Thiết kế và sử dụng các tình huống dạy học nhằm hỗ trợ phát triển trí tưởng tượng không gian cho học sinh trong dạy học hình học ở trường trung học phổ thông TT TIENG ANH

MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY DAU ANH TUAN DESIGN AND USE TEACHING SITUATIONS TO SUPPORT THE DEVELOPMENT OF SPATIAL IMAGINATION FOR STUDENTS IN TEACHING GEOMETRY AT HIGH SCHOOL Major: Theory and methods of teaching Mathematics Code: 9140111 SUMMARY OF PHD THESIS ON EDUCATIONAL SCIENCE NGHE AN - 2021 The work has been completed at Vinh University Scientific instructors: Prof Dr DAO TAM Assoc.Prof Dr NGUYEN CHIEN THANG Referee 1: Referee 2: Referee 3: The thesis will be defended in Thesis Evaluation Committee at University level held at Vinh University Time… , day: month year 2021 Thesis can be found at: - Vinh University Library - Viet Nam National Library Chapter RESEARCH ORIENTATION 1.1 Research problem In the dissertation, we give the research problems including: a, Introduce the concept of spatial imagination (SI) in teaching geometry in high schools b, Elucidate the role of the SI for students' geometric awareness activities in teaching spatial geometry in high schools and the impact of the SI on developing the ability to solve problems in teaching geometry c, The role of SI in the study and explanation of phenomena in reality d, Exploring and exploiting activities that need to be practiced to develop SI for students in the process of teaching geometry in the direction of designing and using teaching situations The research problem of the above thesis comes from the following scientific bases: - First of all, the problem posed by the thesis's research comes from considering the concepts of SI of many domestic and international authors There are many different views on the concept of SI through the introduction of a number of essential properties However, we have not found a clear definition of the concept of SI Therefore, the research problem posed first is to shed more light on SI in a way that can initially visualize the levels of SI - The practice of teaching geometry in high schools according to the current program as well as the current program of innovation in mathematics education, the level of geometrical algebra is quite high, due to the introduction of vector methods, coordinate methods, transformation When the geometry program emphasizes algebra, it will slightly reduce the development of the SI The main cause of this decrease is the lack of importance in teaching geometry about the balanced relationship between the content of general geometry and algorithms using vector tools and coordinate methods in a formal way Since then, there has arisen the phenomenon that many students math on vector and coordinate expressions but they not understand the geometrical nature of the problem solved by vector and coordinate tools More details on this issue will be presented in the practical basis outlined in Chapter - The researches on the development of SI for the students in high school as well as at other school levels have not clarified which key activities to develop the SI for students There have not been theoretical and practical studies to clarify the elemental activities to develop the SI of students in the process of teaching geometry at high schools - There have not been studies in the country as well as abroad on the design and use of situations to organize teaching geometry in order to develop SI for students The design of the above situations contains many difficulties, the outstanding difficulty is to clarify what minimum requirements a situation designed for use in teaching geometry must satisfy The process and steps of designing a teaching situation to use how to develop SI for students are specific in teaching concepts, theorems, geometrical laws, and solving geometry exercises 1.2 The need of research to develop students' spatial imagination in teaching geometry a Stemming from the requirements of the current Math program and the 2018 General Education Math program about the high school goals in the Geometry and Measurement section that emphasizes to develop SI for students b The need to be problem-oriented, creatively solve and develop problems c Requirements of mathematics education in the direction of connection with practice d The need for active teaching The geometric knowledge acquired is the product of active activities of students through interaction with situations, through communication and cooperation between students and students, between students and teachers This poses a requirement for the study, design and use of cases that contain activities towards the development of the SI for students, and poses a need to consider and provide procedures for the design and use of situations in teaching geometry in the direction of developing SI for students 1.3 Research purpose of the thesis Provide a theoretical and practical approach to clarify the concept of SI and related concepts related to the SI, the component activities to develop the SI, the process of designing the teaching methods and the process of using the teaching methods have been designed to organize teaching geometry to develop the SI of high school students 1.4 Scientific hypothesis If the elements of the SI and the activities compatible with those elements can be identified, then it is possible to find opportunities to organize for students to practice the above activities in order to contribute to the development of SI in teaching geometry in high schools 1.5 Research questions a, Based on what theoretical and practical basis to give the concept of SI? b, How is SI expressed in teaching geometry in high schools? How to detect those symptoms? c, What are the main activities that need to be practiced to develop SI for students? d, Based on what basis to develop the design process and the process of using teaching situations towards developing the SI for students? e, What are the levels of SI development of high school students in teaching geometry? 1.6 Research Methods 1.6.1 Theoretical research - Research on psychological perspectives on imagination of domestic and foreign authors - Researching the perspectives on spatial imagination of mathematics educators in the country and around the world 1.6.2 Practical research - Research and design questionnaires to survey students and multiple-choice questions to survey teachers, attend geometry classes in high schools This research activity aims to reveal the manifestations of activities compatible with the characteristics of SI for students - Research the teacher's experiential activities to design teaching situations and use them to organize teaching of typical situations in teaching geometry in high schools - Studying the activities of teachers in the process of design and application of stages, including: Process building activities, discussion organization through seminars of teachers, experiment activities on students to find the feedback for editing to choose the right process for the implementation of teaching situations towards the development of the SI 1.6.3 Pedagogical experiment - Conduct experimental teaching activities according to the process of situations designed to assess the development level of students' SI 1.7 New contributions of the thesis - Systematize and clarify the theoretical basis of the SI, the relationship between the SI with intuition, thinking and knowledge; the typical elements of SI; activities aimed at developing the SI; - Give a conception of the SI by 11 possibilities Propose two levels on the development of SI of high school students in teaching Geometry; - Propose 13 main activities to practice for students in order to support the development of SI; - Develop a 6-step design process for teaching situations and a 5-step process to apply the designed situations in teaching with the development orientation of SI 1.8 Arguments to defend + In teaching geometry in high schools, it is necessary to develop spatial imagination for students; + The concept of spatial imagination of high school students and its characteristics is reasonable and can be developed through the support of the proposed main activities; + The design process, application process and teaching situations of spatial geometry in the direction of developing spatial imagination for high school students designed in the thesis are appropriate and feasible 1.9 Thesis structure The thesis is structured in chapters: Chapter Research orientation Chapter Theoretical basis Chapter Practical survey Chapter Design and use the spatial geometry teaching situations in the direction of the development of spatial imagination for high school students Chapter Pedagogical Experiment Conclusion of the thesis and recommendations References Chapter THEORETICAL BASIS 2.1 Research overview of math educators on topics related to the research topic - Types of situations in teaching Mathematics related to topics and situations that contribute to fostering SI in teaching geometry in high schools - Researches on the concept of SI in teaching geometry - geometrical intuition - The research sheds light on the expression of SI, the need for the development of SI in teaching geometry - Analyzing and synthesizing researches on the components of activities in developing the student's SI Through the research, analysis and synthesis of mathematics educators related to the research topic, we initially obtained the following results: Clarify the relationship between geometric imagination, SI and geometric intuition Be aware of the relationship between SI, intuitive thinking with logical thinking and formal proofs in mathematics: The SI suggests logical thinking, expressions and proofs; On the contrary, if there is good logical thinking, the hypotheses proposed by the SI have a scientific basis Previous studies have not provided a clear definition of the concept of high school students' SI The components constituting SI are shown through studies in different aspects The most typical elements can be mentioned, they are the elements in the composition of the SI: + The ability to visualize the results of shapes, relationships, numbers in geometry learned in high school + The ability to visualize spatial shapes, the relationships of spatial shapes through representations + Ability to orientate space that helps to study geometry and apply it in practice: Problems related to vectors, coordinates, rotation direction, position needed in reality, We find that there are still some contents related to the SI that need to be developed for students but have not been studied fully and deeply For example the following problems: - Estimate the length, magnitude, size of the geometric figures and estimate in reality - The problem of the relationship between shapes, partitioning a shape into familiar shapes, spreading the spatial figure on a plain Clarifying a number of roles of the SI in teaching geometry and in practice - It helps students see the meaning of mathematical knowledge, the meaning of math problems before implementing into proof arguments, explaining problems, reasoning to solve problems - Through the development of SI, it helps students have practical knowledge to help visualize the structure of objects through drawings and designs - It helps students approach to judge math problems, make hypotheses through spatial imagination - It helps to solve problems creatively through visualization of new events, new problems Visualize a number of component activities of the activities of formation and development of SI, including: - Perceiving practical models, geometric models to form correct symbols of shapes, interdependencies and quantitative relationships in that figure to form correct spatial symbols From there, there is a profound SI - Activities of determining direction, direction, determining position from one point to another, from one shape to another - Visualization of shapes, relationships and relationships in pictures through representations; operation that determines the representation of an image For example, we ask students to determine the projection plane and projection so that the representation of a nearly regular tetrahedron is a rectangle with two diagonals added - Visualizing the cross-section of a space figure created by a certain plane - The activity of visualizing the results of solving the problem without using pictures only through imagination Through the review, it is found that the authors have not mentioned the following activities that are meaningful to the formation and development of the following: - The operation of spreading a spatial figure onto a flat figure - The activity of creating a spatial shape according to the given flat parts - Space rendering activities - Estimating the length, area, volume associated with the figures in practice 2.2 Approaching the pedagogical perspective of spatial imagination 2.2.1 Symbols Symbols are forms of perception which are higher than sensations, they give us images of things that remain in our minds after the impact of things on our senses has ceased In psychology, it is understood: "The symbols of memory are images of objects, processes and phenomena that are not currently perceived but have been previously perceived" 2.2.2 Concept of Space The concept of "Space" mentioned in the dissertation is a 2-dimensional, 3dimensional Euclide space in the high school program (Based on real space symbols that people can perceive - Physical space) In the symbols that the SI operates reflect the properties (or signs) of spatial characteristics On that basis, we think that space is understood as a structure including the following sets: - Geometric shapes, objects; - Qualitative properties: shapes of shapes, relative positions between shapes, objects; direction; - Relationships before - after; right - left; - Quantitative factors: Distance, circumference, area; volume of shapes, blocks, etc On the basis of understanding the symbol of memory, we conceive the symbol of space as a symbol of memory about the properties and relationships of spatial objects 2.2.3 Imagination Intelligence is the ability to perceive, remember, think, judge, of human Imagination is creation in the mind an image of something that is not present or has never existed We can understand that imagination is the cognitive ability of people to create images of things that have been perceived but are not present or reflect things that have never been in personal experience by build new icons on the basis of existing images and icons 2.2.4 Spatial imagination As the point of view of space and imagination above, we can understand that the object of the SI is space, that is, the symbols in the process of imagining are spatial symbols Thus, we can understand that spatial imagination is the cognitive ability of people to create images of spatial objects that have been perceived but not have before or reflect spatial objects that unprecedented in personal experience by building new space images and symbols on the basis of existing space symbols 2.3 Features of spatial imagination To give the characteristics of SI, the important bases are: - Derived from the concept both in the country and abroad as well as the concept of SI above; - Based on the research results on the essential characteristics of teaching geometry in high schools, especially the research of Academician A.D Alexandrov about three characteristic elements of teaching geometry are: Reality, logic, imagination; - Be aware of students' mistakes due to not understanding the relationships and relationships between geometric objects in space A common mistake that arises in the process of learning spatial geometry is that they are only interested in manipulating formal operations without visualizing the relationships between geometric objects, especially when studying spatial geometry Geometric studies using vector tools and coordinate methods From the research results mentioned above, in this thesis, we conceive that the SI belongs to the category of geometric intuition characterized by the following capabilities: - The ability to visualize spatial shapes through representations; - The ability to determine the relative position between geometric figures; - Ability to establish dependency relationships between geometric shapes; - The ability to visualize cross-sections, intersect spatial shapes; - The ability to estimate the size of spatial figures; - Ability to transform relationships, relationships into known geometric models convenient for problem solving; - The ability to convert from one geometry language to another to visualize the research model; - The ability to develop shapes convenient for calculations; - Ability to map, coordinate to determine position, size, distance between shapes; - Ability to model real-world phenomena using geometrical language and symbols; - Ability to define new spatial objects on the basis of existing spatial objects With the understanding of the above characteristics, for high school students' SI, there are two levels: Level 1: Deeply understand geometric objects, relationships and relationships between geometric objects, geometrical meanings of formal expressions expressed in algebraic language (Vector language, coordinates) It's essentially understandable the geometrical content through its formal expressions Level 2: Help to create new geometric objects on the basis of transforming existing objects and relationships Example 2.1 Students can construct a rectangular box by using a nearly regular tetrahedron ABCD through the following proposition: “Three pairs of parallel planes pass through pairs of opposite sides of a nearly regular tetrahedron that intersect to form a rectangular parallelepiped" Through this example, we see that: Spatial imagination is not only the activity of building new symbols on the basis of existing symbols, but it is also on the basis of existing knowledge In the above example, the existing knowledge is: There exists only one pair of parallel planes passing through two diagonal lines 2.4 The relationship between intuition, spatial imagination and mathematical thinking in teaching geometry in high schools 2.4.1 Intuition In Soviet teaching theory, intuition is explained: “As a requirement of teaching so that students form symbols and concepts on the basis of vivid perception, objects, phenomena studied of the objective world or its representations” 2.4.2 Logical thinking Because the objects of the SI are relationships and relationships, the mathematical laws need to be verified for correctness and falsehood Therefore, SI needs to be associated with logical thinking According to M.Iu Koliagin: “Logical thinking is characterized by the ability to derive consequences from given premises, the ability to thoroughly separate individual cases, the ability to predict specific results by theoretical means, to sum up generalize the results obtained” Like other types of thinking, logical thinking also has its fulcrum from vivid intuitions 2.4.3 The relationship between spatial imagination and intuitive thinking According to M.Iu Koliagin: “Intuitive thinking is characterized by the absence of clearly defined steps It tends to perceptively reduce the whole problem at once One can get a “right” or “wrong” answer Nowadays, the development of intuitive thinking has attracted many progressive math educators When they talk about the role of intuitive thinking in teaching mathematics, Academician A.N Konmogorov of the Russian Federation wrote: Everywhere It was possible that mathematicians tried to make the problems to be studied by geometrically intuitive everywhere, Geometric imagination or it is said “Geometric intuition plays an enormous role in the study of almost all areas of mathematics, even abstract problems.” From the above considerations, we believe that the SI is a field of geometrical imagination and since then it belongs to the category of intuitive thinking 2.4.4 The relationship between intuition and spatial imagination 2.4.4.1 The First mode From the above analysis, we can see that the transition from visual to SI is done through the symbols of spatial memory The symbol of spatial memory is the product of direct perception of objects, phenomena, processes or their actual visual perception In this case, one of the primary tasks of using visual materials is to form specific symbols in the student's memory Through consciously repeatedly perceiving to learn the properties of objects, the student acquires symbols of memory From the symbols of this memory, through speculative and logical activities to build new spatial symbols, that is the SI 2.4.4.2 The second mode Intuition is also used as a fulcrum for different thinking operations to map out the essential properties of objects, regular relationships between objects, processes and phenomena, thereby forming knowledge in general and knowledge of spatial relationships in particular When they have a fluent knowledge of spatial relationships, students can construct new spatial symbols through indirect perception of the material 2.4.5 The relationship between logical thinking and spatial imagination Because the results of the SI in teaching mathematics are hypothetical statements about the relationships and relationships of spatial objects that need to be tested To confirm that the SI is correct, we need to prove it by logical reasoning, using logical thinking On the contrary, from logical thinking to help the subject dominate mathematical knowledge, this knowledge is the basis for good SI From the analysis of the above studies, we come up with a diagram of the relationship between intuition, logical thinking, Spatial imagination and knowledge as follows (Figure 2.1): Intuition Logical thinking Spatial imagination Knowledge Figure 2.1 The above diagram is made from the analysis of the relationships between the elements: Intuition, SI, thinking, especially logical thinking The diagram highlights the following relationships: - Intuition is the fulcrum for thinking in general, and logical thinking in particular On the contrary, from abstract thinking, in particular, high-level logical thinking is illuminated by richer intuition - Through intuitive construction of space symbols, thereby building new symbols SI On the contrary, if the person has a good SI, the visual perception will be richer and more profound - Because products of SI can be right or wrong, they need to be verified by logical thinking On the contrary, if there is good logical thinking, the hypotheses proposed by the SI have a scientific basis - Thanks to the knowledge of rich spatial relationships that allow students to introduce new spatial symbols indirectly - that is the SI On the contrary, thanks to the propositions that are products of the SI, they will become new knowledge through logical verification - Komensky asserted that: "There will be nothing in the mind about things that have not been perceived before intuitively" This statement means that acquired knowledge begins with visual objects, begins with perception On the contrary, if there is abstract knowledge of spatial relationships, it allows a richer visual perception From the above analysis, it can be seen that the focus of teaching geometry in high schools is to well solve the dialectical relationships between the elements mentioned in the diagram above 2.5 Activities towards the formation and development of spatial imagination in teaching geometry in high schools Based on the results of the overview research, especially the conclusions from the overview research related to the activities oriented to the formation and development of the SI; on the basis of the constitutive features of the SI drawn from the definition of SI mentioned in section 2.4 and based on the analysis of the dialectical relationships between intuition, symbols, mental retardation, logical thinking and knowledge, we give Outlining the following component activities to form and develop SI: Activity 1: Observing and perceiving geometric models taken in practice, representations of spatial figures with the aim for students to analyze, compare, synthesize, generalize, and abstract intuition, thereby obtaining correct representations of geometric figures - their constituent elements and their relationships Since then, SI has been formed for students 11 general, geometry in particular Every concept, every geometric property, no matter how abstract, finds its image and application in practice This is the basis for students to apply knowledge of Geometry and Measurement to life, thereby it helps to develop spatial imagination and logical thinking 2.7 Teaching situation in the direction of supporting the development of spatial imagination 2.7.1 Teaching situation As a fulcrum for this thesis, we choose the following definition of teaching theory by Phan Trong Ngo: “Teaching situation is a situation in which there is a delegation from the teacher This is the process by which the teacher puts the content to be conveyed in the events of the situation and structures the events so that it is suitable for pedagogical logic, when the learners solve it, they will achieve the teaching goal" 2.7.2 Teaching situation in the direction of supporting the development of spatial imagination in teaching geometry at high schools In order to build a definition of the concept of teaching situation to support the development of SI, we mainly rely on the issues explored in the following thesis: a, The key activities for the formation and development of SI in teaching geometry in high schools have been considered in Section 2.5 b, The role of SI in mathematics education in section 2.6 c, Definition of teaching situation mentioned in section 2.7.1 2.7.2.1 Definition: The geometry teaching situation supporting the development of the SI is a teaching situation that contains activities that promote the development of the SI, students need to interact, discover and practice in order to visualize hypotheses about new knowledge and logical steps to solve problems, orienting formal logical arguments in order to acquire knowledge according to the goal of mathematics education 2.7.2.2 Analyze the role of definition The above-mentioned situational teaching helps to foster students: - To develop students' SI through discovering and solving a math problem - To enhance the ability to imagine to visualize judgments, hypotheses and problem-solving arguments, suggestions for problem-solving presentation logic - To fostering the way of finding knowledge, attaching importance to the process of experiencing knowledge discovery - To help students understand the meaning of knowledge Conclusion of chapter - The characteristic components constituting the SI is clarified through the overview study of the works of domestic and foreign authors related to the dissertation Researching the definitions of the SI that can be used in mathematics education and exploiting the inheritance of the definition of the SI Because these definitions are emphasized on psychological characteristics, we have been interested in exploring the key elements, which are characteristic of the concept of SI Typical of these elements are shown in the works of cited authors, such as: A.N Konmogorov, M.Iu Koliagin, Tran Thuc Trinh, Pham Van Hoan, Nguyen Gia Coc, Dao Tam, Le Thi Hoai Chau, Vu Thi Thai, - To elucidate the role of TTKG in mathematics education, we have approached the research: The nature of teaching geometry by A.D Alexandrov In addition, to see more roles of SI, we study to elucidate the dialectical relationship between intuition, SI, 12 logical thinking and geometrical knowledge Determining this relationship, in addition to the above purposes, it also sheds light on the way of fostering mental health through fostering related elements - From giving a definition of SI characterized by 11 component possibilities, we consider the role of SI in mathematics education The above research issues are the basis for giving typical elemental activities to form and develop the SI - In the theoretical approach, we have presented teaching situation This issue and related contents, especially the typical elemental activities, are the basis for giving teaching situation in the direction of supporting the spatial imagination development in teaching geometry in high schools The implementation of teaching situation to support the development of SI is carried out through the use of teaching methods of discovery and problem solving, and constructivist teaching, and self-study teaching methods Chapter PRACTICAL SURVEY 3.1 Purpose of the survey a, To find out the teacher's perception of the following issues: Concept of space, spatial imagination, elements of SI; Their understanding of the approach to the SI; activities towards the development of SI; the role of the development of SI in teaching geometry and in practice for high school students; The symptoms of SI, b, Student's SI ability: The ability to grasp the meaning of geometric problems before starting to solve problems; the ability to intuitively recognize the results of spatial problems; the ability to understand the relationships and the relationships between spatial shapes; the ability to perceive geometrical ideas contained in practical situations 3.2 Survey content Exploiting ideas on education of SI in the content of the current textbook program and the innovative program of teaching geometry at high schools in the future Using exercises and problems in textbook that require to solve by using the SI; Exploiting situations in internal geometry as well as in practice plays a role in developing students' SI 3.3 Survey tool - Use multiple choice questions for teachers The basis for choosing these questions is the theory and practice of teaching in high schools in the direction of approaching the development of SI for students These facilities were presented in Chapter - Attend lessons on geometry by experienced teachers to learn the role of teachers in the formation and development of SI for students - Interview experienced teachers about teaching methods in the direction of developing SI - The questions are raised to achieve the survey objectives - Provide questionnaires; problems in order to observe students' activities in the direction of developing mental retardation for students These problems are either taken from the textbook program or designed to match the students' perceptions 3.4 Survey organization - Have surveys on 30 teachers in Nghe An province at high schools including: schools in mountainous areas, 02 schools in the plains, 02 schools in the city - Attend schools in Vinh City to find out the activities of students and the 13 fostering ability of teachers of high schools in Nghe An province - Interview experienced high school teachers about the concepts and expressions of students and about the role of the development of the SI 3.5 Survey of students - Survey tool: Provide 05 problems and questionnaires, instructions to find out the ability of students' SI according to the current textbook program - Organization and implementation: We conducted the students activities to solve math problems according to the questionnaire of groups of students of grade 11 and grade 12 of high schools in Nghe An province including: Huynh Thuc Khang High School; Ha Huy Tap High School; Do Luong High School; Anh Son High School; Quy Hop High School; Ky Son High School 3.6 Evaluation about the results of the survey on the teachers and the students This assessment aims to draw conclusions about teachers' perceptions, their activities on fostering spatial awareness for students, and at the same time, the advantages and disadvantages of teachers in terms of spatial imagination in teaching geometry in high school Through the activity of solving mathematical problems related to the SI in order to assess the students' ability of the SI, it is the basis for designing and using teaching situations to support the development of the SI of the students in teaching geometry at high schools 3.6.1 Sub-conclusion of the survey on the teachers We conducted a survey to find out teachers' perception of the SI in order to supplement the theoretical research and propose the types of activities towards the formation and development of the SI as presented in Chapter The logic of this inquiry in the order that it proceeds is as follows: To find out the teacher's perception of space symbols, the SI, the path of formation and development of the SI; the expression of SI in teaching geometry at high schools; to find out the teacher's perception of activities that can be exploited to practice for students in order to form and develop SI for students; The role of SI in effective teaching of geometry at high schools Through the results of the survey, qualitative and quantitative analysis, it is necessary to practice for students activities to form and develop SI This requires not only theoretical but also practical aspects of teaching geometry today This necessity does not only arises for the role of the SI in teaching geometry, but it is also for its role in integrated teaching, the ideology that the new program now places the leading position in teaching mathematics education in high schools 3.6.7 Sub-conclusion of the survey on the student - The students have not been not equipped with knowledge to explore and explore different definitions of geometric shapes - The students have not been able to regularly apply various concepts and properties of spatial shapes and in different chapters of geometry - The students have not focused on knowledge of mathematical modeling to describe the properties of the relative positions of shapes as well as about the geometrical quantities present in practice For example, the problems of estimating geometrical quantities in space as well as explaining phenomena in practice related to the student's SI Conclusion of chapter 14 The survey results showed: a, It is necessary to inculcate for teachers an understanding of the characteristics of SI These characteristics have been mentioned in Chapter The SI has two levels: Level 1: On the basis of existing knowledge, deeply practiced, the learners can visualize the relationships and positions between shapes, shapes, quantities and relationships between quantities At level 2: On the basis of being equipped with space symbols, the learners can create new space symbols through their imagination At this level, it requires the SI to move from intuition by observing geometric objects through the use of visual models: representations, real representations, dynamic visualizations, From there, the learners continue get memory symbols through the use of mental manipulations applied in geometry b, When the teachers have the specific characteristics of the SI, they need to know the specific manifestations of the SI, the types of activities to form and develop the SI Especially, the teachers need to know the knowledge and methods to practice for students the activities to form and develop the SI c, Through experiential activities, the teachers need to come up with situations containing activities to help students interact to discover new knowledge and thereby develop the SI It's particular to emphasis here is on the practical situations that are meaningful for the development of the SI Chapter DESIGN AND USE THE SPATIAL GEOMETRY TEACHING SITUATIONS IN THE DIRECTION OF THE DEVELOPMENT OF SPATIAL IMAGINATION FOR HIGH SCHOOL STUDENTS 4.1 Prepare knowledge and skills for teachers on designing a teaching situation in the direction of supporting the development of spatial imagination 4.1.1 In terms of knowledge The teachers need to understand the role of the development of the SI in detecting problems and solving geometrical problems It means that the situations are designed to be set up so that they contain the knowledge for the elucidation of that knowledge by using teaching methods to discover and solve the problems SI also helps to orientate hypotheses, then it conducts testing of that hypothesis whose steps are visualized by learners' SI This shows the situations that can be installed in the process of implementing constructivist theory when teaching geometry In addition to that, the teachers also need to be aware of the activities that need to be practiced for students to develop their SI and they can be installed in self-study situations for students From there, the teachers need to know the knowledge related to teaching methods of self-study Also in the field of knowledge, the teachers need to understand the SI The basic characteristics of SI; the activities that need to be practiced for students to develop SI Understanding these activities is the basis for installing them in the teaching situation of geometry: concept teaching situation, rules teaching situation, theorems and solving geometry exercises teaching The above-mentioned theoretical issues have been presented in Chapter In addition, in terms of knowledge, the teachers also need to have a deep understanding of the relationship between intuition, imagination and logic in teaching geometry This 15 proves that the potential for development of SI in teaching geometry requires teachers to exploit The following is an equally important issue that should be emphasized when we design the teaching situations for the students to interact with in order to form and develop the SI: The spatial symbols are not only in the category of visual objects science, relationships between geometric objects, relative positions between geometric figures, magnitude, size between shapes, but the symbol of space is also present in life, we call it are samples The interest in the above-mentioned specimens is useful for exploring practical situations for students to imagine, generalize and abstract in the process of using modeling to discover knowledge The process of activities mentioned above contributes to the development of SI for students 4.1.2 In terms of skills - The ability to determine the goal of a lesson - The ability to explore and detect situations taken internally in mathematics or in practice contains activities that need to be practiced for students in the direction of supporting the spatial imagination developmen - Lesson research skills in the direction of exploring situations, experimenting, and interacting with other teachers to choose the optimal situation - Carefully prepare teaching skills according to selected situations 4.2 The process of designing and using teaching situations to support the development of spatial imagination in teaching geometry in high schools 4.2.1 Scientific and practical basis for sequential design steps - Clarify the teaching objectives of the lesson content, including knowledge, skills, and components of competence as required by the geometry textbook program in high schools - Clarify the activities that need to be practiced in order to develop the SI through each lesson content Typical activities in teaching geometry such as: Observing the representations of objects and spatial relationships; analysis, synthesis, abstraction and generalization to form correct symbols of geometric objects and relationships; the activity of forming judgments and hypotheses thanks to the SI; the activity of visualizing relationships, relationships, and logic to suggest deductive steps in problem solving; Associative activities to transform images, transform associations, transform information to receive new knowledge - On the basis of clarifying the goals of the above activities, the teachers need to experience and explore situations in order to integrate activities into situations so that students can interact to discover problems and solve problems - When there are situations, the teachers need to conduct lesson research activities to choose the optimal options Such activities, for example: Organize discussions in groups of teachers, exchange with experienced teachers - Experimenting with selected situations through implementing activities to approach problem detection and problem solving for groups of students with questionnaires and instructions from the teachers To conduct this activity, the teachers need to perform observation activities, guide discussions, record audio, and take pictures to understand the behavior of students in the process of detecting and solving problems - The teacher's handling of feedback from students to correct and overcome the disadvantages of the situation for knowledge acquisition activities, activities towards the 16 development of the SI Thereby confirming the design process to apply in teaching geometry 4.2.2 The process of designing teaching situations in the direction of supporting the development of spatial imagination in teaching geometry at high schools Step 1: Research the objectives and content of the lesson and exploit the activities towards the formation and development of the SI to promote cognitive activities in order to acquire knowledge and skills according to the objectives and content of the lesson Step 2: The teacher explores geometry teaching situations that support the development of SI and promotes cognitive activities through specific lessons Step 3: Choose a teaching situation that is suitable to the chosen goal and method Step 4: Discuss and adjust the teaching situation in the direction of studying the lesson Step 5: Experiment with teaching situations Step 6: Confirm the teaching situation 4.2.3 The process of using the designed situations in teaching geometry in high schools in the direction of supporting the development of spatial imagination 4.2.3.1 General thought The idea of this application process is to define the steps to organize teaching according to the designed situations mentioned in Section 4.2.2 The above thought is concretized according to the sequence of steps of the organization of teaching, including the transfer of cognitive tasks to students, the teachers' controlled activities, and the students' activities in the direction of practicing components activities towards the formation and development of SI In addition to the above issues, the process of applying the designed situations needs to master the teaching methods mentioned in section 2.9 of the thesis, these teaching methods help guide the control activities of the teachers that aim to promote the active learning process of students In the process of applying teaching situations, it is necessary to clarify the assessment activities, institutionalize the knowledge found by students, especially attach importance to the assessment of the level of component activities towards the development of the student's the SI 4.2.3.2 Steps of the process Step 1: Identify an active teaching method compatible with the specific objectives and content of the lesson in teaching geometry in high schools Step 2: Transfer cognitive tasks - The situation has been designed to create cognitive needs for students Step 3: The teacher's controlled activities are aimed at directing students into cognitive activities, practicing component activities in the direction of supporting the SI development Step 4: Students' activities aim to interact with situations, experience, make judgments, detect problems, testing hypotheses, and solve problems to confirm knowledge Step 5: Assessment activities, institutionalize knowledge of teachers, assign selfstudy tasks to students 4.3 Apply the design processes and use situations in teaching typical situations in teaching geometry at high schools in the direction of developing spatial imagination Based on the 6-step process of the design and the 5-step process using the above 17 teaching situations, we have designed and used a number of teaching situations to support the development of SI in teaching pictures in high schools through teaching activities of concepts, theorems, rules, solving math problems Especially, while we are applying the steps of the design and the application process above, we not mention the steps of the process but we emphasize the following main issues: - We are interested in exploiting the goal of developing SI through each lesson on teaching concepts, theorems, and rules, we also concretize through the component activities of activities aimed at developing SI that have been introduced in Section 2.6, Chapter These activities are installed in situations designed to transfer cognitive tasks to the students, they are created for the students to operate, experience, and discover knowledge This is an important step of the design process that is indispensable for teachers' teaching activities - We consider activities to form and develop SI in teaching concepts; theorem teaching; rules and teaching math problem solving - For example, in teaching the concepts, we are interested in a number of component activities in the activities towards the development of the SI as follows: Activity 1; 2; 3; 4; 8; 11; 13 In teaching theorems, we pay attention and emphasize some of the following activities: Observation activities, through surveying individual cases taken internally in mathematics, are more likely to develop SI for students to experience, to analyze, to compare to generalize, to discover the rules, the discover theorems and rules Since then, we focus on problem solving stages by using imagination to suggest a formal presentation to solve the problem In the problemsolving stage, we attach great importance to the activity of separating the flat parts related to the problem from the spatial figure to convert to the plane problem This activity has the meaning of fostering the SI; because the students need to visualize the flat parts constituting the spatial shape When they move flat parts out for study, its representations need to be changed to accommodate planar geometry - When we approach the situation of solving math exercises, we emphasize the following component activities of activities aimed at supporting the development of the SI: + Activities of exploiting shapes with similar properties to convert space problems to planar problems + The operation of separating flat parts from the spatial figure to convert the spatial problem into a combination of planar problems For example, the problem of finding the center and radius of a sphere can be reduced to the problem of finding the center and radius of the great circle by considering a plane passing through the center of the sphere + The activity of transforming and associating one picture to another + Activities to transform spatial problems into plane problems that are familiar to the students, they have become acquainted with in middle school and the first part of grade 10 - In the control activities of teachers, we are interested in the questions and the orientations to support the development of the SI such as activities: + Visualize spatial shapes + Associative activity, to convert properties from one shape to another in order to solve simpler problems This activity contributes to supporting the development of SI for 18 the students + Similarization between plane and space shapes For the following, we will concretize the above activities through implementing the design process and applying it to teaching typical situations that are: Teaching concepts, theorems, solving geometry exercises Applying to teaching and solving geometry exercises The design of teaching situations to support the development of SI for the students in teaching geometry problem solving in high schools has different characteristics compared with teaching situations of concepts and theorems because the teaching of concepts, theorems are often organized for the students in an inductive way, but teaching and solving math problems have the main goal of consolidating, applying, and deepening knowledge The characteristics of the design of situations in teaching and solving math problems are mainly shown as follows: - Select problems in order to solve them has the opportunity to practice the component activities of fostering activities of SI for the students Such problems aim to deepen the elements of spatial shape, position relationships between shapes and quantity relationships in space (Length, angle, perpendicular, area, volume, ) Problems that require the perception of space in terms of an aggregate model, a vector model, or a spatial model described by coordinates - Select the practical situations to ask the students to explain and describe by using mathematical models, using language and geometric representations to develop SI for the students Example 4.1 Considering the practical situation: Please observe Figure 4.1, can you describe the lifting and lowering equipment in construction work? Use your mathematical knowledge why when two equal steel bars are attached together in the shape of an X by an axis passing through the midpoints of the two bars, the ends move on two horizontal supports that can be raised Lower objects up and down? To explain this, you should use language and mathematical notation to describe the above lifting device B A I D C Figure 4.1 Figure 4.2 - The teacher asks the question: Which object in mathematics two equal steel bars attached to the axis in the middle relate to? To answer this question, it requires the students to visualize, imagine, think about 19 the two diagonals of the rectangle ABCD (Figure 4.2) - From the lifting device, please tell me in the rectangle representing it, which elements are fixed and which ones change? To answer this question, it requires the students to consider constant quantities, variable quantities: Length of diagonals AC = BD = d (Const), and dimensions AD = x; AB = y variation Then, using the Pythagorean theorem, we have the relationship between the diagonals and the sides: d = x + y (1) - From the equation (1), can you clarify when does the lifting device lift the object to a high floor? To answer this question, it requires the students to visualize and imagine space: When we raise, the endpoints A, B move closer together, then from the equation (1) inferred, when x is smaller the larger y is since the sum of the squares x + y that does not change It can be said that it is the mathematical principle used through the equipment applied in practice Conclusion of Chapter In this chapter, we have given a 6-step design process for teaching situation in the direction of developing SI The process is designed based on the following pedagogical scientific ideas, these scientific bases have been presented in Chapters and 3: - The first emphasized thought is that: Knowledge, skills, thinking as well as correct symbols of space are formed and developed through interaction of the subject with situations and environments The content of the interaction includes observing, comparing, analyzing, synthesizing, generalizing, abstracting, hypothesizing, testing them, - The teaching situations are in the direction of the development of the SI that are the place to install the component activities of the development of the SI for students - The design of situations in use should be compatible with the cognitive characteristics of students, they are accepted situations that must have fulcrums from surveying practical activities, interacting with situations of students, focusing on the component activities of the activities towards the development of the SI The survey of practical activities through the design of questionnaires and instructions of teachers - The situations are discussed through studying the lessons of teachers by subject groups, with the participation of teachers who have experience in teaching geometry in high schools In this chapter, we give a 5-step process to apply to teaching geometry These steps represent selected active teaching methods to orient the control activities of teachers and promote active and self-disciplined learning activities of students Students' learning activities are shown through interaction with situations in order to form and deepen symbols of space The fostering activities of the SI are shown through two levels: the level of deepening the existing symbols and the level of forming new symbols through imagination and generalization 20 Chapter PEDAGOGICAL EXPERIMENT 5.1 Objectives of the experiment - To assess the ability of math teachers in implementing the steps of the process To evaluate the ability to study the lesson of the selected group of teachers; To be mainly interested in debate activities, critical thinking when choosing situations to use - To assess the advantages and disadvantages of students in performing activities towards the development of SI through the interaction with the designed situations mentioned in Chapter 5.2 Experimental content - For teachers We assign tasks to 02 groups of teachers from 02 high schools in Nghe An province: Le Loi High School, Tan Ky district; Dien Chau high school, Dien Chau district; Each group consists of 10 teachers Specific tasks are as follows: Mission 1 Give the process of designing the situation towards the development of the SI for the students when you teach the theorem about the intersection of three planes, Geometry textbook 11, page 57 This process consists of 06 steps; The subject of the experimental teacher has approached first Clarify the process of applying the design case mentioned in part 1: State the steps, clarify the meaning of each step Mission Clarify the student's activities that are meaningful to the practice of SI when you explain the following practical situations by using mathematical knowledge: Figure 4.1, is an image of the actual representation of a device "Raising and lowering" in construction work that brings the equipment up and carries the objects down Please tell us what mathematical ideas are used when lifting equipment and when lowering objects? Use the design process to support the development of SI for the students, lease give an example of designing such a process when teaching the concept: "The center of symmetry of a shape" in lesson "Central symmetry" Geometry textbook 11, page 12 - For the students The experimental content for the students mainly present the situations mentioned in Chapters 2, 3, for the students to interact, experience, make judgments, detect problems and visualize the steps to solve problems to acquire the knowledge The students interact with situations through the teacher's questionnaire and instructions Mission Figure 5.1 - The representation of a football field and Figure 5.2 is a mathematical model of a football field, rectangular in shape A E D I B Figure 5.1 F Figure 5.2 C 21 On the mathematical model of the football field ABCD, there is a rectangle with the mean line EF I is the intersection of EF and the diagonal AC Please clarify the following questions: - Prove that I is the center of rectangle ABCD Let M be certain point on the side of the rectangle Prove through center symmetry I, the image of point M is point M' also on the side of the rectangle above? - The same argument creates an image of the point M' that M is also on the side of the rectangle? - Prove that if point M lies inside a rectangle, then its image M' through central symmetry I also belongs to the rectangle? Let's argue that if image M' is inside the rectangle, make its image through center symmetry I also inside the rectangle? Mission Given a circle (C) with center O Show that O is the center of symmetry of circle (C) - From the two situations mentioned above, give the definition: Point I is the center of symmetry of the figure H? - Give an example to show that if I is the center of symmetry of the figure H, then I can belong to H or not to H? Mission In the orthogonal coordinate system (Oxyz) given two lines d: and d': and the point Find the equation of the line that passes through and intersects the two lines d; d' Mission Estimate the mass of a cylindrical steel pipe with a length of m, a cross-sectional diameter measured on the outside surface of cm, and a cross-sectional diameter measured on the inside surface of 5.4 cm The density of steel is 7,850 kg/ Mission Are there four double straight lines that cross each other and are perpendicular to each other? We assign cognitive tasks to 04 groups of the students, each group of 08 students in 02 high schools in Nghe An province are: Le Loi High School; Dien Chau High School 5.3 Experimental form We conduct practical research by the way of case study, by testing one or a group of subjects on the perception and expression of SI in teaching geometry in high schools During the experiment, we recorded and recorded the exchanges and discussions of the teachers and the students 5.4 Experimental organization The experiment was conducted by the researcher himself with 02 math team leaders of two high schools for teachers: Le Loi High School, Dien Chau High School, Nghe An province When we conducted the experiments for the students, we agreed to ask for the help of the math group teachers: Mr Nguyen Ngoc Hoang organized the group experiment of class 11A1 (08 students); Mr Bui Van Duc, organized the group experiment of class 12A3 (08 students) of Le Loi High School; Mr Thai Doan An, organized the group experiment of class 11A1 (08 students); Ms Vo Thi Bich Ha, organized the group experiment of class 12A1 (08 students) at Dien Chau high school The teachers conducting the experiment have studied the following contents: - The characteristics of the SI in teaching geometry in high schools - Activities that need practice to develop SI - The process of designing situations to organize teaching in order to develop SI 22 for the students; the process of applying the situations that have been designed in teaching in order to develop the SI for the students 5.5 Evaluation of experimental results 5.5.1 A priori analysis The a priori analysis essentially predicts the results of answers, explanations, and performance results of the teachers and the students through assigned tasks in the experiment 5.3 Experimental form We conduct practical research by the way of case study, by testing one or a group of subjects on the perception and expression of SI in teaching geometry in high schools During the experiment, we recorded and recorded the exchanges and discussions of the teachers and the students 5.4 Experimental organization The experiment was conducted by the researcher himself with 02 math team leaders of two high schools for teachers: Le Loi High School, Dien Chau High School, Nghe An province When we conducted the experiments for the students, we agreed to ask for the help of the math group teachers: Mr Nguyen Ngoc Hoang organized the group experiment of class 11A1 (08 students); Mr Bui Van Duc, organized the group experiment of class 12A3 (08 students) of Le Loi High School; Mr Thai Doan An, organized the group experiment of class 11A1 (08 students); Ms Vo Thi Bich Ha, organized the group experiment of class 12A1 (08 students) at Dien Chau high school The teachers conducting the experiment have studied the following contents: - The characteristics of the SI in teaching geometry in high schools - Activities that need practice to develop SI - The process of designing situations to organize teaching in order to develop SI for the students; the process of applying the situations that have been designed in teaching in order to develop the SI for the students 5.5 Evaluation of experimental results 5.5.1 A priori analysis The a priori analysis essentially predicts the results of answers, explanations, and performance results of the teachers and the students through assigned tasks in the experiment 5.5.2 Post-analysis of the results of solving tasks for the teachers and the students 5.5.2.1 Analysis of experimental results for the teachers During the experiment, we observed the spirit and attitude of the teachers when they answered the missions The teachers also expressed their interest in answering questions However, their understanding about the SI is incomplete; especially they not fully grasp the activities towards the development of SI Since then, solving the tasks set for the teachers also reveals some weaknesses due to the lack of knowledge and skills preparation for designing situations and applying them 5.5.2.2 Analyze and evaluate the students' experimental results In the experimental test, we paid the special attention to the practical situations containing activities towards the development of the SI: The activity helps the students imagine and locate the points that are the images of the points lying on the edge of the football ground; The image of the points lying on the football ground through the symmetry of the center is the tee of the ground (On the mathematical model, the center of symmetry is the intersection of two diagonals of the rectangle) In the tasks assigned to test experimentally, the problem solving in those tasks is 23 elucidated through the results of SI For example, the students need to imagine to calculate the volume of a rotating cylindrical steel pipe, as the basis for estimating its mass Conclusion of chapter - Assessing teachers' understanding (Through case studies for two groups of teachers, each group of 10 teachers from Le Loi High School, Dien Chau High School, Nghe An province) on the process of designing situations towards Formation and development of SI for high school students This process consists of steps with the requirement of teachers to explain and clarify the content of the steps, to clarify the activities installed in the situations that are meaningful to the development of the SI in teaching geometry It requires to test teachers on their awareness of the steps of the application process In which, it specially appreciates the role of steps for the development of SI for the students - The experimental test for the teachers is focused on the tasks of concretizing the steps of the design process and the application process in teaching concepts, theorems, solving geometry exercises in the direction of supporting development SI for the students in teaching geometry in high schools - In the experimental test for the students, we are interested in installing the following activities into situations designed by the teachers and the situations where the process is applied to teaching concepts, theorems, and math exercises - Some assigned tasks require a high level of SI To solve those tasks requires an imagination to associate with logical thinking, with descriptive intuition, modeling activities taken from real situations - The empirical assessment is mainly through qualitative analysis, observing students' behavior, listening to students' discussions, recording, and communicating activities to see the expression of students' efforts, interest in doing assignments and Refer to the results through the quality of the answers, the quality of solving specific problems Conclusion of chapter - Assessing teachers' understanding (Through case studies for two groups of teachers, each group of 10 teachers from Le Loi High School, Dien Chau High School, Nghe An province) on the process of designing situations towards Formation and development of SI for high school students This process consists of steps with the requirement of teachers to explain and clarify the content of the steps, to clarify the activities installed in the situations that are meaningful to the development of the SI in teaching geometry It requires to test teachers on their awareness of the steps of the application process In which, it specially appreciates the role of steps for the development of SI for the students - The experimental test for the teachers is focused on the tasks of concretizing the steps of the design process and the application process in teaching concepts, theorems, solving geometry exercises in the direction of supporting development SI for the students in teaching geometry in high schools - In the experimental test for the students, we are interested in installing the following activities into situations designed by the teachers and the situations where the process is applied to teaching concepts, theorems, and math exercises - Some assigned tasks require a high level of SI To solve those tasks requires an imagination to associate with logical thinking, with descriptive intuition, modeling activities taken from real situations 24 - The empirical assessment is mainly through qualitative analysis, observing students' behavior, listening to students' discussions, recording, and communicating activities to see the expression of students' efforts, interest in doing assignments and Refer to the results through the quality of the answers, the quality of solving specific problems - The conduct of qualitative analysis to evaluate the results of the teachers and the students based on a priori and a posteriori analysis Through empirical research, especially analysis of experimental results, it has been shown that the thesis has answered the research questions a, b, c, e: + What theoretical and practical basis does it base on to give the concept of weight loss? + How is the SI expressed in teaching geometry in high schools? How to detect those symptoms? + What are the main activities that need to be practiced to develop the SI for the students? + What are the levels of development of the SI of high school students in teaching geometry? Most of the activities towards the development of SI have been demonstrated by the students through solving the tasks in this experimental part Through experiment, it can be concluded that most of the fostering and development of the SI for the students is the deep exploitation of level of the SI Thus, the scientific hypothesis of the thesis has been tested and the research task of the dissertation has been completed CONCLUSION The dissertation has achieved the following main results: Synthesize theoretical issues from the educational aspects of mathematics, psychology, philosophy to serve as a scientific pedagogical basis for clarifying key concepts in the thesis: SI; the relationship and the relationship between the SI, intuition, logic, knowledge; typical elements of the SI; activities towards the development of the SI Give the concept of the SI characterized by 11 possibilities It can be pointed out that there are 02 levels of development of the SI of high school students in teaching geometry The research and the design survey questionnaires on students and multiplechoice questions to survey the teachers to reveal manifestations of activities that are compatible with the characteristics of the SI for students Provide 13 main activities to practice for the students in order to support the development of the SI The role of fostering mathematical thinking for the students in teaching geometry in high schools Research and survey to clarify the difficulties and the limitations of teachers and students in teaching and learning pictures in high schools in the direction of developing the SI Give the 6-step design process for teaching situation and a 5-step process for applying the designed situations to teaching geometry in the direction for the development of the SI Specific situations have been designed to be taken internally in mathematics, especially to find out situations taken from samples taken in real life in order to develop mathematical memory for high school students in teaching geometry Organize a pedagogical experiment to examine the feasibility and effectiveness of the design processes, the process of applying the designed situations to teaching geometry at high schools in order to support the development of the SI Publish scientific articles related to the thesis topic 25 LIST OF PUBLICATIONS Dau Anh Tuan, Ngo Tat Hoat (2016), The capacity of students to organize cognitive activities needs to be fostered for Math students at pedagogical colleges, Today's Journal of Teaching and Learning, June 2016, pp 17 - 18 Dau Anh Tuan (2016), Design and use teaching situations to support the development of spatial imagination for students in teaching geometry at high school, Today's Journal of Teaching and Learning, May August 2016, pp 69 - 72 Dao Tam, Dau Anh Tuan (2018), Spatial imagination and its role in mathematics education, Vietnam Journal of Educational Science, No 02, 02/2018, pp 50 - 54 Dao Tam, Dau Anh Tuan (2019), Exploiting the relationship between intuition, spatial imagination and mathematical thinking in teaching geometry at high schools, Scientific Journal of Ha Noi University of Education, Vol 64, No 9, September 2019, pp 158 - 164 Dau Anh Tuan (2020), Research overview on spatial imagination in teaching geometry in high schools, Scientific Journal of Vinh University, Vol 49, No 3B/2020, pp 70 - 78 ... 12 of high schools in Nghe An province including: Huynh Thuc Khang High School; Ha Huy Tap High School; Do Luong High School; Anh Son High School; Quy Hop High School; Ky Son High School 3.6 Evaluation... on 30 teachers in Nghe An province at high schools including: schools in mountainous areas, 02 schools in the plains, 02 schools in the city - Attend schools in Vinh City to find out the activities... Research and design questionnaires to survey students and multiple-choice questions to survey teachers, attend geometry classes in high schools This research activity aims to reveal the manifestations