Phát triển năng lực trực giác toán học cho học sinh trong dạy học toán ở trường trung học phổ thông tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION VO XUAN MAI DEVELOPING MATHEMATICAL INTUITIVE COMPETENCE FOR STUDENTS IN TEACHING MATHEMATICS AT HIGH SCHOOL Major: Theory and Methodology of teaching Mathematics Code: 9.14.01.11 A SUMMARY OF DOCTORAL THESIS IN EDUCATIONAL SCIENCES HANOI – 2020 THE WORK WAS COMPLETED IN HANOI NATIONAL UNIVERSITY OF EDUCATION Science Instructors: Prof PhD DAO TAM PhD NGUYEN PHUONG CHI Reviewer 1: Assoc Prof PhD Nguyen Trieu Son Reviewer 2: Assoc Prof PhD Trinh Thanh Hai Reviewer 3: Assoc Prof PhD Tran Trung The thesis will be defended before the university council at Hanoi National University of Education, No 136 Xuan Thuy, Cau Giay, Hanoi At 2019 The thesis can be found at: - National library of Vietnam - The library of Hanoi National University of Education LIST OF SCIENTIFIC WORKS OF THE AUTHOR RELATED TO THE THESIS Vo Xuan Mai (2016), Applying imagination in teaching geometry to help students get knowledge, Vietnam Journal of Education, 382, pp 55-58 Dao Tam, Vo Xuan Mai (2016), Toward an understanding of intuition and its role in mathematics teaching, Vietnam Journal of Education, 389, pp 46-49 Dao Tam, Vo Xuan Mai (2016), Ensuring balance between intuitive and analytical thinking for student in mathematics teaching, Vietnam Journal of Educational Sciences, Vietnam national Institute of Educational Science, 114, pp 46-49 Nguyen Phuong Chi, Vo Xuan Mai (2017), Learning by intuiting – The way to solve unforeseen problems in mathematics education, Vietnam Journal of Science, Hanoi National University of Education, June 2017, pp 3-8 Vo Xuan Mai (2018), Building teaching situations with support of visualization to improve student’s mathematical intuition in problem solving, Vietnam Journal of Education, 431, pp.36-40 Vo Xuan Mai (2018), Developing students’ mathematical intuitive competence based on learning meaningful knowledge in mathematics education, Proceedings The first International Conference on Mathematics Education, An Integrated Approach in Mathematics Education and Teacher Training, Hanoi National University of Education, September 18-19th, 2018, Hanoi, Vietnam Vo Xuan Mai (2018), Developing students’ mathematical intuitive competence based on learning meaningful knowledge in mathematics education, Vietnam Journal of Education, No 5, December 2018, pp 88-93 Vo Xuan Mai (2019), The concept and some characteristics of student’s mathematical intuitive competence in teaching mathematics at high school, Vietnam Journal of Education, 448, pp.42-47 Vo Xuan Mai (2020), Organizing cognitive activity to develop intuitive thinking for students in teaching geometry at high school, Vietnam Journal of Education, 474, March 2020 PREFACE Reason for choosing the topic 1.1 Developing mathematical thinking competence is one of the essential requirements in teaching Mathematics at high school: Nowadays, education with oriented the development of individual learners’ competence has been become an essential requirement Mathematical problem-solving competence and creative competence are defined as specific competences that need to be created for students through mathematics teaching at high school In teaching, teachers could encourage learners how to discover and solve problems, how to think and gain creative thinking, skill training, ability to solve problems of real life through teaching knowledge The author Tran Kieu also emphasized “the need to pay attention to the capacity of logical thinking in inference, reasoning; at the same time attach to critical thinking, creativity, as well as predictive facts, mathematical intuition and spatial imagination” Therefore, developing mathematical thinking competence for students is one of the requirements that be recommonded in teaching Mathematics at high school 1.2 Identify the role of intuition and the research issues on intuition: The problem on intuition which has been studied extensively in the world, is concerned with its meaning, diverse role and manifestations Some authors claimed intuition as the source of innovation and is a first and necessary step for education, such as the works of the authors: Wild (1938), Poincaré (1958), Bunge (1962), Bruner (1965), Descartes and Spinoza (1967), Westcott (1968), DiSessa (1982) Some researchers argued that intuition can be trained and applied in the education process such as Tall and Vinner (1980), Fischbein (1987), Tieszen (1989), Jagla (1994), Hogarth (2001), Giardino (2010), Young Hoan Cho and Seo Yon Hong (2015) Although there are many studies on intuition and mathematical intuition in Vietnam about the concept in the material on the development of mathematical thinking for students, no researches on the application of mathematical intuition in mathematical teaching 1.3 The reality of teaching Mathematics has focused on deductive inference and formal reasoning, not interested in the development of mathematical intuitive competence: In mathematics teaching at high school, when students face unfamiliar situations or new problems, teachers often spend much time to provide students with procedural knowledge, most of them have a slight opportunity for trying to think, discovering ways to solve problems This led to many students only learn the formal knowledge, mainly using without knowing use of mathematical knowledge to solve problems in real life On the other hand, in mathematics teaching, most of the teacher's teaching methods and the presentation in the textbook content with theorems, rules, and proofs are often presented as are available products While intuition plays a significant role in the process of developing students' perceptions and can help them become more active and creative in making judgments, discovering new knowledge, finding strategy of problem-solving and making decision by themselves before embarking on specific manipulations Therefore, if students have mathematical intuitive competence, it will help them gain curious habits, such as exploring, thinking and finding ways to solve problems, connections between visualization with abstract It leads to them apply those ways in different situations or in unfamiliar contexts Thus, mathematical intuitive competence (MIC) is one of the important competences that needs training for students to know how to grasp knowledge, recognize problem solving strategies, toward to develop mathematical thinking For these reasons, we choose the topic “Developing mathematical intuitive competence for students in teaching Mathematics at high school” Purposes of the study Propose the procedure of organizing cognitive activities toward to develop students’ MIC and the ways of the developing each components through teaching Mathematics at high school, so thereby contributing to improve the effectiveness of teaching Mathematics Research tasks: The thesis finds the answers for the following questions: 3.1 What is students’ MIC? What are the characteristics of students’ MIC? What are the components of the MIC in learning mathematics? What opportunities are found to develop students’ MIC through teaching Mathematics at high school? 3.2 How is current teaching situation of the development of the MIC for students at some high school? 3.3 What are the steps in the procedure of organizing cognitive activities toward to develop students’ MIC in teaching Mathematics at high school? How to organize cognitive activities to develop each components of the MIC in teaching Mathematics at high school? 3.4 How to conduct to check the feasibility and appropriateness of the proposed procedure during the pedagogy experiment? Object and scope of the study 4.1 Object of the study: the teaching process toward to the developing students’ mathematical intuitive competence through teaching Mathematics at high school 4.2 Scope of the study: the content of teaching in the mathematics program in grades 10 and 11 and the teaching process to acquire those mathematical contents at high school Scientific theories: If the thesis determines students’ characteristics of the MIC and proposes the procedure toward to the organizing cognitive activities for students in accordance with the teaching practice at high school, students would gain mathematical knowledge more actively and creatively, and contribute to develop their MIC, so thereby meet the requirements of innovation to improve of teaching Mathematics in oriented developing learners’ competence Research methods: Theoretical study method, observation - investigation method, professional method, pedagogical experiment method and study - case method Contributions of the dissertation 7.1 In theory - Clarifying the students’ characteristics of the MIC and the components of students’ MIC in teaching Mathematics at high school - Proposing the procedure of the organizing cognitive activities toward to develop students’ MIC through teaching some mathematical contents at high school - Proposing the ways of the organizing cognitive activities toward to develop the components of the MIC through teaching Mathematics at high school 7.2 In practice - Proposing the procedure for teacher to organize cognitive activities developing students’ MIC through teaching Mathematics - Contributing to innovation of the mathematics teaching method toward to focusing on developing the MIC for students Theses that were protected - The characteristics and components of MIC in learning mathematics at high school - The procedure of the organizing cognitive activities toward to develop students’ MIC in teaching Mathematics at high school - The ways of the organizing cognitive activities toward to develop the components’ MIC through teaching Mathematics at high school Structure of the dissertation Chapter Theoretical foundation Chapter Practical foundation Chapter Organizing cognitive activities toward to develop students’ MIC in teaching Mathematics at high school Chapter Pedagogical experiment CHAPTER THEORETICAL FOUNDATION 1.1 An overview of relevant researchs 1.1.1 The researchs of intuition and mathematical intuition 1.1.1.1 Problem in intuition and mathematical intuition in the world In the world, the problem of intuition has been studied through works in many fields: philosophy, psychology, mathematics and education Some intuitive theory in the field of philosophy have representatives such as Kant, Hilbert and Bernays, Husserl, Godel, Parsons, Brouwer, Bergson In cognitive psychology, psychologists have devoted to studying the process of insight into things, immediate understanding of things and experiences "aha" after the time of solving problem failed The conception of intuition by mathematicians such as Poincaré, Descartes, Hadamard, Koliagin, Konmogrorop and in the field of education with some writers’ work on using intuition and mathematical intuition in the teaching process such as Tall and Vinner, Fischbein, Tieszen, Jagla, Hogarth, Giardino, Young Hoan Cho and Seo Yon Hong 1.1.1.2 Problem in intuition and mathematical intuition in domestic In Vietnam, some authors also refer to the concept of intuition in the literature on the development of mathematical thinking, mathematical competence such as Nguyen Van Loc, Pham Gia Duc and Pham Duc Quang, Tran Luan, Nguyen Phu Loc 1.1.1.3 Some observations from the results in mathematical intuition on domestic and foreign: From the researches in the world, we find that the problems of mathematical intuition are deeply studied with different aspects and has been widely used in many areas of mathematics and Mathematical education But the great roles and characteristics of mathematical intuition in teaching Mathematics in Vietnam is still untapped and has no many researches aimed to using in practice of teaching Mathematics at high school 1.1.2 The researches of organizing mathematical cognitive activities The researches on teaching some mathematical contents in high school towards positively activating cognitive activities such as the doctoral thesis of the authors Pham Minh Tien, Nguyen Manh Chung, Tran Trung; research on the process of organizing awareness activities for students through Dao Tam’s works were also interested in teaching Mathematics However, there is no specific research materials on organization of cognitive activities in order to develop the students’ MIC through teaching Mathematics at high school 1.1.3 The receptive results In short, the problem of developing students’ MIC in teaching Mathematics in Vietnam has not been exploited, so we absorb the following results: - Studying on the concept, factors related to the shape and development of the MIC - Studying the application of the MIC during the teaching process - Studying the process of organizing cognitive activities for students through teaching Mathematics at high school From these results, we will study to propose the concept and clarify some characteristics of the MIC, identify some components of the MIC towards building the process of organizing cognitive activities for students to develop the MIC through teaching Mathematics at high school 1.2 Students’ mathematical intuitive competence in learning mathematics at high school 1.2.1 Intuition, mathematical intuition 1.2.1.1 The concepts of intuition: Although there are many concepts in the different mentioned categories, but most of the concepts have the following two characteristics: “directly grasp knowledge quickly” and “do not require clear arguments” In the process of understanding and analyzing, we chose to approach the concept of intuition according to “cognition” Because, there is a very close relationship between awareness and thinking, awareness is also associated with activities to be effective in solving problems that have a positive impact on the process The learners' awareness towards organizing the cognitive activities to develop students’ capacity in the process of teaching in high school Therefore, we have acknowledged the approach of viewing intuition as human perception and carrying the two above characteristics Therefore, inheriting from the concept of author Piaget, we claim “Intuition can be understood as the immediate and implicit cognition of objects without analytical reasoning in the cognitive process” 1.2.1.2 Classification of intuition + According to Fischbein, based on origins of intuitions: primary intuition and secondary intuition Based on roles, classification of intuitions may be grouped into affirmatory intuitions, conjectual intuitions, anticipatory intuitions and conclusive intuitions + According to Piaget, intuition insist on empirical intuitions and operational intuitions + Bahm has mentioned three types of intuition: Objective intuitions, subjective intuitions and organic intuitions + According to Baylor, classification intuitions may be grouped immature intuition and mature intuition 1.2.1.2 Mathematical intuition: Mathematical intuition can be understood as the immediate and implicit cognition of mathematical objects, relations, and facts which occurs the reduction of analytical reasoning or without analytical reasoning in the cognitive process 1.2.2 Intuitive thinking in teaching mathematics According to author Koliagin, Intuitive thinking is one of the basic components of mathematical thinking, in which “Intuitive thinking is a special method of awareness, characterized by finding truth directly, related to the phenomena such as unexpected, lightning-fast problem solving, non-compliance with logical requirements, results obtained by this method very quickly” Nguyen Van Loc said that “Mathematical intuition is an element of a thinking method called Intuitive thinking, that is an type of thinking based on the perception of the whole problem immediately, capable of performing in the form of sudden change, shortened process of argument Therefore, Intuitive thinking is characterized by the absence of clear reasoning steps, this type of thinking is characterized by the perception collapsing quickly, immediately and helping them to have direct conclusions about the problem, completely bypassing the intermediary stage, is the leap forward thinking process, the person with creative deposit immediately responds to the question, the ability to visualize the outcome of an issue without absolutely a full process 1.2.3 Students’ mathematical intuitive competence 1.2.3.1 Competence and students’ competence in learning mathematics: Competence is a combination of unique attributes of an individual, including knowledge, skills and attitudes, in accordance with the requirements of a given activity, ensuring that the activity is effective Students’ competence in learning mathematics is the ability to mobilize their knowledge, skills, experiences and attitudes such as the interest, belief, willing to effectively implement activities in the process of learning mathematics 1.2.3.2 Thinking competence, mathematical competence in learning mathematics “Thinking competence is considered to be the level of applying intellectual activity, it is very diverse in using information to achieve results It includes factors such as problem solving, making decisions, critical thinking, developing arguments and using evidence to support their arguments It also is the core of many intellectual activities” According to the author Krutexki, “mathematical competence is understood as personal psychological characteristics (firstly the characteristics of intellectual activity) to meet the requirements of mathematical learning activities, and in the same solid conditions, it is the cause of the success in mastering mathematically creatively as a subject, especially mastering the knowledge and skills quickly, easily and deeply in the mathematical field” 1.2.3.3 Students’ mathematical intuitive competence in learning mathematics We understand that “Students’ mathematical intuitive competence is the thinking capacity who apply intellectual activities to realize the internal characteristics and attributes of mathematical objects, relationships and problems quickly in specific cognitive situations in the process of learning mathematics, on the basis of students performing the reduction of reasoning or without analytical reasoning in the cognitive process” Students’ MIC is related to thinking skills such as association, imagination, judgment and generalization It is considered as a combination of many mathematical thinking competencies, and is the result of the intellectual activity process smoothly without relying on clear explanations or strict sequences of deductive process Moreover, it also depends on the process of accumulating knowledge, experience and skills system along with attitudes, emotions and qualities of each individual learner When facing the same problem, for this student, the possibility of generalized learning can occur quickly but other students must take place more slowly Therefore, in teaching Mathematics, teachers should focus on training mathematical thinking skills to create a foundation for them a) Deduction and imagination: According to Nguyen Van Loc, “In geometric study, geometric intuition is understood as imaginary competence such as imaging shapes and their transformations” MIC is represented by imaginary abilities such as imaging shapes and properties, seeing changes, relationships between objects, it is also the ability to visualize the way of the solving problem, force see the result of the problem In mathematics, mathematical intuition is expressed in the ability to visualize the transformations of mental expressions by shortening some of the transformational steps without fully realizing it It is possible to visualize the mathematical knowledge that is being considered with the actual model or the ability to visualize the problem solving approach before proceeding b) Association: Association plays a special role in the process of developing knowledge cognition as well as the process of developing students' thinking Without the ability to associate and mobilize knowledge, there is no intuition, so it will be poor in ideas and ability to solve problems is also limited However, to associate and to mobilize knowledge effectively as well as to have a valuable intuitive basis, there must be reflexive association Furthermore, learners must have the capacity to associate quickly and precisely to facilitate effective intuition The building this capacity contributes to developing, expanding knowledge and fostering exploration to discover new knowledge for students on the basis of previous experience c) Judgement and generalization: Mathematical intuition is the direct realization of the mathematical object and relationship in the process of cognition without the need for detailed steps and justification, perceiver can make judgments on mathematical object and relations through thinking abilities such as generalization, inductive inference without the need for rigorous sequencing of the inference process, in some cases, a student can give predictions about the nature, predicting the outcome of the problem 1.2.4 The roles of mathematical intuition in teaching Mathematics Intuition plays a decisive role not only in the realization of discoveries but also in scientific activity and in education Wilder mentioned “Intuition plays a basic and indispensable role in mathematical research and in modern teaching methods”, to provide a conceptual foundation that suggests the directions Burton said “The research mathematicians provided clear directions towards a mathematical pedagogy that could be as engaging, exciting and rewarding for learners, as it was for them in their own practices These directions were firstly towards valuing and nurturing intuitions and also recognising the importance of making connections or links in the building of mathematical meaning” In our opinion, the role of mathematical intuition for modern approachs in mathematics education is as follows: using mathematical intuition as a way for making predictions or judgments, scientific hypotheses for the detection of mathematical problem, guiding the way of mathematical problem solving and clarifying the meaning of mathematical knowledge 1.2.5 Some students’ characteristics of mathematical intuitive competence in teaching Mathematics at high school 1.2.5.1 Characteristic 1: Characterized by the immediately direct cognition of mathematical objects, relations and problems: MIC is characterized by direct perception of mathematical objects, relationships and problems, which help students perceive quickly and immediately the intrinsic properties of mathematical objects, relationships and problems in the process of learning knowledge Mathematical intuition as the glow in the problem solving process, so students’ mathematical intuition competence expressed that they can grasp and deal with problems or see the results of problems immediately 1.2.5.2 Characteristic 2: Characterized by a shortened process of argument or without using the full logic steps: MIC is the capacity which the perceiver in the aware process of mathematical object and relations without the need for rigorous sequencing of the inference process, in some cases seeing the result of the problem that can not be explained The MIC’s rapid cognition of the mathematical object, relationship and problems is due to intermediate steps in the process of reasoning, interpretation and analysis have been reduced and shortened 1.2.5.3 Characteristic 3: Characterized by the ability to activate associations and mobilize knowledge flexibly through the process of accumulation, reflection in observation, previous problem solving: The other characteristic of students’ MIC is the product of the process of accumulation, reflection in observation, research and problem solving, with the flexibility of knowledge, the speed of association It demonstrates at the higher level of cognition such as the rise of cognitive processes and the deepening of intellectual thinking 1.2.5.4 Characteristic 4: Characterized by creativity, breakthrough in making ideas 10 questions and requirements, students are discovered problems and judgments about the way to solve and described about perceiving problems immediately before taking reasoning steps Teachers’ encouragement helps students to make different judgments for the problem with many solutions, different aspects of the problem, finding breakthrough and creative ideas from learners Sometimes teachers have to accept their naive, wrong ideas (3) Focusing on developing students with pre-logical thinking skills in mathematics: through activities, students are able to use imagination, association, generalization and inductive reasoning (4) Shaping a habit for students to grasp the nature of the problem and the solutions, to ignore detailed reasoning and intermediate steps: Teachers should instruct to students how to present thinking outline, how to describe the problems’ structure through their reasoning reduction, and Deepen the meaning, nature of mathematical knowledge for students (5) Emphasising the intuitive activities implemented before the deductive activities: In teaching Mathematics, teachers focus on mathematical intuition as the goal that needs to be realized first to orient the problem-solving strategy, while logical and deductive reasoning as the means to examine the results of intuition 1.4.3 Opportunities to develop mathematical intuitive competence for students in teaching Mathematics at high school In studying mathematics, the content and basic learning activities of students are linked together, aiming at forming their competence Students' MIC in teaching Mathematics is demonstrated in the ability to link the content of knowledge of mathematics through learning activities to visualize the problem, grasp the problem, judge the problem solving situation, transport situation applied mathematical knowledge From the above requirement, we found that the following characteristics need to be considered in order to find opportunities to develop MIC for students through teaching Mathematics at high school: - Opportunities to develop competence of immediate generalization and competence of judgment and decision making are expressed through organizing for students to consider individual and specific events in the receiving situations be aware of the problem, know how to generalize the problem from individual cases, know how to generalize from the general known to the general unknown; able to judge, detect the general rules of events in math content learning, be able to offer problem solving orientations, make hypotheses and predictions for math problems, then choose choose appropriate decisions on problem solving strategies - Opportunities for developing competence of association and visualization knowledge is shown through students' ability to visualize and relate math problems with existing knowledge and experience , know how to connect and visualize visual images or detect compatible models of abstract math problems, situations that require learners to discover knowledge and solutions related to problems new, the ability to imagine their images and properties, visualize the problem solving path in the face of new situations - Opportunities for developing the competence of the reduction of reasoning when students learn to reason and estimate quickly, have the ability to briefly discuss how to change 11 the problem, reduce intermediate steps in problem solving process,visualize deductive diagrams to see the problem solving approach CHAPTER PRACTICAL FOUNDATION 2.1 The purpose and object of the survey 2.1.1 The purpose of the survey - Research the awareness of maths teachers at high school about the concepts and some activities of mathematical intuition, the elementary components of MIC, the levels of aware and application intuitive activities through teaching Mathematics at high school - Find the ability of using mathematical intuition of students in the learning mathematics at high school - Find out some else activities or competences expressed such as the elementary components of MIC through teaching Mathematics at high school - Research and analyze the advantages and disadvantages of teachers’ activities in the process of teaching students access to knowledge in the development of MIC 2.1.2 The object of the survey: 98 mathematics teachers and 142 students grade 10 at some high school in Dong Thap province 2.2 The content and organization of the survey 2.2.1 The content of the survey - Through observation, interview: observe some maths lessons at high school - Through teacher survey: design survey forms for teachers - Through students’ questionnaire: design mathematical exercises for students grade 10 2.2.2 The organization of the survey - Design survey forms for maths teachers - Design math homework exercises for grade 10 students - Time of survey fom October 10th 2017 to November 12th 2017 - Collect questionnaires, summarize and analyze data - Conduct qualitative and quantitative assessments Draw initial conclusions about the development of teacher and teacher's MIC after the survey 2.3 Result of the survey 2.3.1 Result of the survey: Through observation of some lessons, survey forms for maths teachers and results students’ questionnaire 2.3.2 Analyzing result of the survey  Advantages: The majority of teachers have more than 10 years of teaching experience and some of them have postgraduate qualifications with approaching new research directions Teachers have not only suitable contact in teaching but also have effective problem solving and class organization skills  Limitations: Teachers are still applying in traditional ways They are familiar with introducing and presenting mathematical knowledge without paying attention to explaining the meaning to help students understand the nature of mathematical knowledge They are not 12 required to motivate their students to find solutions, instead of, they usually provide immediate solution, indicating the step-by-step procedure for new mathematical problem Many exercises only require the use of formulas, skills of calculation and manipulation There are lacks of questions and exercises visual observation training and prediction detection, quick reasoning skills, not yet fully exploited the situation can develop the ability out of school  Reason: - For teachers: Teachers are influenced by traditional teaching methods, so they often teach with their psychology of attempt in short-time to convey more knowledge for students Especially in teaching assignments, they only focus on solve as many exercises with applying the available procedure as possible The selection of content to develop pre-logic thinking, especially intuition and imagination is still difficult and not interested properly - For students: Students have generally quite low knowledge level, so they are also afraid of differences and making mistakes When facing to solve the new problem, their study habits is expected on solutions or direct instructions from their teacher Most students still think in the stereotypes way, learning by memorized knowledge - In terms of teaching materials: The literature on mathematical intuition and its application to the teaching process in Vietnam is not much There are no research to show how to implement teaching for the development students’ MIC at high school CHAPTER ORGANIZING COGNITIVE ACTIVITIES FOR DEVELOPING STUDENTS’ MATHEMATICAL INTUITIVE COMPETENCE IN TEACHING MATHEMATICS AT HIGH SCHOOL 3.1 Orientations for the organizing cognitive activities for developing students' mathematical intuitive competence in teaching Mathematics at high school: Orientation in contents, teaching methods, orientation with teacher and learner 3.2 The procedure of organizing cognitive activities for developing students’ mathematical intuitive competence in teaching Mathematics at high school 3.2.1 Science foundation of proposing the organizing cognitive activities for student: Philosophical foundation, Piaget’s Theory and Vygotxki’s Theory, orientations of innovation in Program education Mathematics 2018 3.2.2 The procedure of organizing cognitive activities for developing students’ mathematical intuitive competence in teaching Mathematics at high school: including the following steps: (1) Creating cognitive situations contained new knowledge, (2) Organizing for imagine the ideas and conjecture the way to solve the problem, (3) Using deductive inference to test the results of intuition, (4) Drawing conclusions about new knowledge, (5) Selecting apply to new situations Teacher’s activities Targeting, selecting, motivational learning The procedure of organizing cognitive activities for developing students’ MIC Step Student’s activities Identify problem space, associate and mobilize knowledge 13 Identify students’ problem Selecting approciate teaching methods Organizing cognitive activities Guide to use the manipulation of inductive Creating cognitive situations contained new knowledge Step Organizing for imagine the ideas and conjecture the way to solve the problem Step Using deductive inference to test the results of intuition Step Assert new knowledge Assess students’ the use of new knowledge Drawing conclusions about new knowledge Step Selecting apply to new situations Conduct corresponding intellectual activities Detect problems, appear oriented solutions Fail Prove, explain the problem Implement the solution Find new knowledge and way to solve problem Apply to solve in the new situation 3.3 Some ways of organizing cognitive activities for developing students’ the components of MIC in teaching Mathematics at high school 3.3.1 Some ways of organizing cognitive activities for developing students’ competence of association and imagination problem in teaching Mathematics at high school 3.3.1.1 Purpose of organization: The organization of training activities to build association and imagination in order to build a solid foundation to create opportunities for the appearance of the MIC when the subject realizes before a specific consideration problem In new knowledge discoveries, the practice for students to relate to ideas and mobilize knowledge quickly and effectively helps students have good and accurate intuition from which to form and develop MIC in teaching Mathematics 3.3.1.2 Some methods to train competence of association and imagination for students in teaching Mathematics at high school - Exploiting similar mathematical properties and objects to help students quickly identify associate related to the problem - Using learning situations to help students develop spatial imagination in order to detect associations suitable to the problem quickly 14 - Exploiting the cause-effect relationship of mathematical knowledge, the relationship between mathematical knowledge helps visualize the way of solving mathematical problems - Using the learning situation for students to transform associations from one object to another to help them appear ideas and discover new knowledge - Developing situations contained a given visual image to intuitively discover the nature of a mathematical problem or imagine a way to solve a mathematical problem - Developing situations to help students visualize the visual image of the problem, thereby finding a way to solve mathematical problems 3.3.1.3 Illustrative examples: Example 3.4 Tthrough teaching the problem solving: “Give A(-1; 2; 3), B(3; 0; -1), C(1; 4; 7) the plane (P) x – 2y  2z   Find the coordinates of point M of (P) such that T  MA2  MB  MC has the smallest value” 1) Step 1: Create cognitive situation contained new knowledge: The problem of determining the coordinates of a point on a plane satisfies the given condition, which is quite familiar to students, but the difficulty of expressing a given minimum value is quite complex 2) Step 2: Imagine the idea and conjecture the way to solve the problem + Students can visualize the position of the point M on the (P) plane relatively close to A, B, and C so that the small-valued of T, since the point M on (P) goes away from the points A, B, C the longer the distance from M to those points, the greater the value of T + With the three given points A, B and C, the middle point G of triangle ABC are fully defined Observation of the expression has given students the ability to associate to MA  MB  MC  3MG Thus, the expression contained G must be transformed into a simpler expression that defines the point M + Students associate and mobilize the familiar problem solving method “Let point A not belong to the plane (P) Find the point M on (P) for the smallest MA2” Find the point M on (P) such that the smallest MA2  M on (P) such that the MA has smallest value  M is the projection of A on the plane (P) + Students discover intuitively problem that the point M needs to find can be the projection of point G of the triangle ABC on the plane (P) Thus, the solution of the problem 3) Step 3: Use deductive inference to test the results of intuition 2 2 2  T  MA  MB  MC  3MG  GA  GB  GC  2MG GA  GB  GC  With G(1;2;3) is the center of the triangle ABC We have T  MA2  MB  MC minimum  MG minimum  M is the projection of the point G on (P) Finding the projection M of A on (P), we get M  0;4;1 MG  d (G,(P))  thus Tmin  3.32  48  75 15 4) Step 4: Draw conclusions about new knowledge and apply to new situations: Students develop method knowledge by associating the familiar rules to solve the problem Teachers can train students with the ability to associate students with a view to solving problems in a variety of perspectives on problem solving, the ability to develop new problems from the original problem by associating the object with the other object 3.3.2 Some ways of organizing cognitive activities for developing students’ competence of conjecture and decision making in teaching Mathematics at high school 3.3.2.1 Purpose of organization: The organization of cognitive activities to develop the competence of judgment and making decisions to help students can immediately make predictions about the nature of mathematical problems, to judge right away the solution or the results of the problem, it leads to select their appropriate decisions for understanding and grasping the problem, and the ways to solve problems, contributing to the shape and development of the MIC for students 3.3.2.2 Some methods to train competence of judgment and making decisions for students through teaching Mathematics at high school - Using situations for students to conduct intellectual activities such as comparison, similar, generalization, specialization to judge hypotheses and make decisions - Organizing situations to help students use different kinds of reasoning to judge hypotheses, ways to solve problems and make appropriate decisions - Using situations for students to visualize visual images of problems to make judgments about how to solve problems - Creating opportunities for students to briefly use reasoning arguments to quickly make effective judgments and decision choices 3.3.2.3 Illustrative examples: Example 3.7 Through teaching of solving the equation: x    x  x  6x  11 1) Step 1: Create cognitive situation contained new knowledge: Obviously, it is not possible to use equivalent transformations to solve the above equation because it will raise the exponent significantly, therefore the problem becomes more complex 2) Step 2: Imagine the idea and conjecture the way to solve the problem + Judgment 1: Students have comments on the right side of the equation is always greater or equal From which the student makes the judgment by evaluating the two sides of the equation + Judgment 2: Observe the formulas in the radical in the left side of the equation It is obviously related to the expression of the right side Therefore, students can make judgments by using the sub-set method - Make the appropriate decision for the choice of how to solve the problem + For judgment 1: Find x    x  k , with constanst k by using Bunhiacopxki inequality 16 + For judgment 2: The ability to hide subtracts the solving of the old equation solving equation solvers can be solved + For judgment 1: Find out VP  x  6x  11  (x  3)2   Students can discover x    x in the equation, therefore they can associate to use Bunhiacopxki inequality + For judgment 2: Intuition with using the sub-set method We have u  x  2, v   x , u, v  Therefore VP  u 2v  So, they can u  v  solve the system of equations:  with u  0, v  2 u  v   u v   3) Step 3: Use deductive inference to test the results of intuition + For judgment 1: Condition of examined equation  x  Ta có VP  (x  3)2   (1) Using Bunhiacopxki inequality, we have:  x    x   1   x    x    VT  2 x    x  (2)  x  Therefore   x  x 2  4x   + For judgment 2: Put u  x  2, v   x , u, v  We have system of equations 2  u  v  (I) with u  0, v   2 u  v   u v    S  P  (1) Put S  u  v, P  uv we have  Thus (P  1)(P  P  5P  7)  (*)  S  2P  (2)   Solution of the system of equations is u  1, v  so solution of the equation is x  4) Step 4: Draw conclusions about new method knowledge and apply to new situations Students form method knowledge through solving problems of complex equations 3.3.3 Some ways of organizing cognitive activities for developing students’ competence of immediate generalization in teaching Mathematics at high school 3.3.3.1 Organizational purpose: Intuitive activity immediately recognizes the problem, the mathematical relationship or the discovery of the nature of the subject matter's mathematical problem, obtained through the rigorous process, inductive deductive in a way quick to catch the problem In teaching Mathematics, teachers need to organize training for students in awareness activities towards helping to observe, compare similar factors, individual things and analysis to detect things in specific phenomena, using inductive reasoning to detect the rule of the problem, capture the important, the nature of the problem 3.3.3.2 Some methods to train competence of immediate generalization for students in 17 teaching Mathematics at high school - Creating situations stemming from specific cases, individual events of mathematical subjects for students to quickly implement the problem - Developing situations using inductive reasoning so that students can discover mathematical rules and nature of the problem - Focusing on grasping the nature of the problem, visualizing the core elements in the problems through students' deep understanding of knowledge, understanding the meaning of the problem to create a foundation for the quickly effective implementation of the MIC - Organizing situations to help students discover the content and meaning of knowledge in accordance with the given form, to create a harmonious balance between syntax and semantics of mathematical knowledge in teaching Mathematics 3.3.3.3 Illustrative examples: Example 3.9 After the student finishes the lesson “Rotation”, the instructor gives the following problem “Let acute triangle ABC Outside of the triangle make equilateral trial ABM , BCN ,CAP satisfied AB  AM , NB  NC , AC  AP Prove that AN  BP  CM ” Solve the problem after that comment and generalize the problem 1) Step 1: Create cognitive situation contained new knowledge: choosing the appropriate transformation to solve problems makes difficult for students 2) Step 2: Imagine the idea and conjecture the way to solve the problem - Determine the specific elements of the problem to discover the suitable transformation used to solve the problem How does this relate to the appropriate displacement? - Problem solving activity: The activity of grasping and visualizing the nature of the problem by using inductive reasoning quickly Teachers can organize activities through the following questions: + What condition of the problem can we use the means of rotation to solve? + Can still use the rotation to solve the problem when changing the condition? + Using inductive inference detects similar or generalized factors to see new problems? + Proposing new problems can be replaced by similar properties - The activity uses inductive inference quickly to detect the rule, the mathematical nature of the problem: replace the triangular construction outside the triangle has given shape to form the result to be unchanged? - Students intuitively discover the nature of the rotation from which to derive alternative shapes satisfying the requirement: Problem 1: Give acute triangle ABC Outside of the triangle make equilateral triangles ABM , BCN ,CAP Prove that AN  BP  CM Problem 2: Give acute triangle ABC Outside of the triangle make isosceles triangles ABM , BCN ,CAP satisfied BAM  CAP  BNC   Prove that AN  BP  CM 3) Step 3: Use deductive inference to test the results of intuition Solve two new problems by using the same solution 3.3.4 Organizing cognitive activities for developing students’ competence of the 18 reduction of reasoning in teaching Mathematics at high school 3.3.4.1 Purpose of organization: competence of the reduction of reasoning aims to image a solution for new problems or visualize the results of unfamiliar problems, thereby helping subjects get immediate math problems Therefore, students need to have a tendency to think quickly and briefly about the proof or tend to ignore the steps of analyzing logical reasoning, students must have the ability to quickly grasp the nature and go deep into problem, simplify the stage of intermediate reasoning, not focusing on long-term form changes to easily visualize the outcome or problem solving path 3.3.4.2 Some methods to train competence of the reasoning reduction for students in teaching Mathematics at high school - Training students to change quickly and briefly, and omit intermediate arguments in the process of familiar problem solving - Organizing for students to visualize the general reasoning diagram or visualize the results of the problem by reducing the analysis of arguments - Developing situations to help students quickly transform the form of mathematical knowledge into different forms to realize the content of mathematical problems - Emphasizing the meaning of mathematical knowledge, focusing on semantic interpretation of formal expressions 3.3.4.3 Illustrative examples: Example 3.13 Solve problem: “Give positive real numbers  a, b, c satisfied a  b  c  Find the smallest value of P      1 1        ” a  b  c  1) Step 1: Create cognitive situation contained new knowledge: The finding the smallest value of an expression that contains the factors satisfying the given condition makes difficult for students to use appropriate solutions to estimate the expression 2) Step 2: Imagine the idea and conjecture the way to solve the problem - The initial intuition of the expression and visualize the possible result of the problem: positive real numbers satisfied a  b  c  , thus, students can conjecture that a  b  c  so P  64 may be the smallest value Students can associate to Cauchy inequality Rapid change of mathematical problems by eliminating the steps of reasoning, intermediate analysis, students can occur some ideas in their minds as follows: + Express conversion expressions P   + Using Cauchy inequality for 1 1 1       a b c ab bc ca abc 1 1 1 , , , , a b c ab bc ca Pmin  64 3) Step 3: Use deductive inference to test the results of intuition + Visualize the results: a  b  c  19 Detailed explanation of the steps of reasoning and analysis 4) Step 4: Draw conclusions about new method knowledge and apply to new situations: By using the way of solution, the general problem can be generalized CHAPTER PEDAGOGICAL EXPERIMENT 4.1 The purposes and tasks of experiment 4.1.1 The purposes of experiment Pedagogical experiment is conducted for the purpose of testing the proposed scientific hypothesis, thereby initially confirming the feasibility and effectiveness of the process of organizing cognitive activities to develop the MIC for students through teaching Mathematics in high school and ways to train each component element of the MIC with some corresponding activities Specifically: - Test the rationality of the process of organizing cognitive activities towards developing general capacity for students through teaching Mathematics at high school - Test students' progress in using a number of activities corresponding to each of the component competencies of general students through teaching Mathematics at high school 4.1.2 The tasks of experiment - Design teaching projects toward to develop the MIC in the lessons - Organize teaching activities according to the designed teaching plans - Observe the process of teaching in the class of students when performing the task of teaching according to the process of developing the MIC for students - Design experimental round and - Analyze empirical experiments round and - Comment and evaluate the results of the experiment, student data processing - Evalue results of experiment 4.2 Time and objects of experiment 4.2.1 Time of experiment: Round 1: from November 15th 2017 to December 28th 2017 and Round 2: from March 14th 2018 to April 28th 2018 4.2.2 The objects of experiment - Classes in experiment: Class 11CB8, Thong Linh high school, Cao Lanh district and Class 11CB3, Thien Ho Duong high school, Cao Lanh city - Teachers in experiment: Nguyen Van Dop, Mathematic teacher in Thong Linh high school and Nguyen Thi Minh Tam, Mathematic teacher in Thien Ho Duong high school 4.3 Process of experiment: The process of experiment inclucde in steps: preparation step, experimental step and experimental assessment step 4.4 Content of experiment: Experimental content was designed to test some activities proposed components of MIC by the procedure of the organizing cognitive activities with teaching plans and experimental contents for student grade 11 through two rounds: 20 4.4.1 Experiment in round 4.4.1.1 On experimental teaching in teaching project + Geometry grade 11: “Review Isometry” (2 periods) + Algebra and Analysis grade 11: “Sequence of numbers” (2 periods) + Special subject: Using visualization in support mathematical problem solving (2 periods) after finish lesson “Geometric progression” 4.4.1.2 The activities of the components of MIC are practiced through the lessons: In experimental teaching in teaching project round 1, we tend to test the organizing cognitive activities for developing the components of the MIC such as competence of generalization, competence of linking between visualization and astract, and competence of judgment and decision making with the procedure in diagram 3.1 4.4.1.3 Content of experiment in round 1: Content insists on exercises • Analyze the intended use of compatible activities with the level of expression of the components - In Question 1, we aim to consider the possibility of using activity to visualize visual images of mathematical problems (mathematical associations and visualizing competence) and activities to make judgments for familiar situations belonging to (judgment and decision making competence), specifically students must visualize or imagine to recognize the corresponding model of the problem from which to deduce possible cases of the area of a tetrahedron with a any plane is a polygon that does not need to be drawn - In Question 2, we aim to consider the possibility of using activity to detect general and general factors of the object from the consideration of individual cases (quickly generalization competence) Students must recognize and detect factors with equal sides and the alternating angle between inertia construct an isosceles triangle and use rotation - In Question 3, we aim to consider the possibility of conducting activities using visual connection of mathematical problems to detect and solve problems (mathematical associations and visualizing competence) Students must know how to use visual images to show the results without going through the calculation and reasoning steps, students have the opportunity to show an understanding of the nature of the problem through imaging when observed and then re-test the results by applying the formula • Analyze pre-test in experiment round 1: We anticipate the results of the student's work before organizing the experiment as follows: The questions in the empirical part not focus on lengthy analysis, rigorous arguments because these are questions with short answers So in a short period of time, students need to immediately comment, immediately imagine the problem posed to solve Through quick questions with pedagogical intentions directed to cognitive activities organized to develop some components of the MIC, students must conduct reflection, visualization and judgment to intuitively detect the problem, make comments about the process and then show the results after visualizating process 4.4.1.4 Experimental Results Round 21 • Assess the results of the student's work • Assess through observation in the classroom: We observed that, students have to spend time reflecting on the problem and visualizing the results before writing a clear explanation or answer on the answer sheet Some of them have the ability to use illustrations, some sketchy drawings in addition to scratch paper Almost every time they are asked, they can come up with some results for questions through their initial comments on the problem and results • Interview students: In order to clarify some information that is difficult to determine the connection between visualization and abstraction, the ability to make judgments in students' results, we study conducting interviews, exchanging with some students Tram, An and Thanh • Assess through post-test in experiment round 1: Some student's activities are manifested in the first round experiment: The activity visualizes the visual image of the mathematical problem, The activity uses visual connection of mathematical problems to detect and solve problems (competence of association and visualization) The activity makes judgments for familiar situations (competence of judgment and making decision) The activity of detecting general factors of the object from the consideration of individual cases (competence of immediate generalization) 4.4.2 Experiment in round 4.4.2.1 On experimental teaching in teaching project + Geometry grade 11: “Straight line perpendicular to the plane” (2 periods) + Geometry grade 11: “Distance” (3 periods) + Algebra and Analysis grade 11: “Definition of Derivatives” (3 periods) 4.4.2.2 The activities of the components of MIC are practiced through the lessons: In experimental teaching in teaching project round 1, we tend to test the organizing cognitive activities for developing the components of the MIC such as competence of association and mobilization knowledge, competence of linking between visualization and astract, competence of the reduction of the reasoning, and competence of judgment and decision making with the proposed procedure in diagram 3.1 4.4.2.3 Content of experiment in round 2: Content insists on exercises • Analyze the intended use of compatible activities with the components of MIC - In Question 1, we aim to comment on the possibility of abolishing the intermediate argument steps in the process of solving familiar problems or using associations to give some make judgments and select decisions on how to solve known problems by calculating the distance between two diagonal lines in space when students have known a number of methods solution - In Question 2, we aim to consider the expression of the activity of transforming the form of mathematical knowledge into another form to realize the content quickly and inculcate the meaning of knowledge for students - In Question 3, we aim to examine the manifestation of the activity of discovering the 22 rule, the nature of the problem and explaining the meaning of the problem, recognizing the given number sequence can be performed in other forms to help students better understand knowledge and the ability to use visual images to detect students' math problems - In Question 4, we aim to identify some activities of understanding and explaining the meaning, understanding the geometric meaning of the given theorem and performing it in another form thereby helping students capture and understand deeply knowledge • Analyze pre-test in experiment round 2: We design experimental content with exercises that require students to solve problems quickly, shorten argument steps to infer results or ways of solving problems, students need to know how to use images to contact, discover and represent problems, represent mathematical contents in various forms; understand the meaning of mathematical knowledge in the form of visual and geometric languages 4.4.2.4 Experimental Results Round 2: • Assess the results of the student's work • Assess through observation in the classroom: Observing students' class activities and notebooks, we found that students have the ability to think and use illustrated images to produce results for questions They know how to apply the known method in specific situations, represent mathematical content in many different forms Some average, weak students are also able to make initial judgments with new ideas when participating in speeches • Interview students group (Tram, An, Thanh) • Assess through post-test in experiment round 2: Some student's activities are expressed in this experiment: - The activities of visualizing images or models of mathematical problems, the activities of using visual images to detect problems, the activities of finding and coordinating associations about the relationship between events and achievements knowledge knowledge to solve problems (competence of association and visualization) The activity of discovering the meaning of mathematical knowledge, the activity quickly recognizes the content of mathematical knowledge through different forms of representation (competence of immediate generalization) The activity of eliminating the intermediate argument steps in the process of solving familiar problems that have known solutions The activity of using intellectual manipulation for making judgments (competence of judgment and making decision) Summary of the student group’s response level through some activities of each component of the MIC with levels (not good, average, good) Students’ response level The Some activities of each components Student Student Student component of the MIC Tram An Thanh The activity of association and Not good Average Good Competence transformation of associations of association The activity of visualizing images or Good Good Good and models of mathematical problems visualization HĐ hình dung đầu hình Good Good Average tính chất chúng 23 The activity of detecting general factors from the consideration of Average Average Good individual cases The activity of discovering the Competence Average Average meaning of mathematical Average of immediate knowledge generalization The activity quickly recognizes the content of mathematical knowledge Average Average Good through different forms of representation Competence The activity of using intellectual Good Good Good of judgment manipulation for making judgments and making The activity of using visual Good Good Good decision situations for making judgments The activity of eliminating the intermediate argument steps in the Good Good Good process of solving familiar problems Competence with known solutions of the The activity of seeing changes and Average Average reasoning relationships between mathematical Average reduction objects The activity of eliminating the Average Average argument steps to image on thinking Good outline or the result of the problem 4.5 Results of experiment 4.5.1 Qualitative assessment: The levels of expression of the component of students’ MIC through the results of two experiments, we achieve through following chart: The levels of expression of component competences of students’ MIC through round and 60 40 20 Level Level Round Level Level Round 4.5.2 Qualitative evaluation 4.5.2.1 Through observation students’ expressions of activities and attitude: students have many opportunities to participate in activities such as expressing their ideas, describing and explaining their problem, discussing together to find the problem, and finding a solution with "flash" the idea 4.5.2.2 Through interview and case study: Through interviewing some students about teaching steps in the procedure, most of students said they have more time to think, discuss 24 and clarify the problem They feel more active with many activities such as a predicting, generalizing, finding, exploring, and describing the solved way for problems, especially for non-exemplary problems and unfamiliar situations Sometimes they can discover new creative ideas that are not stereotyped when solving problems Although the assessment of a certain competence is a long-term process that can be verified, but through the experiment process above, we can see that the students have made progress on their new approach to knowledge, know how to understand the meaning of mathematical problems in learning, grasp knowledge and know how to find strategies to solve problems before carrying out specific analysis activities This is shown through some manifestations of cognitive activities of the activities compatible with the component competencies of the MIC as detailed in the above sections CONCLUSION Through research process, the dissertation has achieved the following major results: The dissertation has clarified the concepts of intuition, mathematical intuition , proposed the concept of students’ MIC, and determined four students’ characteristics of MIC, and proposed four the components of MIC in teaching Mathematics at high school The dissertation has studied the reality, and analyzed the advantages and disadvantages through conveying the current situation of teaching Mathematics toward to the development of the MIC with teachers and students The dissertation proposed the procedure of organizing cognitive activities toward to develop students’ MIC for and proposed some ways of the organizing cognitive activities to develop each components of the MIC with some mathematical contents at high school Initial testing of the feasibility of the procedure of organizing cognitive activities toward to develop the MIC by the implementation of experimental pratice Some research results of the dissertation have been published in science journals and on proceedings the International Conference on Mathematics Education The results of this thesis allow to draw the following conclusions: Mathematical intuition also has an active role in the development of pre-logic thinking, the discovery and creation of new knowledge, thereby contributing to improving the quality in mathematics education The organizing cognitive activities for the development of the MIC help students achieve another form of mathematics with prediction, exploration, intuition and creativity beside the form of the deduction with rigorous rules of reasoning However, in teaching Mathematics, it is recommended teachers to organize awareness activities for students so as to balance between intuition and deduction to help students know how to reasonably use presentation ability, reasoning and ability of judgment and making appropriate decisions, intuitively deduce and experience more in solving problems in a new context The results from theoretical and practical research have initially shown that the scientific hypothesis of the dissertation is acceptable, the research purpose and task of the dissertation has been completed./ ... Thong Linh high school, Cao Lanh district and Class 11CB3, Thien Ho Duong high school, Cao Lanh city - Teachers in experiment: Nguyen Van Dop, Mathematic teacher in Thong Linh high school and Nguyen... develop the ability out of school  Reason: - For teachers: Teachers are influenced by traditional teaching methods, so they often teach with their psychology of attempt in short-time to convey... 2020 1 PREFACE Reason for choosing the topic 1.1 Developing mathematical thinking competence is one of the essential requirements in teaching Mathematics at high school: Nowadays, education