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MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF EDUCATION NGUYEN TIEN TRUNG DESIGNING GEOMETRY TEACHING SITUATIONS AT HIGH SCHOOL FOR HELPING STUDENTS CONSTRUCT NEW KNOWLEDGE Major: Theory and methods of teaching Mathematics Code: 62. 14. 01. 11 SUMMARY OF PEDAGOGICS DOCTORAL THESIS HA NOI – 2013 2 Science instructor: Prof, Dr. BUI VAN NGHI Judge 1: Prof, Dr. of Sci. NGUYEN BA KIM, Dr. Judge 2: ASS.P Assoc. Prof, Dr. NGO HUU DUNG Vu Duong Judge 3: Dr. Ngu Assoc. Prof, Dr. TRAN VUI n Van Thuan Thesis will be defended in front of Council at University: Hanoi University of Education, date:……………………. Thesis can be seen at National Library and Library of Hanoi National University of Education. 3 REFERENCES 1. Books 1 Bui Van Nghi - Nguyen Tien Trung (2009), Teaching based on standards of knowledge, skills of mathematics 12 th grade, University of Education Publishing house. 2 Bui Van Nghi - Tran Trung - Nguyen Tien Trung (2010), Teaching based on standards of knowledge, skills of mathematics 11 th grade, University of Education Publishing house. 2. Journals 1 Nguyen Tien Trung (2009), Recommendation engine in teaching the cosine theorem in the triangle (Geometry for 10 th grade), Journal of Educational Science, No 206, p. 35-37. 2 Nguyen Tien Trung (2011), Adidactic situation and mathematics teaching situation, National Scientific conference of young teacher in pedagogical universities, the first 2011, Hanoi National Univesity of Education, pp. 588-593. 3 Nguyen Tien Trung (2011), Designing teaching situation: Developing formula to calculate distance from a point to a line (Geometry for 10 th grade), Journal of Educational Science, No 275, pp. 34-35. 4 Bui Van Nghi - Nguyen Tien Trung (2012), Designing a teaching situation: the cross product of two vectors concept, Journal of Science of Hnue, Vol. 57, No. 1, pp. 3-7. 5 Bui Van Nghi - Nguyen Tien Trung (2012), Designing teaching situation: Defining plane section of puramid (original line of intersection method), Journal of Educational Science, No 290, pp. 49-51. 6 Bui Van Nghi - Nguyen Tien Trung (2013), Designing teaching situation: Defining plane section of puramid by the way of using the radial projection (the alignment method) (Geometry 11th grade, Chapter 2), Journal of Educational Science, No 301, pp. 45-47. 7 Bui Van Nghi - Nguyen Tien Trung (2013), Some opinions on design the situation of mathematic teaching in high school towards improving the positive student perceptions, Journal of Educational Science, No 312, pp. 45-47; 50. 8 Bui Van Nghi-Nguyen Tien Trung (2013), Designing a teaching situation: Developing formula to caculate the distance from a point to a plane in space (Geometry for 12 th grade, Chapter 3, Lesson 2), Journal of Science of Hnue, Interdisciplinary Science, Vol. 58, No. 5, pp. 47-52. 3. Scientitic project 1 Nguyen Tien Trung, Designing some geometry teaching situations in high school, No SPHN-10- 591NCS, Grade school Science project of Science and Technology, Hanoi National Univesity of Education, acceptance on 7/12/2012. 4 1 INTRODUCTION 1.1. REASONS FOR CHOOSING TOPIC OF THE THESIS * Approach in reasearch problems from published thesis. The thesis builds upon success of physiology, pedagogy and teaching methods applied by Vietnamese lecturers in 1960s of the 20 th century. There have been some well- known theories such as: theory of cognitive development by J.Piaget, theory of pschological activity; cultural historical theory of psychological development of high level psychology funtions by L.X. Vygotsky, theory of didactical situation by Guy Brousseau , …At that time, Vietnamese education scholars who had been experienced in teaching match made some innovations in teaching method. One of the most focused issues were how the theory of teaching method could be adapted to realities of teaching in Vietnam . Conducting research on view on action based teaching, professor Nguyen Ba Kim was interested in action based learning , [50], [49], associate professor Trinh Thanh Hai had his opinion on how to apply action learning to technology training sessions at high school (2009)[35], doctor Nguyen Huu Hau (2012) focused on finding more actions, training students and facilitating their learning [40]. Professor Nguyen Huu Chau who had dedicated himself to give insights into effective action learning emphasized the importance of enhancing students’motivated cognitive abilities [11]. Professor Dao Tam who had taken interest in classification of learning activities conducted research on constructivist teaching in action, “hoạt động biến đổi đối tượng” and cognitive activities.… [83], [85], [86], [88] . With a view to study on theories of teaching and pschological theories of Maths teaching, there have been well- know researches such as constructivist approach in learning and teaching Maths conducted by professor Nguyen Huu Chau (1996) [12], application of constructivist view on geometry teaching at high school by doctor Cao Thi Ha (in teaching solid geometry) [13], [33]. Theory of didactical situation had been main concern of many education scholars who employed it in teaching maths. There have been some few more findings of application of theory of didactical situation to teaching for example researches done by Do Thi Chau (2008) [7] by Vu Dinh Phuong (2008) [72]. From the best of my knowledge, in order to make theory of didactical situation become practical applications in Vietnam, a number of considerable adaptation would be required. 2 There has been a growing trend of application of teaching methods to math teaching. The most common issues could be raised[103], interactive teaching method or collaborative teaching [23], [59], [60], …. * Research demand characteristics: The upcoming orientation of education innovations in the 21 st century includes: renewed teaching method, surmounting single impartion of information, improved critical thinking skills and applying advanced education method to teaching, enhanced self-study Geometry which plays the most part in high school level of Maths, provides students with the basis of plane geometry and solid geometry (including knowledge of the method of coordinates in the plane and the method of coordinates in space). Moreover, geometry enables teacher to develop student’s logical thinking. Consisting largely of abstract knowledge, geometry requires teacher to take an action based teaching which aims at inspires student’s positive thinking and enriches the lessons. After years of investigating issues, we have found that a lot of attempts have been made to apply active perspective as well as constructivism and theory of didactical situations to mathematic teaching at high school. However, very few researches on teaching situations especially geometry teaching situations which we are searching for could be found. 1.2. RESEARCH PROJECT From the reasons above, we have chosen project “Designing geometry teaching situations at high school for helping students construct new knowledge”. 1.3. SCIENTIFIC HYPOTHESIS If applying the active perspective, idea of constructivism and theory of situations (the theory of tidactical situations in mathematics), we can design teaching situations in geometry at high school in which student can construct their new knowledge. 1.4. RESEARCH OBJECTIVE AND RESPONSIBILITY Research objective: Designing some specific example of geometry teaching situations at high school which inspire students and enable their constructivist learning. THDH, teaching plan and designs of THDH based on constructivist learning would be recommended. Responsibilities: 3 +) Investigation of maths teaching and learning and practical teaching +) Proving theoretical frame of math teaching situations in general and geometry teaching situations in particular at high school. +) Clarifying the viewpoint and design of geometry teaching situations based on contructivist learning at high school +) Recommending plan to design geometry teaching situations at high school. +) Designing and experimenting with some examples of geometry teaching situations, completing some geometry teaching situations at high school based on clarified viewpoints. 1.5. RESEARCH OBJECTIVE Geometry teaching situations at high school and students’s constructivist learning. 1.6. RESEARCH SCALE Content, geometry program and common geometry teaching situations at high school. 1.7. SCIENTIFIC VALUES AND FEASIBILITY The thesis have recommeded several geometry teaching situations for helping students construct new knowlegde at high school. The mentioned teaching situations that were practically designed and experimented would be possible for mathematic teachers to apply at high schools. In the thesis, author has showed his opinion on a teaching situations, plan and design of geometry teaching situations for helping students construct new knowlegde at high school. Thus, the thesis highly supports teachers who are devoted to study and design geometry teaching situations particularly and mathematic teaching situations generally in the hope of producing higher teaching quality at high school. 1.8. RESEARCH OUTLINE The thesis consits of 3 parts dividing up into 5 chapters (excluding index of abreviations, photos, figures, reference and appendix.…): INTRODUCTION RESEARCH OVERVIEW RESULTS CHAPTER 1: THE FIRST RESEARCH RESULT 4 CHAPTER 2. THE SECOND RESEARCH RESULT CHAPTER 3: THE THIRD RESEARCH RESULT CHAPTER 4: THE FOURTH RESEARCH RESULT CHAPTER 5: THE FIFTH RESEARCH RESULT CONCLUSION AND RECOMMENDATION RESEARCH OVERVIEW 2.1. DEFINITION OF TERMS 2.1.1. Learning actions 2.1.2. Teaching actions 2.1.3. Situations 2.1.4. Teaching situation Teaching situation is a circumstance in which teacher gives lesson and student learns in a session or some sessions to gain some certain teaching objectives. 2.2. HISTORY OF RESEARCH PROBLEMS 2.2.1. Reference and approach (1) Lesson study: Lesson study is aimed at enhancing pedagogical profession. It requires that the joint research on teaching and learning in the certain classes has been done systematically to improve teaching and student’s critical thinking. We intended to design highly practical teaching situations approved by high school teachers. As a result, we would be able to form theory as well as giving out suitable designs of identical teaching situations. 5 (2) Constructivism and constructivist teaching: Based on viewpoint of constructivist teaching, student gains knowledge when they was immersed in an active learning environment where they find out their problems and solving them by applying given knowledge and experience to new situation. (3) Theory of didactical situations: Through document and knowlege gained at conference on theory of didactical situations and didactic Maths, we would like to apply the idea of a learning situation structure consisting of 3 basic situations: situation of action, situation of comunication and situation of validation. We were not to study on “adidactic” presented in theory of didactical situations but to apply this kind of that situation to designing teaching situations. (4) Methods and teaching strategies: Based on suggestions of teaching strategies, teaching situations should be student centered learning which enables students to judge, to anticipate, to gain knowledge and to practise skills. In teaching situations, we always anticipate, provide opportunities and support for students to solve the problem from the their knowledge and skills. It also is a way to improve the teaching situations. (5) Thesis on geometry teaching with different approaches: Some research papers which shared the same concern were conducted by professor Nguyen Huu Chau - Cao Thi Ha [33]; geometry teaching applied mathematics by professor Bui Van Nghi [63]; Development of logical thinking in elementary geometry teaching by associate professor Vu Quoc Chung [15] and application of information technology to geometry teaching by associate professor Tran Vui [109], [110]; geometry skills building for 10 th graders by professor Pham Duc Quang [75]; application of constructivism in geometry teaching by doctor Cao Thi Ha [12]. From the reasons mentioned above, it is a certain that our thesis is practical. 2.2.2. Innovations of method of teaching mathematic 2.3. LITERATURE REVIEW 2.3.1. Active perspective 2.3.1.1. Phylosophy and pschology 2.2.1.2. Applying a ctive perspective in teaching Active perspective in teaching means that teaching is done by actions from beginning to the end. It also means students would be taught how to think and solve problems rather than learning theory only. From that viewpoint, it is 6 highly recommened that teaching situation should be designed in the way that encourages cognitive development. 2.2.2. Theory of constructivism Jean Piaget (1896  1980) was a psychologist who collaborated to conduct a well-known psychology theory named theory of constructivism. His theory mentioning the development of cognitive process in children provided a solid foundation for child care and education. Two types of constructivism in teaching can be listed here: radical constructivism and social constructivism. Individual constructivism emphasizes personal learning process. Social constructivism focuses on how factors like culture, society influence on pernal development of knowledge. From conclusion and application of constructivism to teaching process, we deeply think that teachers should follow the model of learner centered teaching in which students acts as information searchers through their activities (like signals, clauses, map and model, …). By this way, under the guidance of teachers, students can construct their knowledge. 2.2.3. Theory of didactical situation 2.2.3.1. Concepts and definition of theory of didactical situation and applications to teaching process a) Concepts b) Four scientific hypothesise about theory of didactical situation c) Concepts of theory of didactical situation d) Application of theory of didactical situation to teaching Thanks to researches on theory of didactical situation and application of theory of didactical situation to teaching mentioned as [7]; [72]; [83]; [90]; [113], we believe that, teacher needs to assign students challenging tasks which not only stimulate student’s thinking ability but also to improve their growing knowledge. This type of teaching and learning requires both teacher’s creativity and student’s activeness. 2.4. OTHER RELATED RESEARCH RESULTS 2.4.1. Teaching environment 2.4.2. Teaching methods and effective teaching methods 2.4.2.1. Teaching methods 2.4.2.2. Effective teaching methods [...]... learning from general to particular (teaching cosine theorem in the triangle, Geometry 10th grade) 1.1 RESEARCH PROCESS Process of designing the teaching situation: the cosine theorem in the triangle: Step 1: Investigating content overview, teaching objectives, anticipating advantages and disadvantages during lesson (i) The cosine theorem in the triangle (Geometry for 10th grade) plays an important... that theorem a Students: a 2  b 2  c 2 if ABC is rigth triangle at A (squared length of the hypotenuse A of a right triangle is equal to the square of the length of the two other side) b C Figure III.1.1 Teacher: What things will happen if ABC is not a right triangle? Can you calculate the length of a term b, c and angles A or not? would the formula look like? Students: Yes, we can Because a triangle... the cosine theorem in the triangle Step 3: Ask for advice on designed teaching situation from mathematic teacher Step 4: Result statistic, reference to teachers and edit designed teaching plan Step 5 If getting no more 80% of approval, return step 2, 3,4 1.2 RESULT 1.2.1 DESIGN OF TEACHING SITUATION Activity 1: Warm-up: th B Teacher: At 9 grade, students all know the Pytago theorem in right triangle... 10th graders from Nguyen Tat Thanh high school a question: When it comes to lesson of cosine theorem, how many students already know it? Results: at least 3 students each class are taught the lesson ahead of schedule We asked 80 students who did not know the cosine theorem, have not read the textbook, in 10 two of the 10 classes above the following problem: “ ABC is triangle with AC  b, AB  c and... theory of situations which includes learning situations: situation of action, situation of communiation, situation of validation Orientation 3: Designing typical teaching situations: notional teaching situation, theorem teaching situation, methodological teaching situation, theorical teaching and problem-solving teaching situation This is necessary because it is practical base for implementation, theoritical... some theorical notions relating to application of active perspective, theory of situations and constructivism to teaching math in high schools 2 Suggest and experiment successfully some teaching situations of geometry at high school These are practical references which prove theory and answer research questions 24 3 Experimental teaching (experimental procedure) is also considered as an important contribution... each group’s worksheet 11 Teacher: Expected outcomes of students: Case 1: ABC is triangle with angle A = 600: a 2  BC 2  BH 2  HC 2  AB sin 600   AC  AH   c 2  b 2  bc 2 2 Case 2: ABC is triangle with angle A = 300: a 2  BC 2  BH 2  HC 2  AB sin 300   AC  AH   c 2  b 2  bc 3 2 2 Case 3: ABC is triangle with angle A = 1200: 2 2 a 2  BC 2  BH 2  HC 2  AB.sin600  AB.cos600... (3) During lessons, student might have to cope with some difficulties : do not know how to use or contact the existing knowledge: Pythagorean theorem in right triangle Teachers should give them suggestions to revise what they are taught in previous lessons (4) Theorem can be proved by an other way For example, consider the case A acute angle or obtuse angle and prove each these cases (5) Due to spend... situation and theorem teaching Moreover, high school teachers of maths could have theorical and practical base in: implementing designed teaching situations in teaching at high school, and applying those into others in the way of constructivist learning B.3 For students of mathematic education B.4 Some further researches from research result B.4.1 Research on the application of constructivism, theory of... the content of the theorem at different levels 1.3 CONCLUSION 1.4 CONCLUSION FROM CHAPTER 1 Teaching situations presented above is an example of the organization of learning activities towards helping students construct their knowledge through active thinking generalize Teaching as this teaching situation, teacher lead their student calculate the length of a term b, c and angles A in triangle ABC for . Pytago theorem in right triangle. Presentation and writing formulas of that theorem. Students: 2 2 2 a b c   if ABC is rigth triangle at A (squared length of the hypotenuse of a right triangle. century. There have been some well- known theories such as: theory of cognitive development by J.Piaget, theory of pschological activity; cultural historical theory of psychological development of. Concepts of theory of didactical situation d) Application of theory of didactical situation to teaching Thanks to researches on theory of didactical situation and application of theory of didactical