MẪU BÌA LUẬN ÁN CÓ IN CHỮ NHŨ Khổ 210 x 297 mm DEPARTMENT OF EDUCATION AND TRAINING FORMATION AND DEVELOPMENT OF SOME OF INTELLECTUAL ADAPTATION SKILLS FOR SENIOR HIGH SCHOOL STUDENTS THROUGH THE TEAC[.]
DEPARTMENT OF EDUCATION AND TRAINING FORMATION AND DEVELOPMENT OF SOME OF INTELLECTUAL ADAPTATION SKILLS FOR SENIOR HIGH SCHOOL STUDENTS THROUGH THE TEACHING OF GEOMETRY Major: Theory and Method of the Teaching of Mathematics Major Code: 62 14 01 11 SUMMARY OF PHD THESIS OF EDUCATIONAL SCIENCE 2014 This work was completed at Vinh University Name of Supervisor: Prof PhD Đào Tam —Vietnamese Society of Teaching of Senior High School mathematics Objection 1: Objction 2: Objection 3: The thesis was defended before the school-level Thesis Assessment Council gathered at Vinh University (Month)….(Day)…., (year)…., At (hour)… The thesis can be found out at: - National Library of Vietnam - Information Center - Nguyen Thuc Hao Library, Vinh University LIST OF THE WORKS OF THE AUTHOR HAVE BEEN PUBLISHED THAT DIRECTLY RELATED TO THE THESIS Nguyễn Viết Dũng (2009), Some of the methods of choosing coordinate system in solving solid geometric problems by means of the method of coordinates Science Magazine, Ho Chi Minh City University of Pedagogy, No 16 (50), 4/2009, Natural Science, pp 131-138 Nguyễn Viết Dũng (2010), Mathematics teaching in senior high schools according to the viewpoint of intellectual adaptation Postgraduate’s Scientific Seminar Yearbook, 2010,Vinh University, 12/2010, pp 43-51 Nguyễn Thị Hồng Lam, Trương Thị Dung, Nguyễn Viết Dũng (2009), Using the system of questions and exercises in teaching geometry to activate the cognitive activities of in senior high school students Vinh University Journal, No 1A, vol 38-2009, pp 50-56 Nguyễn Viết Dũng (2011), Applying the viewpoint of “intellectual adaptation” in the study and teaching of Mathematics in senior high schools Journal of Education, no 254, issue (1/2011), pp 45-48 Nguyễn Viết Dũng (2011), Training intellectual adaptability in teaching mathematics according to the constructive theory Vinh University Scientific Journal, no 4A,vol 40, 2011, pp 33-39 Nguyễn Viết Dũng (2011), Teaching the solving of mathematics excercises for senior high school stucents according to the viewpoint of “intellectual adaptation” Journal of Education, no 266, issue (7/2011), pp 39-43 Nguyen Viet Dung (2012), Teaching intellectual adaptability to mathematics students using activity theory Journal of Science of Ha Noi Pedagogical University, Vo 57, No 1,2012, pp 8- 15 Nguyễn Viết Dũng (2012), Training intellectual adaptability for senior high school students according to the method of detecting and solving problems Jornal of Science&Education, no 84, 9/2012, pp 21-24 Nguyễn Viết Dũng (2012), Training intellectual adaptability for senior high school students according to the theory of discovery Journal of Education, no 294, issue (9/2012), pp 42-44 INTRODUCTION The reason for choosing the topic 11th National Congress of the Communist Party of Vietnam has confirmed: “The developing of education and training along with the developing of science and technology are the first national policies” At the same time, it emphasized: “Radical and comprehensive innovating Vietnamese education towards standardization, modernization, socialization, democratization and international integration” Approving the project “Innovating radically and comprehensively the system of education and training to meet the requirements of industrialization and modernization in the conditions of market-oriented economy towards socialism and international integration”, Resolution No 29-NQ/TW on November 4, 2013 of the 8th Conference of the 11th Central Committee of the Communist Party of Vietnam has pointed out: “Building open education, practical learning, practical profession, good teaching, good learning, good management; having knowledge, basic skills and creative abilities to master oneself, have good living, and work efficiently.” Education Act 2005 ( Amendment 2009) stated: “The purpose of education is to train Vietnamese people to develop comprehensively, having morality, knowledge, health, artistic ability, and profession, faithful to the ideals of national independence and socialism; formation and nurture the personality, qualities and abilities of citizens to meet the requirements of national construction and defense ” (Chapter I , Article 2), and defined: “Methods of general education must promote in students positive, self-disciplined, self-motivated attitude, and creative thinking, in accordance with the characteristics of each class, each subject; feed up self-study methods, train skills to apply knowledge to reality, impact on student’s emotion, bring joy and excitement to student’s learning” (Chapter II , Article 28) We must renovate teaching methods based on current curriculums by strengthening the intellectual activities and mathematical abilities of students Current general education program manifests clearly the tendency of applying teaching methods that are highly dynamic and highly socialized, have the ability of making learners become active, encouraging learning, and developing social skills of learners This is the basis for students to shape and develop effective learning abilities, life skills, and the abilities of operating in reality Life skills and ability to apply knowledge to reality are regarded by world community as key factor of the quality of education Skills training process is a process of adaptation In his theory of cognitive development, J Piaget asserted: “Adaptation is the process of creating a balance between actions of the body and the surrounding environment It is the process of interaction between the body and the environment” When talking about intelligence — brainpower, J Piaget said: “All intelligences are processes of adaptation; all adaptations imply assimilations of things of the mind as well as the process of complement of accommodation” Similarly, author Nguyen Ba Kim reckoned: “If they can apply available knowledge and concepts to new objects, it is assimilation; if the new objects affect subject, compel it to adjust available knowledge and concepts to solve emerging issues, it is adjustment (or accommodation) Assimilation and application are together referred to as adaptation to environment” Thus, in the cognitive process, subject must be positive, must think proactively to reject old ideas that are no longer appropriate, must adjust old knowledge and attitudes through activities that help transform objects in order to make them reveal their properties, gradually penetrate into the objects to interpret, understand, and apply them This cognitive process is and adaptive process through the interaction with the environment Applying this to the task of teaching and learning is very important in the innovation of teaching methods This is a new and very difficult task, attracting the attention of educational researchers The innovation of teaching methods have been carrying out in a synchronous manner, among them is the innovation of mathematical program in senior high schools Geometry is a subject that has a very important role in helping to complete the mathematical knowledge and develop thinking ability for students In fact, students are often afraid of geometry, especially solid geometry Their ability of imagination about space is limited Their ability to draw, “read image”, reason logically is not coherent enough, still confused between plane geometry and solid geometry in their abstract thinking Students are not given due attention so that they can shape and develop intellectual skills in the process of problem-solving Meanwhile, the content of geometry in senior high schools contains suitable elements to establish and develop intellectual adaptability in the teaching process Moreover, as Pham Minh Hac wrote in the preface of the book “A Collection of J Piaget’s Psychological Works [Tuyển Tập Tâm Lý Học J Piaget]”: “All average students can get the ability of mathematical reasoning if we know how to exploit this capability based on their activity as well as eliminate emotional inhibitions that make they have inferiority complexes while studying this subject.” When assessing the theory of cognitive development of J Piaget, Paulpraisse remarked: “From now to the end of the century, I fear that the psychology of the world still can not fully develop the ideas of J Piaget” Currently, the formation and development of intellectual adaptabilities in mathematics for students is a subject that attracts the attention of many scientists and educators In the fourth edition of his mathematics teaching book, Nguyen Ba Kim has repeatedly refers to learning skills based on adaptive method: “Learners build their knowledge by adapting themselves to the environment with contradictions, difficulties and imbalances”, “It is important to establish the circumstances that have pedagogical intentions in order to help learners to apply adaptive skills to their learning activities.” Author Dao Tam, head of ministerial key research in 2008 “Fostering elements of the intellectual adaptation ability for students of mathematical pedagogy at universities through approaching modern teaching views”, subject code B2007-27-38-TĐ, and the works [81], [82], [83], [84], [85],[86], has analyzed some issues of intellectual adaptation and applying the view of intellectual adaptation to mathematics teaching Author Nguyen Phu Loc has schematized the process of intellectual adaptation and put forward pedagogical measures that support assimilating and accommodating processes in teaching Doctor of Science thesis in pedagogy of Do Van Cuong entitled “Fostering intellectual adaptability for students in order to enhance effectiveness in teaching solid geometry at senior high school level” and the works [ 18 ] and [ 19 ] have many explanations of the concept of intellectual adaptation, building measures of fostering intellectual adaptability for students in order to enhance effectiveness in teaching solid geometry at senior high school level However, there is not any fully and systematically research work for finding solutions for cultivating intellectual adaptabilities in learning mathematics at senior high school level For the above-mentioned reasons, we choose the research topic: “Formation and development of some of intellectual adaptation skills for senior high school students through the teaching of geometry” The Purpose of the Research This study is aimed at building fostering measures and training intellectual adaptabilities for senior high school students through the teaching of geometry, thereby contributing to the innovation of teaching mathematics in the current period Target and Object of the Research 3.1 Target of the Research: forming and developing intellectual adaptabilities in teaching geometry in senior high schools 3.2 Object of the Research: The content of the teaching of geometry in senior high schools Scientific Hypothesis If we can identify a number of intellectual adaptabilities and, through the teaching of geometry, put forward appropriate measures to establish and develop a number of intellectual adaptabilities for senior high school students, we can improve the efficiency of the learning of geometry Task of the Research 5.1 Do the investigation to further clarify the connotation of the concept of intellectual adaptation 5.2 Identify some intellectual adaptabilities in the teaching of geometry 5.3 Find out the status of teaching in order to establish and develop intellectual adaptabilities for students through the teaching of geometry 5.4 Put forward practice solutions, form and develop intellectual adaptabilities for senior high school students through the teaching of geometry 5.5 Organize pedagogical experiment using proposed pedagogical measures Methods of the Research Using the methods of theoretical study, empirical research, investigation, observation, and pedagogical experiment Contributions of the thesis 7.1 Further clarifying the connotation of the concept of intellectual adaptation 7.2 Identifying some intellectual adaptabilities , building stages of formation and development of intellectual adaptabilities 7.3 Successfully rendering the manifestations of the process of intellectual adaptation, forms of activities to practice intellectual adaptabilities , the manifestations of intellectual adaptation through active teaching methods and through the teaching of concepts, theorems, and how to solve mathematical exercises 7.4 Successfully building six measures to establish and develop intellectual adaptabilities in order to enhance effectiveness of the teaching of geometry in senior high schools Contentions being defended 8.1 Views on intellectual adaptation 8.2 Some of intellectual adaptabilities , stages of formation and development of a number of intellectual adaptabilities 8.3 Forms of activities to practice intellectual adaptabilities that have been identified as appropriate 8.4 Measures to establish and develop a number of intellectual adaptabilities in teaching geometry in senior high schools The structure of the thesis Apart from the Introduction, Conclusion, Appendix, and the list of references, the content of the thesis is presented in chapters: Chapter 1: The basis of reasoning and the reality Chapter 2: Training measures for establishing and developing some of intellectual adaptabilities for senior high school students through the teaching of geometry Chapter 3: Pedagogical experiment Chapter THE BASIS OF REASONING AND THE REALITY 1.1 Ability and skill 1.1.1 The concept of ability: From the viewpoints on the issue of ability, we asserted: ability is the sum of individual attributes that meet the requirements of a certain operation and ensure high performances for that operation 1.1.2 The concept of skill: From the researches on the issue of skill, we realized: Skill is the ability to successfully perform a certain operation in certain conditions by choosing and applying available knowledge and experiences 1.1.3 The relationship between skill and ability: Skill is an important component of ability, it realizes ability, whereas ability is an indispensable foundation of skill Skill is an ability to a job with a certain efficiency while ability ensuring high quality for that job People with good skills is not necessarily have great abilities, but through skill training they can develop their own abilities 1.2 Intellectual adapting skill 1.2.1 Intelligence: From the mentioned concepts of intelligence, we realized: intelligence is the ability to resolve new situations through operations of the subject This ability is transformed and developed in the process in which the subject absorbs these new situations and transforms himself to capture knowledge in new situations 1.2.2 Intellectual adaptation 1.2.2.1 The concept: as we understand, intellectual adaptation is an operation of subject to tackle new situation awareness through employing or adjusting available cognitive structure In other words , intellectual adaptation is characterized by the ability to apply available knowledge to new situations or renew available knowledge in order to make new knowledge appropriate to new situations 1.2.2.2 Some of the viewpoints on intellectual adaptation a The viewpoint of theory of cognitive development: From psychology studies of theory of cognitive development, we understand: intellectual adaptation is the balanced state between the processes of assimilation and adjustment Intellectual adaptation is established by diagrams, in which, intellectual diagram is the most important b The viewpoint of operational psychology: From the viewpoint of operational psychology, we understand: intellectual adaptation is the ability to convert outside psychological function to inside one through symbolic tool as psychological tool that defines the nature of society – history and through cooperative activities between the subjects of cognition c The viewpoint of association psychology: From the viewpoints on association psychology, we understand: Intellectual adaptation is the adaptability expressed in the process of converting, from this situation to other ones, the associations that are hidden in the objects to make them being suitable to available knowledge in order to solve the proposed issue 1.2.3 Intellectual adapting skill From the above-mentioned studies, we identified a number of intellectual adaptabilities as follows : 1.2.3.1 The skills of Reading comprehension, information gathering, and information selection: To solve a problem, students must firstly find out about it, must have the ability of analyzing the content of it, distinguishing where is the main information, where is the supplemental information, and where are the informations that likely to cause confusion And students must have the ability of answering the questions such as: Where is the unknow? Where is the data? What is the condition? Whether it can satisfy the conditions of the problem or not? Whether the conditions are sufficient for identifying the unknown or not? Whether there is any information that is superfluous or deficient? Is there any contradiction? To get this ability, students must regularly perform the assimilation tasks and exercises to form their skills in oder to adapt to proposed situation 1.2.3.2 Analogical reasoning skill in problem solving: Analogical reasoning or analogy is the method of calculation that based on some similar properties of two objects to draw a conclusion about similar of different properties of these objects We can also understand that analogy is the method of comparing between things that are different in general but similar in some appropriate respects 1.2.3.3 Skill to overcome obstacles and difficulties when available knowledge is not compatible with the proposed issue: We say that there is a difficulty if the problem is solved without requiring the review of viewpoint of theory being discussed or of the current concepts We say that there is an obstacle if the problem is resolved after we have reconstructed the concepts or changed the viewpoint of the theory When talking about why students often make mistakes when they encounter difficulties in learning mathematics, author Bui Van Nghi suggested: “In mathematics program at senior high school level, we study solid figure through representing figure From representing figure, students need to imagine the given picture and investigate relationships and their properties in representing figure This is a difficulty for students when they transfer from plane geometry to solid geometry, from concrete thinking to abstract thinking Students often have misconceptions and mistakes” 1.2.3.4 The skill of Transforming forms of the objects, creating favorable conditions for subject to penetrate the objects in order to understand, interpret and make use of them a The skill of restructuring the problem: Usually, when solving a problem, students identify its form and use appropriate methods to solve, but sometimes form of problem obscure its underlying content to such an extent that the corresponding method can not satisfy the task b The skill of restructuring the content: There are many problems with the forms of which are not compatible with the knowledges and methods of students These problems are often referred to as “atypical” There is no specific way or method for solving this kind of problem and student must accommodate themselves at high level to be able to restructure one or more contents in order to solve the proposed problem c The skill of restructuring the language: This activity includes language switching within one field of mathematics and switching from one language to another In a field such as geometry, we can solve a problem by using synthetic method, vector method, and coordinate method 1.2.3.5 The skill of logical reasoning: Logical reasoning or logical thinking is the ability to draw conclusions from given premises, the ability of dissecting a case into separate smaller cases in oder to completely survey the event being studied, the ability to predict results of the theory, generalize conclusions received 1.2.3.6 The skill of turning practical situation into mathematical language (mathematizing practical situation) and vice versa: Mathematics arises from reality, and practicalness in mathematics is often overshadowed by its abstractness 1.2.4 Manifestations of intellectual adaptability in mathematics 1.2.4.1 The manifestation of intellectual adaptability through concept learning a The manifestation of intellectual adaptability through the processes of identifying and representing the concept: Identifying and representing are two different operations but they are closely related and often interlocking “Representing” was a concept that involves the concept of “identifying” in the role of checking operation b The manifestation of intellectual adaptability through the process of converting the language of the concept: Language converting is the process of assimilation and adjustment to change the form of the concept in order to disclose the content that should be identified, this is the manifestation of intellectual adaptation c The manifestation of intellectual adaptability through the dividing of the concept: The process of concept dividing must adhere to a united standard This task requires the subject (student) must have necessary skills and a sound basis to perform This is an adaptation through the processes of assimilation and adjustment to ensure that the division does not violate the definition of concept dividing and ensure the most beneficial result in cognitive process d The manifestation of intellectual adaptability through the activities of generalizing, specializing, systematizing to find out the concept through examining individual cases e The manifestation of intellectual adaptability through the activity of finding out different applications of the concept: Mathematics is widely used in its mathematical field, in other branches of learning and in multifold real life multifold 1.2.4.2 The manifestations of intellectual adaptability through theorem learning: The skill of judging of rules through the activities of specializing and generalizing to fine out the theorem; The skill of judging of rules through the process of deducing to find out the theorem ; The skill of judging of rules through the process of similarizing to find out the theorem 1.2.4.3 The manifestation of intellectual adaptability through the process of mathematical exercise solving The manifestation of intellectual adaptabilities through mathematical exercise solving to discover knowledge and through the ability of linking the exercises in the textbook One of the manifestations of intellectual adaptabilities when solving mathematical exercise is the ability of “linking” the exercises in the textbook according to the standards of knowledge and skill, and the ability of making use of original mathematical problem 1.3 Forming and training intellectual adaptability for students 1.3.1 The formation of intellectual adaptability 1.3.1.1 The stages in the formation of intellectual adaptability Stage 1: Students apply theories to solve simple problems, it is considered as assimilating activity Stage 2: Students use their knowledges to solve basic problems with high level of assimilation and low level of accommodation Stage 3: Students use their knowledges and skills in solving basic problems to solve more complex and diverse problems Thus, in this stage, students must have the ability of accommodation to establish new cognitive schema at a higher level through the formation of intellectual manipulation in order to transform new problem into a familiar one 1.3.1.2 Factors that affect the formation of intellectual adaptability: Knowledge system, technical system, manipulation of operation, measures and means of operation, processes of inspection and evaluation of each student personally 1.3.1.3 Some psychological factors that affect the process of practicing intellectual adapting skill Students get their skills and techniques mainly through the processes of activities; levels of progress in terms of skills and techniques of students in the course of activities is unequal; each activity method creates only a certain moderate result; a skill or technique if not being used regularly will weaken and may eventually be lost completely; in the process of activity, there must be an interaction between teacher and students Through the analysis of the above-mentioned researches, we identify two characteristics of the process of forming intellectual adaptability as follows : Firstly, the general process has stages: applying theories to solve simple problems solving basic problems with high level of assimilation and low level of accommodation accommodating available knowledge to solve more complex and diverse problems Secondly, about the measures: making use of mathematical problems from simple to complex associated with technical means ( if needed ) The process of using measures depends on structural characteristics of each type of skill 1.3.2 Forms of activities to practice intellectual adaptability 1.3.2.1 Activities of assimilating and accommodating: Based on the above explanations, we believed: Activity of assimilation is the process in which the subject receives information and associates it with available knowledge to expand and consolidate his (or her) knowledge The activity of the accommodation is the process of adjusting received information that is not compatible with available knowledge and concepts to make it being suitable to new situation, creating new adaptive step 1.3.2.2 Activity of modifying the object: Activity of modifying the object is a system that consists of intellectual thinking manipulations of the subject to penetrate into the object change the structure of the object in oder to make it being suitable to available knowledge and revealing the results to be obtained, in order to form new knowledge 1.3.2.3 Activity of penetrating into object: Activity of penetraing into object is an intellectual activity of students It is adjusted by the knowledge that has been accumulated through the modifying operation that unfolds the object of the activity by means of logical deduction 1.3.2.4 Activity of association: In mathematics, from a particular situation we can relate to similar problem, similar transformation, from plane geometry problem we can relate to solid geometry problem, 1.3.2.5 Activity of language conversion: Language conversions are the intellectual activities that associate the immediate problem with the methods of synthesis, geometry, vector, coordinate, and transformation 1.3.2.6 Activity associated with the reality: Even though being highly abstract, mathematics can not lose its practicality completely because it arises from reality Mathematics is commonly used in its mathematical field and in other sciences 1.4 The training of intellectual adaptability is presented in a number of active teaching methods 1.4.1 Intellectual adaptation according to the viewpoint of activity theory: According to the activity theory, intellectual adaptation is a process of forming the ability to penetrate into an object to make the object gradually revealed through the available knowledge This Activity is an act of thinking through situations that evoke motivation of the needs of students in order to expose the attributes, relationships, and mathematical laws 1.4.2 Intellectual adaptation according to the the teaching of theory of exploration: intellectual adaptation adheres to the method of exploration is the ability of modelizing object classes through the abilities of description, conparison, analysis, synthesis, generalization, abstraction, relocating action functions by transforming the objects of action It is the skill of manifesting dialectical perspective of mathematical reasoning in the process of discovering new knowledge 1.4.3 Intellectual adaptation according to the method of finding out and solving problem: In the teaching of finding out and solving problem, to learn is the process of accommodation to deal with new situations Thus, this method requires efforts in individuals to resolve contradictions, overcome obstacles in their awarenesses The purpose of this method is to practice the ability of penetrating into the problem and solving it in a scientific way, give the subject the ability of associating the ways of thinking, exploration, prediction, and then proposing hypotheses and tests in the process of discovering new knowledge 1.4.4 Intellectual adaptation according to the construction theory 1.5 The Survey of real situation of teaching involving the formation and development of intellectual adaptability in senior high school students We have conducted a survey to find out the status of the formation and development of intellectual adaptability among teachers and students at senior high schools of the provinces of Dong Thap, Long An, Dong Nai, Binh Phuoc, Binh Thuan, and of Ho Chi Minh city (Appendix 1) We have interviewed many mathematics teachers and attended some classroom observations of their classes at some senior high schools and conducted a survey to find out common advantages and difficulties of teachers and students in the teaching and learning of Mathematics in general and geometry in particular, and survey activities of accommodation that were carried out in the processes of teaching geometry 1.6 Conclusion of Chapter This thesis found out some contents of the theoretical basis of the concept of intellectual adaptation, the concept of intellectual adaptability, identifying some intellectual adaptabilities, some viewpoints and manifestations of intellectual adaptation process that are being used in the cognitive process of students From that points it formed the basis of construction and development of intellectual adaptability and factors that affect the process of practice intellectual adapting skill of students Concurrently, through practical surveying, it has confirmed that teaching follow the direction of practicing intellectual adapting skill to senior high school students should be studied more deeply to contribute to the process of changing new methods of teaching and learning mathematics in senior high schools in the current period Chapter MEASURES OF FORMING AND DEVELOPING INTELLECTUAL ADAPTING SKILL FOR STUDENTS IN TEACHING MATHEMATICS IN SENIOR HIGH SHCOOLS 2.1 The role of geometry in the formation and development of intellectual adapting skill 2.1.1 Some characteristics of Geometry textbook in senior high school 2.1.2 The role of geometry in senior high schools 2.1.3 Some characteristics of geometry in senior high schools that are related to intellectual adaptation 2.1.4 The bases of the formation and development of intellectual adaptabilities in geometry We render levels of formation and development of intellectual adaptabilities in geometry: assimilation - assimilation combined with the low level of accommodation - accommodation 2.2 The orientations in the process of setting forth measures for the formation and development of intellectual adaptability for students 2.2.1 Orientation The formation and development of intellectual adaptability must be consistent with the proposed purpose, the textbook program, the principles and the educational methods of teaching mathematics at senior high schools 2.2.2 Orientation 2: The formation and development of intellectual adaptability must be based on the orientations of renovating teaching methods, heighten the roles of the qualites of selfteaching, positive, and creative of students under the guidance of teachers 2.2.3 Orientation 3: The formation and development of intellectual adaptability must be based on the characteristic elements of intellectual adapting skill necessary for training students 2.2.4 Orientation 4: The formation and development of intellectual adaptability must ensure the strengthening of its association with reality 2.2.5 Orientation 5: The formation and development of intellectual adaptability must be based on the basis of training the ability of three-dimensional imagination to overcome mistakes, obstacles, and difficulties when transferring from plane geometry to solid geometry 2.2.6 Orientation 6: Carrying out the process of formation and development of intellectual adaptabilities so as to apply to the teaching of geometry, mathematics, and a number of other subjects in senior high schools 2.3 The measures of formation and development of intellectual adaptability for students in teaching geometry in senior high schools 2.3.1 Measure 1: Training students to practice the activity of assimilation to use their knowledges and appropriate methods to solve the learning situation, and at the same time, through the activity, the operation of association to overcome the difficulties and obstacles in their minds in the process of receiving new knowledge 2.3.1.1 The purpose of the measure Helping students know how to use their knowledges of methods with algorithmic nature to carry out activities of assimilation to solve basic mathematical forms, thereby consolidating and expanding their knowledges At the same time knowing how to accommodate to convert and restruct the problem when encountering difficulties and obstacles in order to make it easier for for them to solve the proposed problem Through the activity of association help students overcome the difficulties and obstacles to practice, step by step, three-dimentional imagination gradually exercise imagination and practice space This measure is to contribute to form and develop the skills of 1.2.3.1, 1.2.3.2, 1.2.3.3, 1.2.3.4, and 1.2.3.5 2.3.1.2 Scientific basis and the role of the measure a About the knowledges of the methods; b Obstacles and difficulties; c The activity of association; d The role of the measure 2.3.1.3 Organizing the implementation of the measure a Having students perform the activities of assimilation to apply and make use of the knowledges of methods to solve the learning situations in order to reinforce and extend their knowledges The activities of assimilation and accommodation are often interwoven There are problems with average level of difficulty that students can apply their existing knowledge to assimilate in order to expand their cognitive schemas But there are also many problems with higher level of difficulty that can not be solved by means of available knowledges of the students so they must carry out the activities of accommodation to solve their tasks We analyze the assimilating activities through specific examples as follows: Let us suppose that the cognitive schemas of students had mastered the theorem of sine function, then they will perform the activities of assimilation to solve the following problem: Example 2.1: Calculate the area of triangle ABC, knowing that B 300 , C 450 , side AC = x A Students can carry out the activity of assimilation to x calculate side BC according to x based on the formula AC BC , in which, A = 1800- (B + C) = 1050, then apply the 300 450 sin B sin A B C Figure 2.1 formula S AC.BC.sin C to calculate the area of the triangle ABC Students can also perform the activity of assimilation by means of calculating the side AB, then apply the formula for calculating area S AB AC.sin A However, there are also students perform the acitivity of assimilation through drawing the vertical height AH, relying on the right triangle AHC to calculate the height AH, continously calculate the side BC and then apply the formula S AH BC to calculate the area of the triangle After solving the problem above, students’ knowledges of calculating sides and interior angles of a triangle, and of formulas for calculating its area, have been consolidated They also know about drawing additional factor AH to solve the problem Thus, assimilation activities help students to consolidate and expand their knowledge b Training students to perform the activities of accommodation , apply the knowledges of the methods with the nature of speculation to adjust the process of finding new knowledge through the activities of association There are links between knowledge of method, activity, and new knowledge, and they are illustrated in the diagram below: Knowledge of method Activity New knowledge Here we analyze the activities of accommodation through a number of examples as follows: Example 2.4: Calculate the volume of the tetrahedron ABCD with pairs of opposite sides are equal : AB = CD = a, AC = BD = b, AD = BC = c The calculation of the volume of the tetrahedron is converted to the calculation of the volume of the pyramid according to the process of calculating the base area and the length of the height However, there is an obstacle in this problem, namely the fact that students cannot yet identify the location of the foot of the height being drawn from a certain peak of the tetrahedral; this foot of height does not belong to any known point This is an obstacle When solving this problem according to the process of algorithm of calculating volume of a pyramid, students will meet with many difficulties in determining the length of the height of the pyramid being drawn from any vertex of a given tetrahedron To overcome this difficulty, students must restructure the problem, associate it with the existing knowledge in order to solve it in the A most favorable way This can be done through the process of N associating the relationship between tetrahedron and parallelepiped: M B “Every tetrahedron can be inscribed in a parallelepiped, and the volume of that tetrahedron is equal to one third of the parallelepiped circumscribing it.” P Based on this association, students can carry out the D following activities: Let the tetrahedron ABCD be circumscribed by the parallelepiped AMBN.PCQD (AP//MC//BQ//ND) Due to the C Q fact that pairs of the opposite sides of tetrahedron are equal so Figure 2.3 students can easily prove that the parallelepiped AMBN.PCQD is a rectangular parallelepiped Perform the activity of accommodation to restructure the problem: Calculate part of volume of the rectangular parallelepiped AMBN.PCQD with AM = x; AN = y, AP = z The calculation of x, y, z are based on right triangles with their factors have been identified, and we can easily calculate the volume of the rectangular parallelepiped, from which we can find out the volume of the tetrahedron On the other hand, teachers can suggest students to think of the relationship between the median lines og lateral surfaces of the trirectangular trihedron with the length of the line segments connecting the feet of that median lines Suggest students to think about the questions such as: is there any problem that is similar to this problem? In a trirectangular trihedron, is there any relationship between median lines of lateral surfaces and line segments that link the feet of that median lines? Students easily find out that the length of median line of lateral surface is equal to the length of the line segment that links the feet of remaining median lines A From the given problem, students can apply the insight of guessing and finding method to restructure the given problem by means of drawing tetrahedron ABCD inscribed in square tetrahedron APQR as follows: At each vertex in the plane ( BCD ) draw the straight lines that are parallel to the opposite sides, these lines intersect at the D points P, Q, R Then, APQR must be a trirectangular trihedron Q P with vertex A B 1 C Indeed, DC AB QR , DB AC PR , 2 Figure 2.4 R BC AD PQ Fron this point, students can easily find out that, in the triangle AQR, median line AB is equal to half of the length of bottom side QR Therefore, the angle A of AQR is a right angle Similarly, the triangles ARP and APQ also have right angles at vertex A Thus, tetrahedron APQR has three right angles at vertex A, consequently, 1 the volume V AP AQ AR We can easily find out VABCD VAPQR AP AQ AR 24 c Training students in performing the activities of applying the knowledge of the method to overcome mistakes and obstacles in their minds when transferring from plane geometry to solid geometry There is a relationship between plane geometry and solid geometry Plane geometry is a basis for building solid geometry Nevertheless, there are some differences between them The most obvious difference is that when a three-dimentioned figure being represented on a plane, it is no longer true as in plane geometry Thus, solid geometry will cause some difficults and obstacles 2.3.1.4 Some points should be noted when implementing the measure - Students need to identify these difficults and obstacles in order to find out the suitable ways of solving - To cause associations to appear, we need to replace suppositions and modify the form of the problems regularly 2.3.2 Measure 2: Training students to perform the activities of assimilation and accommodation to find out similar issues, from plane geometry to solid geometry and vice versa, to resolve geometric issues 2.3.2.1 Purpose of the measure Helping students know about the relationships due to similar properties between plane geometry and solid geometry, relying on that they may know how to turn plane geometric problem to the geometry of space and vice versa When encountering with a solid geometry which they not yet know how to solve, students can try to convert it into plane geometry problem (if possible) to solve it and apply this activity to the processes of solving problems in solid geometry Regularly performing the activities of converting problems in solid geometry into plane geometry and vice versa will help students enhance their abilities of three-dimensional imagine This is the process of adaptation in the process of discovering new knowledge This measure is to contribute to the formation and development of the skills 1.2.3.1, 1.2.3.2, 1.2.3.4 2.3.2.2 Scientific basis and the role of the measure The purpose of using similar concepts is to convert source (familiar) knowledge into destination (unfamiliar/new) knowledge If they share a number of characteristics, properties, then we will draw some corresponding signs Diagram of finding corresponding signs: _Source knowledge _ _Destination Corresponding signs knowledge _ This measure is mainly to perform the activities of assimilation We gathered from the texbooks some solid geometry problems and exercises that have some similarities to plane geometry problems in order to convert solid geometry prolem to plane geometry and vice versa This is to help sudents on the one hand know how to find corresponding signs between plane geometry and solid geometry, through wich they can develop a deeper understanding of geometry; on the other hand, through the process of finding corresponding problems, student can apply their knowledge of plane geometry problems to complex and diffuse solid geometry poblems Praticing these activities will help student form and develop the skills of analogic reasoning, receiving and choosing relevant informations, and converting the form of the problem 2.3.2.3 The implementation of the measure On the basis of corresponding process of Glynn, we propose a similar process as follows: Step 1: Introducing destination knowledge and source knowledge; Step 2: Finding the relationships between destination knowledge and source knowledge to establish correspondences and search for similarities; Step 3: Searching for similar problem; Step 4: Solving the similar problem a The similarity (correspondence) between plane geometry and solid geometry There are many similar theorems and properties between plane geometry and solid geometry Many theorems and properties in solid geometry are built on the basis of plane geometry b Some similar knowledges between plane geometry and solid geometry Solid geometry is built on the basis of plane geometry There are many knowledges in solid geometry being extended from plane geometry, such as vector, image transformation, circle and sphere , the equation in plane geometry and in solid geometry, c From a plane geometry problem, searching for similar problem in solid geometry and vice versa In geometry program, there are many problems of solid geometry have been built on the basis of problems of plane geometry This gives students the opportunity of associating in order to, through plane geometry problem, predict and present similar problems in solid geometry and vice versa d Using similarity as a means to perform the activity of assimilating knowledges In the above items of this measure, when looking for a similar problem, students must tabulate corresponding things in order to, from a plane geometry problem, present a similar problem in solid geometry and vice versa But there are many problems in which the application of similarity encounters difficulties because the knowledges of methods of plane geometry and solid geometry are not yet compatible, so students must perform the activities of assimilation in order to get new cognitive schemas to resolve the proposed situation 2.3.2.4 Some points should be noted when implementing the measure - Teachers must help students realize the similarities and the differences between plane geometry and solid geometry -Pay attention to some of the counter-examples to help the students avoid common mistakes in solving solid geometry problems due to analogizing from plane geometry 2.3.3 Measure 3: Create opportunities for students to apply their knowledges of geometry to reality In which, they know how to improve the activities of assimilation to apply their knowledge of geometry to explain practical situations and know how to accommodate to bring back the practical situations to the problem with practical content 2.3.3.1 The purpose of the measure - Through the activities of assimilation, students can see that mathematics is very close to real life Relying on that we can gradually stir up enthusiasm for students to learn geometry in particular and mathematics in general - Through the activities of accommodation, students can incorporate situations from real life (if possible) into the form of mathematical problem with practical content, and know how to use geometric tools to solve the problems that are not geometry problems in a more convenient way This measure is to contribute to the formation and development of the skills of 1.2.3.1, 1.2.3.4, 1.2.3.6 2.3.3.2 Scientific basis and the role of the measure Mathematics derives from real life, and it is the “key” in most human activities It is present everywhere 2.3.3.3 Organizing the implementation of the measure Skills are formed and developed through activities, every skill is attached to a certain specific activity The formation and development of a skill require the participation of subject in activities suitable to that skill The skill of mathematizing of students being formed and developed through mathematics activity a Regularly combine the process of knowledge transmission with the process of associating with surrounding reality in order to inspire students a.1 Regularly present suggestive questions in order to get students to explain the practical situations in surrounding reality by means of mathematical knowledge a.2 Give students opportunities to discover practical situations through classroom lessons b Create interesting situations in the form of expression and in the content of mathematics teaching, inspire students in the process of mathematizing practical situations In fact, students often think about concepts, properties, and theorems of mathematics when solving mathematical problems and rarely wonder about practical situations that are relevant to mathematics in specific cases So, when teachers create interesting situations in the form of expression and in the content of mathematics, they will inspire students in the process of mathematizing practical situations c Applying mathematical knowledges to the process of solving mathematical problems with practical content c.1 Some theories we refer to are the basic knowledges about the nature of similarity, the angles, the cosine theorem, sine theorem , for solving extremum problems in geometry c.2 Mathematics is used a lot in physics- an experimental science Laws and formulas of physics are usually built on mathematical expressions consistent with experimental results c.3 Geometric transformation is used a lot in reality In our surrounding life, there are things, tools, machines, constructions works using geometric transformations such as axial symmetry, translation, central symmetry, Today, information technology is growing and there are a lot of operations on computer using image transformations such as zoom in, zoom out, rotate an image or a character through an arbitrary angle, move an image to an arbitrary position d Using geometric tools to solve some problems with no geometric content d.1 Solving algebraic and analytic problems by means of geometric methods; d.2 Solving trigonometric problems by means of geometric methods; d.3 Solving arithmetic problems by means of geometric methods 2.3.3.4 Some points should be noted when implementing the measure Strengthening activities of assimilation to adjust gradually Only this when having favorable opportunity 2.3.4 Measure 4: Training students in performing the activities of entering object to modify the object in order to discover the relationships that are contained inside through available knowledge 2.3.4.1 The purpose of the measure Training students on the skills of entering object to modify it, such as: modify the form of the object by means of restructuring the content of the problem; convert the inside of content of a problem; convert the language… to find out relationships inside the object in such a way that students easily mobilize their available knowledges to enter the object in order to solve the problem more favorable Practicing the skill of modify object will help students form and develop the ability of threedimensional imagine This measure is to contribute to the formation and development of the skills 1.2.3.2, 1.2.3.3, 1.2.3.4, 1.2.3.5 2.3.4.2 Scientific basis and the role of the measure This measure is mainly to carry out the activities of accommodation to solve complex situations 2.3.4.3 Organizing the implementation of the measure a Training students on the skill of converting between form and content of object to find out relationships between new knowledges and available knowledges, then relying on that finding out the way to solve the problem This converting process is the process accommodation that help subjects (students) adapt their knowledges in order to gradually master the skill of penetrating objects, identifying the underlying properties of the objects, and thus they can interpret the objects through their available knowledge This can be illustrated by the following example: Example 2.26 Let S.ABCD be a pyramid with the base being a right trapezoid ABC BAD 900 ; BA BC a, AD 2a Lateral edge SA is perpendicular to the base, and SA a We call H the perpendicular projection of A on SB Calculate ( according to a) the distance between H and the plane ( SCD ) The difficulty of this problem is the determination of the foot of perpendicular line from H to ( SCD ), this is a very complicated task because its location has not been determined To overcome this difficulty, students can perform the process of transforming object to find the relationship between hypothesis and conclusion First of all, teachers instruct students to think of other “reasonable” way rather than locating the foot of the height from H to ( SDC ) to calculate the height of the pyramid That “reasonable” way is borrowing the process of calculating the distance between B and (SDC) to calculate the height of the pyramid Then, based on the fact that H and B are both located on SB, we can use the property of similitude (uniformity) to infer the distance between H and ( SDC ) However, the calculation of the distance between B and ( SDC ) based on assumptions only possible if we it through the process of calculating the volume of the pyramid B.SDC according to the formula VB.SDC VS ABCD VS ABD This is a complex approach When following this approach, students will encounter difficulty So students can change form and content of the problem into the other direction, that is due to Ax, Ay, Az perpendicular to each other in pairs, thus, the point H is completely defined on SB, the plane ( SDC ) is also completely defined Thus, the distance between H and (SDC) will be completely defined This will make students to think of coordinate method Then, they can use coordinate method to solve the problem succinctly as follows: Because AB, AD, AS are perpendicular to each other in pairs, so we choose coordinate system in such away that A(0;0;0), B(a;0;0); D(0;2a;0); S(0;0; a ) C a;2a;0 AH SB We call H h1; h2 ; h3 , because H SB and AH SB , so SH t.SB 2a a a 2 H ;0; Therefore, d H ;( SDC ) 3 b Training students on the skill of modifying form of objects to create favorable conditions for subject to penetrate into the objects in order to understand, interpret and make use of them In practice, there are many geometic problems with complex forms that cause difficulties for students in choosing appropriate knowledges and methods to solve On the other hand, the complexity of the displayed form may conceal the content and bring about mistakes or “deadlock” that impossibilitates the way of finding out the solution “Disturbing property” of displayed form of the problem create the impression that known methods and available knowledges are unfamiliar to its content of the students already have past alien to its content Example 2.29 Find a set of points M in the tetrahedron ABCD in such a way that the total volume of the tetrahedron MABC, MACD, MDAB equal to k times of the volume of the tetrahedron MBCD For students, this is a difficult problem Initially, the object is unfamiliar to students because their available knowledges are not yet linked to the requirements of the problem, it is clearly not yet disclosed Modify the object by letting: V1, V2, V3, V4, V is by turn the volume of MABC, MACD, MADB, MBCD; ABCD has the relationship: V1+ V2+ V3 = k V4 A (1) Since M is in the tetrahedron, so students easily know that V1 + V2 + V3 + V4 = V Hence, base on the expression (1), Adding two sides to V4, students can infer that V (k 1)V4 or M V (2) V4 k 1 B Thus, the process of modifying object has convert the D relationships between the volumes V1, V2, V3, V4, into the relationship between the volume V4 and the volume V of Hình 2.26 C tetrahedron ABCD according to the expression (2) The process of changing expression (2) has revealed the object, hence, based on the fact that BCD is the common base of the tetrahedrons ABCD and MBCD, students can use their available knowledges to solve the given problem through the process of considering the relationship of common height of two tetrahedrons with a common base The process of changing the form of the problem has led to a new problem: Let ABCD be a tetrahedron with h being the distance between A and the plane (BCD) Find the set of points M in the tetrahedron so that the distance from M to the plane ( BCD ) being equal to h This is the process of accommodation for adapting the knowledge k 1 c Training students in the skill of language converting in the process of solving geometric problems The process of researching and solving geometric problems is done by several different methods, in which, the methods such as synthetic method, vector method, coordinate method, and volumetric method (for solid geometry) are used the most 2.3.4.4 Some points should be noted when implementing the measure - When converting the language in the process of solving problem, teachers must help students identify the appropriate association - We should avoid the modification that unnecessarily complexifying the problem 2.3.5 Measure 5: Training students in performing the activities of assimilation and accommodation through a number of techniques of using knowledge of plane geometry to solve the problems of solid geometry 2.3.5.1 The purpose of the measure The purpose of this measure is to use geometric tools to solve the problems of solid geometry more favorable The practicing of techniques of using knowledge of plane geometry knowledge to solve problems of space geometry is conducted through activities in order to gradually clarify the objects, the properties should be studied, leading to the process of adaptation This measure is to contribute to the formation and development of the skills 1.2.3.2, 1.2.3.3, 1.2.3.4 2.3.5.2 Scientific basis and the role of the measure In plane geometry, students are familiar with ways of looking at “real image”, but when it is the issue of solid geometry, they have to work with three-dimensional images being represented on flat surfaces So they are difficult to "see " the flatness of these flat images This is the most basic obstacle that often lead to confusion 2.3.5.3 Organizing the implementation of the measure a The Correspondence between plane geometry and solid geometry b Some grounds for converting the process of solving some problems of solid geometry into solving problems of plane geometry and vice versa c The activity of separating flat components from three-dimensional image Activity of separating flat components from three-dimensional image is performed when objects, the geometric relationships that should be surveyed, is hidden in a certain flat image of a three-dimensional image This activity helps students to change the form of the problem with concealed contents to explore it d Activity of spreading out all parts of three-dimensional image on the same plane (image flattening) Step 1: Analyzing the problem in order to convert the problem of solid geometry into the problem of plane geometry by means of spreading out all faces of three-dimensional image on the same plane Step 2: Performing the process of image flattening Bước 3: Performing the process of converting the given problem into the problem of plane geometry and then performing the activity of solving e Performing the activity of solving the problem by applying parallel projection Steps to practice to apply parallel projection to the solving of the problem of solid geometry: Step 1: Identifying the given factors and the factors that should be found of the problem Step 2: Choosing appropriate projection plane and projection direction so that the projection plane can represent all of given factors and the factors that should be found Step 3: Defining the projections on the projection plane, finding the relationship between the given factors and the known factors in order to express the corresponding problem in plane geometry Step 4: Solving the corresponding problem in plane geometry 2.3.5.4 Some points should be noted when implementing the measure We can only convert the problem of solid geometry into the problem of plane geometry and vice versa when there is an appropriate correspondence 2.3.6 Measure 6: Strengthening cooperative learning activities in order to create an interactive environment, in which, students can impact, support and promote each other, and create many trends of activities of assimilation and accommodation when solving the problems of geometry 2.3.6.1 The purpose of the measure Using cooperative learning activities to develop the skill of enquiring about the problem, finding the way to solve, representing solution and profoundly studying the solution Performing cooperative learning activities to help and support each other in the process of learning whereby students can construct a friendly and solidary environment This measure is to contribute to and develop the skills 1.2.3.1, 1.2.3.2, 1.2.3.3, 1.2.3.4, 1.2.3.5 2.3.6.2 Scientific basis and the role of the measure Four steps to train students on the skill of cooperating are: Step 1: Creating the cooperative context, that is helping students being aware of their interdependence and caring about the states of minds of each other Step 2: Constructing and organizing the occasions of knowledge debate Creating cognitive conflicts in order to provide students the chance to practice critical thinking Step 3: Teaching students the skill of negotiation Step 4: Teaching students the way to reconcile Three steps for performing cooperative activities between student groups:1/ Individual selfstudy (activity of independent thinking) 2/ Group discussion (activity of conversation with critical thinking) 3/ Presenting the outcomes of the group (activity of aggregate thinking) The cooperation between the groups include: joining and/or unifying learning outcomes; mutual learning between groups, aggregate thinking, criticizing; the cooperation between teachers and students includes the activities of analysis and aggregation of knowledges, assessment and self- assessment 2.3.6.3 Organizing the implementation of the measure We delve into the fact that through cooperative activity we can help students know the ways to assimilate and accommodate in order to adjust their knowledges in the processes of improving their abilities of solving mathematical exercises a Perfoming cooperative activities to inpress on the memory the methods of solving mathematical exercises b Performing cooperative activities in order to develop the ability of inquiring into the problem from many different viewpoints c Performing cooperative activities to practice the skill of finding the way to solve the problem 2.3.6.4 Some points should be noted when implementing the measure When applying the method of cooperative learning, the role of the teacher is very important Teachers need to design learning situations through the system of suggested questions with pedagogical intent but the answers must be within students’ capability in order to make them interested in participating in the process of discussion 2.4 Conclusion of Chapter In this Chapter, we have done some works as follows: - General introduced the content of geometry program in senior high schools, - Highlighted the role of geometry in the formation and development of intellectual adaptability to students in senior high schools - In the main content of this chapter, we have brought out six constructing orientation to implement the measures - Proposed six pedagogical measures to train students on intellectual adaptability in the process of teaching geometry in senior high school In each measure, we present four main issues: the purpose of the measure, the basis and the role of the measure, the guidance for implementing the measure, and some points should be noted when implementing the measure Chapter PEDAGOGICAL EXPERIMENT 3.1 The purpose and requirement of the pedagogical experiment 3.1.1 Purpose: The purpose of the pedagogical experiment is testing scientific hypothesis of the topic in order to evaluate the feasibility and effectiveness of the measures of fostering, forming, and developing of intellectual adaptability for senior high school students through the teaching of geometry 3.1.2 Requirement: Pedagogical experiment must be objective, suitable to capability of students, and close to the actual teaching situation 3.2 The objects of the pedagogical experiment The objects of the pedagogical experiment are the students of senior high schools Thap Muoi and Phu Dien in Dong Thap province, and Tran Hung Dao senior high school in Go Vap district, Ho Chi Minh City 3.3 Methods of pedagogical experiment 3.3.1 Method of inquiry 3.3.2 Method of observation 3.3.3 Method of mathematical statistics 3.3.4 Constructing method and criterion for evaluating 3.4 The content of pedagogical experiment In order to achieve the goal and requirement have been proposed, the selection of the teaching lessons is aimed at forming and developing intellectual adaptability We selected lessons, each of which has teaching plans belonging to advanced and fundamental programs for the classes 10, 11 and 12 3.5 Training teachers to teach experimental class 3.6 The process of pedagogical experiment 3.6.1 Round of the pedagogical experiment, school year 2010-2011 3.6.2 Round of the pedagogical experiment, school year 2011-2012 3.6.3 Interview, survey and observe pedagogical experiment classes to help to qualitatively evaluate the possibility to form and develop intellectual adaptability in students 3.7 Conclusion of Chapter Round of pedagogical experiment has been carried out at Thap Muoi and Phu Dien senior high schools in Thap Muoi district, Dong Thap province, to examine the feasibility of the measures of forming and developing the intellectual adaptability in students by means of simultaneously applying measures have been proposed in teaching geometry for students in grades 10 and 11 The result is that the proposed measures are in accordance with the content and form of the teaching of geometry, however, there are some issues that need to be modified, such as the implementation of some measures needs to be reexamined so that they can be closer to the capability of the students Round of pedagogical experiment was conducted on a large scale at senior high schools, in Dong Thap provice and one in Ho Chi Minh City, with the measures have been adjusted to suit the learning goals and characteristics of junior high school students The results of round of the pedagogical experiment have allowed us to assert: The measures proposed in Chapter are highly feasible, satisfying the requirements of teaching innovation today The proposed measures have helped students become more active in learning, helping them prove their abilities; helping them form and develop their intellectual adaptabilities better From those bases, teachers and students are forced to adjust teaching and learning process to improve the quality of training The pedagogical experiment have confirmed the relationship between these measures We must apply these measures synchronously in the process of forming and developing intellectual adaptability, can not overlook one measure while esteem another The pedagogical experiment has confirmed the correctness of relevant scientific hypotheses According to current educational goals, the simultaneous application of these measures has positively enhanced the effectiveness of the teaching of geometry in particular and innovated the teaching of mathematics in general CONCLUSION From the results of the thesis, we can draw some conclusions as follows: We can search for models of teaching and other applications to introduce into the teaching of geometry in particular and mathematics in general on the basis of approaching teaching methods in view of intellectual adaptation This thesis has systematized the theoretical basis, contributing to further clarify the connotation of the concept and the skill of intellectual adapting; proposing some skills of intellectual adaptation; constructing the stages of forming and developing of intellectual adaptability; highlighting the manifestations of the process of intellectual adaptation, the forms of activities for practicing the skill of intellectual adapting The manifestations of intellectual adaptation through positive teaching methods and through the teaching of concepts, theorems, school exercise solving, are used as a basis for improving the effectiveness of the teaching of mathematics in general and geometry in particular in senior high schools This thesis has proposed orientations and constructed measures to form and develop a number of intellectual adapting skills in students, contributing to improve the effectiveness of the teaching of geometry in senior high school The thesis has drawn some results through the processes of pedagogical experiment at many senior high schools in different areas in order to check the reliability of the subject The teaching of mathematics in accordance with the viewpoint of intellectual adaptation should be more concerned by the researchers because the results of the research works are educational models and applications with pedagogical values, they will contribute to improving the effectiveness of teaching mathematics in senior high schools All the research results of the thesis have proved that the scientific hypothesis of the thesis is acceptable, the purpose and task of the research have been completed, and the arguments being defended have been affirmed