research questions posed are: - In what activities students use metacognitive skills? - How metacognitive activities affect students' mathematical thinking and reasoning competence, mathematical problem-solving competence and other components of mathematical competence? 4.3.1.2 Research subjects During the case study, we observed how the metacognitive activities of 18 students engaged in maths were revealed when they faced with and needed to deal with less familiar situations 4.3.1.4 Methods of data collection and analysis The data obtained from these worksheets will be processed using conventional statistical tools to make judgments about the actual use of students' metacognitive skills in the process of solving mathematical problems 4.3.2 Research results 4.3.3 Findings For relatively familiar problems It can be seen that for familiar situations, students can solve them quickly by flexibly applying previously learned knowledge or seeking relevant knowledge from other support such as the Internet 22 However, it can be seen that for familiar situations, students will easily find a solution, therefore, the role of metacognition in guiding the process of solving problems of students has not been clearly demonstrated Impact of cognitive and metacognitive activities on mathematical competence During the process of students performing cognitive and metacognitive activities (corresponding to the 5-step process built in chapter 3, in which reading the problem and understanding the problem are integrated into step of the process) 4.4 Evaluation after the pedagogical experiment After the process of pedagogical experiment, including survey, pedagogical experiment and case study, the author found that the four measures proposed in the thesis have been effective in developing students' mathematical competence 4.5 Summary of chapter This chapter presented about the pedagogical experiment to test the feasibility and effectiveness of the proposed measure in applying metacognitive theory to develop mathematical competence for students in teaching Mathematics The results showed that the surveyed math education experts and teachers confirmed the necessity and feasibility of the three proposed measures In addition, there are some suggestions to help the implementation of measures in teaching practice achieve high efficiency From the results of the pedagogical experiment, it can be seen that the purpose of the pedagogical experiment has been completed; the proposed pedagogical measures have promoted the development of students' mathematical competence by applying metacognitive theory in teaching Mathematics at the secondary school level 23 CONCLUSION In the research process, the thesis has done theoretical research on metacognition and metacognition with student's learning activities Research results show that there are similarities with previous research results such as Flavell (1979), (Schneider and Artelt, 2010), Palincsar and Brown (1984), Schneider & Pressley (1997), Ghatala and research team (1986), or some works of Hoang Xuan Binh (2019), Le Binh Duong (2019), Hoang Thi Nga (2020), Phi Van Thuy (2021) However, regarding math learning activities of secondary school students in Vietnam, especially when general education is being implemented under the 2018 General Education Program, the research has shown achievements and shortcomings when applying metacognitive theory in teaching Mathematics Specifically, after studying the current situation of applying metacognitive theory into the development of mathematical competence for middle school students as shown in Chapter 2, the research results show that in the actual situation of applying metacognitive theory in teaching and learning activities of teachers and students, there are still many limitations Many teachers applied metacognition in the teaching process, and many students had metacognitive activities; however, the activities of applying metacognition by both teachers and students are not regular; there is no habit of using metacognition as a thinking tool Therefore, the students have not made significant progress in mathematical competence, and their learning results in Mathematics are low The thesis identified and clarified the role of metacognitive activities in the development of students' mathematical competence in the process of teaching Mathematics at the secondary school level Provided a table of cognitive and metacognitive manifestations of secondary school students in the five components of mathematical competence Analyzed the influence and role of metacognition in math learning and in the process of forming and developing students' mathematical competence Assessed the actual situation of applying metacognitive theory to teaching Mathematics at secondary school in the direction of developing mathematical competence for students Built a procedure of organizing cognitive and metacognitive activities in teaching Mathematics and proposed measures to apply metacognitive theory in teaching Mathematics at secondary school in the direction of 24 developing mathematical competence for students These measures are tested by the author for practicality and effectiveness through the process of experimentation Experimental research results have confirmed the necessity, feasibility and effectiveness for the process of teaching and learning Mathematics Thus, the thesis has answered the original research questions The results obtained in theory and practice allow to conclude that the research purpose has been achieved; the proposed scientific hypothesis is acceptable Further research directions: The application of metacognitive theory in teaching secondary school mathematics in the direction of developing mathematical competence for students has contributed to the formation and development of components of mathematical competence in particular and general competencies for students; at the same time improve the quality of teaching and promote the positiveness of students However, the topic has only stopped at the secondary school level It is suggested that further research works continue to concretize metacognitive theory to apply in teaching Mathematics at all levels, gades and classes in order to foster the necessary competencies for students THE AUTHOR'S RESEARCH WORKS RELATED TO THE THESIS Nguyen Thi Huong Lan (2019), “Metacognition in react teaching strategy” Annals Computer Science Series 17th Tome nd Fasc - 2019 pp 24-29 Nguyen Thi Huong Lan (2019), Research on solving mathematics proplems of secondary school students, HNUE JOURNAL OF SCIENCE, Educaitional Sciences, 2019, Volume 64, Issue 12, pp 184-190 Nguyen Thi Huong Lan (2020), Metacognitive Skills with Mathematical ProblemSolving of Secondary School Students in Vietnam - A Case Study, Universal Journal of Educational Research 8(12A): 7461-7478 Nguyen Thi Huong Lan (2022) Cognitive and metacognitive activities in the process of developing mathematical competence for secondary school students HNUE JOURNAL OF SCIENCE, Educaitional Sciences, 2022, Vol 67, Issue 3, pp 220-227, DOI: 10.18173/2354-1075.2022-0060 Bui Van Nghi, Nguyen Thi Huong Lan (2022) Applying Metacognition in Teaching and Learning in Math Lessons in Secondary School a Study in Tuyen Quang Province, Vietnam NeuroQuantology | July 2022 | Volume 20 | Issue | Page 462-469 | doi: 10.14704/nq.2022.20.8NQ 44052 ... locations: secondary schools in Tuyen Quang province (Ỷ La Secondary School; Phan Thiet Secondary School; Hung Thanh Secondary School, Le Quy Don Secondary School, Nong Tien Secondary School) Survey... metacognitive manifestations of secondary school students in components of mathematical competence Cognitive and metacognitive Cognitive and metacognitive manifestations in manifestations in mathematical... Maths in secondary schools is necessary For the above reasons, the researcher chose the research topic: “Applying metacognitive theory in teaching Mathematics in secondary schools in the direction