Purposes of the study: Introduce and clarify the concept the competency of exploring and obtaining knowledge through elemental competencies and compatible activities in teaching maths for high school students; Find out the reality and need of improving the competency of exploring and obtaining knowledge in teaching maths for high schools students majoring in physics, chemistry and biology; Build and test the feasibility of pedagogy method applied in teaching maths at high school classes majoring in physics, chemistry, and biology to foster this capacity for students.
i MINISTRY OF EDUCATION & TRAINING VINH UNIVERSITY NGUYEN TRAN LAM TEACHING MATHEMATICS FOR HIGH SCHOOL STUDENTS MAJORING IN PHYSICS, CHEMISTRY, BIOLOGY IN ORIENTATIONS TOWARDS IMPROVING THE COMPETENCY OF EXPLORING AND OBTAINING KNOWLEDGE Major: Theory and Method of Math Teaching Code: 9.14.01.11 SUMMARY OF THE THESIS NGHE AN – 2019 i The dissertation is finished at: Vinh University Names of supervisors: Dr Nguyen Van Thuan Prof Dr Dao Tam Reviewer 1: Assoc Prof Dr Tran Kieu Reviewer 2: Assoc Prof Dr Vũ Vu Duong Thuy Reviewer 3: Dr Thai Huy Vinh The research will be defened at Ministerial Thesis Evaluation Committee held at Vinh University Time: , date: month The Thesis can be found at: - Vietnam National Library - Vinh University Library year 2019 INTRODUCTION Justification of the study 1.1 Nowadays the educational systems of all countries in the world have been facing challenges and practical requirements: being required to actively innovate to catch up with current changes because of the rapid development of technology, and the growth in the quantity and quality of human knowledge The distance between theory and practice is on the increase, so the task of Education and Training (ET) is said to create active students Together with the orientation towards action, the orientation of developing ability has been able to fulfill this important task 1.2 The concep “ability”, especially the term “ability of teaching maths” have been found in many researches in Vietnam and in the world However, the topic Developing the competence of exploring and obtaining knowledge in teaching maths for students, especially at high school level, has not been deeply studied In fact, there are not any studies of teaching maths for high school students oriented towards improving the competency of exploring and obtaining knowledge 1.3 The students majoring in physics, chemsitry and biology are chosen to take part in this research since practical surveys have found that these students have a habit of studying and discovering new things, and they are often interested in what they studied by themselves Moreover, the students majoring in physics, chemistry and biology are usually eager to learn maths as it is considered as a good tool to support them to studies their majors more effectively 1.4 The concept Teaching maths through activities, which can shape and develop ability, have been researched by a large number of educators Innovating teaching method towards learner-centered has been implemented in schools Intuitively, the term "the competency of exploring and obtaining knowledge" more or less expresses the ambition as well as active participation of learners in approaching and mastering the knowledge Basing on theoretical and practical basis, the paper has set requirements and facilitated the research on ability of discovering and acquiring knowledge on the basis of giving conception, proposing pedagogy methods to foster this capacity in teaching maths at high school and the gifted high schools Purposes of the study 2.1 Introduce and clarify the concept "the competency of exploring and obtaining knowledge" through elemental competencies and compatible activities in teaching maths for high school students; 2.2 Find out the reality and need of improving the competency of exploring and obtaining knowledge in teaching maths for high schools students majoring in physics, chemistry and biology; 2.3 Build and test the feasibility of pedagogy method applied in teaching maths at high school classes majoring in physics, chemistry, and biology to foster this capacity for students Scientific hypothesis Basing on the research to identify the elemental capacity of the ability of creativity and the knowledge –grasping competennce, if methods can be given to create effective teaching activities for the high school students majoring in physics, chemistry and biology, they can improve the competence of discovering and the knowledge –grasping competence Besides, it can boost the effectiveness of teaching maths at the gifted high schools Research task: The thesis will answer the following questions Questions 1: What are the viewpoints of researchers in regard to the topic of this thesis? What are the reasons to choose the topic? Questions 2: What are activities or cognitive activity defined? What are definitions of discovery activities and knowledge acquisition; discovery competence and knowledge acquisition? How can compatible activities of the elemental ability, and its levels be displayed?; Questions 3: What are characteristics of the high shcool students majoring in physics, chemistry and biology? What are the method of teaching maths suitable for encouraging these students to improve the competency of exploring and obtaining knowledge?; Questions 4: What is the reality of improving discovery ability and the competency of exploring and obtaining knowledge in teaching maths for high school students majoring in physics, chemistry and biology? What are the reasons and solutions of this issue? Question 5: How is the discovery ability and knowledge acquisition of the students majoring in physics, chemistry and biology improved after being taught maths basing suggested methods in the thesis? What are the results of pedagogical experiment? Research method 5.1.Theoretical methods 5.2 Practical methods 5.3 Experimental methods The significance and meaning of the thesis 6.1 Theoretical viewpoints 6.1.1 The paper has introduced the connotation concept of the term discovery activity and knowledge acquisition; discovery ability and knowledge acquisition as well as elemental ability and its levels in teaching maths for the high school students in general and the gifted high school students in particularly 6.1.2 The paper has given directions and six pedagogical measures to improve the competency of exploring and obtaining knowledge for the students through teaching maths 6.2 Practical viewpoints 6.2.1 The paper has built learning topics that can be used in teaching maths to improve the competency of exploring and obtaining knowledge for the students majoring in physics, chemistry and biology at the gifted high schools 6.2.2 The paper has initially tested the feasibility of the pedagogical measure built through pedagogical experiments and expected to be successful to be implemented on a large scale 6.2.3 The paper can be used as a reference for math teachers, which aims to improve the effectiveness of teaching maths at high schools and gifted high schools The contents of the paper defended 7.1 Viewpoints on improving the competency of exploring and obtaining knowledge based on elemental competencies in teaching maths for gifted high school students (these elements are really necessary and can be transfered to students in teaching maths at high schools) 7.2 The system of proposed measures in teaching maths is appropriate and feasible in training compatible activities to improve the competency of exploring and obtaining knowledge for high school students in general and high school students majoring in physics, chemistry and biology in particularly 7.3 Some developed teaching topics connect with the use of pedagogical measures proposed by the thesis are specific ways to improve the competency of exploring and obtaining knowledge in teaching maths at the gifted high schools Design of the paper In addition to the Introduction and Conclusion, the paper consists of three chapters Chapter Literature review Chapter Some pedagogical measures to improve the competency of exploring and obtaining knowledge for high school students majoring in physics, chemistry and biology Chapter Pedagogical experiment CHAPTER 1: LITERATURE REVIEW 1.1 Overview 1.1.1 Competence 1.1.1.1 Previous studies in the world Perrenoud's definition of competence can be considered to be the most meaningful andaccurate among given definitions of this concept According to Perrenoud, a competence is the ability to act before an assumption, the situation that someone can master because they have both the requisite knowledge and the ability to consciously mobilize this knowledge to identify and solve real problems [62] 1.1.1.2 Previous studies in Vietnam There are some main characteristics in studies of competence Firstly, the ability express individual psychological, physiological characteristics that are different from that of a biological genetically inherited factor, developed or limited due to other conditions of the habitat Secondly, the ability or competence is not only an innate factor, but also develops in activities, by activities, exists and manifests itself in activities of individuals In other words, the term “competence” should be mentioned in a specific type of activity 1.1.2 Studies relating to the thesis topic With regard to effect of teaching through discovery activities, there were a large number of researches by researchers, such as Heather C Hill, Merrie L Blunk, Charalambos Y Charalambous, Jennifer M Lewis, Geoffrey C Phelps, Laurie Sleep & Deborah Loewenberg Ball, [206] Besides, Martin C Libicki, Stuart E Johnson [209], indicated important roles and trends in improving the knowledge- grasping competence while Daeka [169] studied the model that was based on experience and collaboration in work group to develop the creativity of students J Bruner [164] studied teaching basing on discovering and methods of discovery activities; Malin Brännback, Patricia Wiklund[211] studied principles of improving the knowledge – grasping competence; and Peter Sullivan [214] refered teaching maths by owning the knowledge In Vietnam, Nguyen Huu Hau [62] has researched activities to dominate knowledge in teaching Algebra and Calculus The authors Dao Tam and Le Hien Duong [112] mentioned knowledge -grasping competence in general and proposed some pedagogical measures to support students to develop this capacity in teaching elementary geometry at the University of Pedagogy However, there have not been any researches on improving the knowledge – grasping competence, which can be studied in full and in detail, especially in term of teaching for high school students majoring in physics, chemistry and biology basing on this orientation 1.1.3 Teaching perspective-oriented capacity development 1.1.3.1 In the world Teaching from the perspective of capacity development has been paid attention to and conducted in many nations in the world because it not only encourages students to concentrate on intellectual activities but also enhances them to develop problem - solving skills within situations of life and career; it can also link intellectual activities with practical activities Summarizing theories on competencybased teaching approaches in education, training and development, Paprock (1996) outlined five basic characteristics of this approach Firstly, competency-based approach on the learner's philosophy is the center Secondly, competency-based approach meets the requirements of professional activities Thirdly, competencybased approach is oriented to real life and real career activities Fourly, competencybased approach is very flexible and dynamic Finally, competency-based approach is formed in learners clearly 1.1.3.2 In Viet Nam In summary, there are several core factors of the teaching from the perspective of capacity development: Teachers organize activities to promote positive and proactive learning of students; create a supportive learning environment (linked to the real context); encourage students to reflect ideas and actions, encourage communication; enhancing the learning responsibilities of learners; creating favorable conditions for learning, sharing, exchanging, debating, connecting to learning; provide full opportunities for learners to explore, experience and create; and teaching is like a learner's learning process 1.2 Improving the competency of exploring and obtaining knowledge 1.2.1 Activity and cognitive activity 1.2.1.1 Activity An activity includes folowing characteristics Firstly, it always has objects and it is done by a subject Moreover, it always has its own purpose, and can operate according to indirect principles According to A N Leonchev [82, p 579], activity is a process of performing the mutual exchange between two poles: the subject - the object The author described the macro structure of the activity with components that have dialectical relationship with each other 1.2.1.2 Cognitive activity The term Cognitive activity is defined as the following basic characteristics Initially, it is the subject of discovering and recreating the world, thereby forming and developing the understanding of subject of the world and its methods of movement The subject does not directly affect objects but indirectly through labor tool The Cognitive activity often takes place in direct or indirect interactions between individuals It is said to have many levels, which depends on the involvement of both cognitive and rational cognitive functions It is a dynamic, positive and creative process of the subject; which starts from unknown to knowing, from superficial to internal attributes, that is, from sensory, intuitive, separate to the complete, stable, regular, and deeper substance of a whole class of objects, phenomena and finally return to reality Cognitive activities in general and mathematical cognitive activities in particular have three levels, which includes being aware of the problem, understanding and clarifying it 1.2.2 Activity of exploring and obtaining knowledge We propose the concept of connotation of the concept of discovering and acquiring knowledge in accordance with that satisfies students' requirements to positively study learning as the following viewpoints Discovering and acquiring knowledge is a series of activities that actively experience, discover, think, attempt, and use intellectual abilities (observation, comparison, analysis, synthesis and experience, verification ) along with personal qualities such as desire for learning, perseverance, flexibility, critical thinking as well as endurance towards issues that need to be addressed without specific knowledge, to capture and master the problem more fully, to be more independent, creative as well as to gradually expand it with related issues, adding to their existing knowledge Thus, it can be understood that discovering and acquiring knowledge is a sequence of activities done by the subject, which can transform the logic of the object into the thinking process due to the subject’s active, positive participation; thereby grasping, mastering, thoroughly understanding the subject and including the expansion of research and mastering of other related subjects Discovery not only indicates what exists inside each object or phenomenon of the problem, but also shows an extension of the mastery of the subject being studied beyond what is inside the thing, the phenomenon of that problem 1.2.3 The role and meaning of organizing activity of improving the knowledge – grasping competence for students in teaching maths According to B Maskey, J Collum: concept card (Nepal Training Institute), using verbal methods and encouraging student self-organization, combining methods to mobilize multiple senses at the same time to participate in the learning process is limited Studies show that students want to gain in-depth understanding and longterm memory, it is important to encourage them to participate in activities directly because, they will be able to use the acquired knowledge in a better way due to their own experience This can make the process of discovering and acquiring knowledge more fully and deeply Moreover, it is essential to give students joy and excitement because it is the motivation to promote their self-discipline and positive learning, especially the joy and excitement of a person who finds the truth on his own It is said that students should be put into real-world situations, and required to directly observe and make experiment, discuss and to solve problems in their own way Thereby, students will acquire new knowledge, learn new skills and methods as well as find out that knowledge and skills not necessarily have to follow the existing patterns to reveal their potential creation [12, p 3] 1.2.4 The concept of improving the competency of exploring and obtaining knowledge From psychology and education perspectives, from studies on activities, awareness, competence, which are being applied in teaching mathematics at high schools, and the concept that considers discovering and acquiring knowledge as an important competence to be develop, the competency of exploring and obtaining knowledge can be defined as personal psychological characteristics, through a combination of elemental competencies expressed in activities of individuals in order to actively explore, comprehend, master and address concerned issues flexibly and creatively 1.3 Expression and elemental competencies, as well as compatible activities of the competence of exploring and obtaining knowledge in teaching maths The ability of discovering and acquiring knowledge of students in learning maths can be evaluated according to the following criteria Firstly, they can Mobilize math knowledge related to discovering and mastering a specific math content Secondly, they can be able to conduct activities such as doing exercises, understanding mathematical concepts and proving theorems properly, and gaining results consistent with required purposes For example, in proving the theorem, students' ability to discover and gain knowledge is shown through their understanding of theoretical proof, independently conducting theoretical proof In addition, learners also know how to apply creatively in many other related exercises, or other topics of which a higher level is applied in life Students have clear attitude and emotions with problem solutions, such as finding mistakes and correcting mistakes, realizing the good and profound sides of each solution Moreover, students learn to be persistent in expanding assignments and problems The competence of exploring and obtaining knowledge of learners in studying mathematic can be described basing on the following six elemental competencies and 21 suitable activities 1.3.1 The first elemental competence: The ability to experience, explore situations, detect problems, detect contradictions appearing in situations and problems This component competence includes compatible activities, such as hands-on experience, consideration of ongoing issues It also includes experiencing, discovering signs of nature, general properties as well as mathematical relationships of things and phenomena Another important characteristic is to limit problems, identify the hypotheses and the conclusions of the theorem, and the relationships of those elements as well The last point is said to discover and detect contradictions that appear in the problem 1.3.2 The second elemental competence: The ability to observe, intuitively solve problems, formulate and implement problem solving in common logic ways for similar problems This component competence includes compatible activities, such as discover and detect visual symbols and their relationship that appears in the problem and know how to choose good images to support the problem domination The second criterion is to analyze and explore knowledge through research and observation of visual images or animation Besides, it is necessary to make good use of the mathematical software or visual images, which can create symbols and images relating to the problem as well as point out mistakes to solve them Finally, a suitable approach should be given to solve problems in an appropriate sequence 1.3.3 The third elemental competence: ability to predict and infer reasonably, associate and mobilize knowledge, perform thinking manipulations in the process of acquiring knowledge This component competence includes compatible activities, such as making reasonable predictions and inference, selecting reliable predictions Besides, it is required to associate, mobilize knowledge to the "nearest development zone" to facilitate in finding a way to solve the problem Finally, it is essential to implement systematization as well as infer effectively in the process of discovering and mastering the problem 1.3.4 The fourth elemental competence: Ability to use language in mathematical communication There are some compatible criteria The first requirement is to express mathematics content in different ways accurately to have better access to the problem to be solved The second requirement is to be able to "interpret" from the common language form of maths clauses to mathematical terms and symbols, and vice versa Thirdly, it is required to use language of set theory and mathematical logic along with mathematical symbols and terminology accurately and rationally to express solutions Moreover, it is necessary to use the right rules of proof, reciprocal clauses Last but not least, it is required to explain the nature of a formula or the final steps of an exercise Below is the example of this competence Example 1.1 (cited [139, p 109 - 110]) According to the design, the original height of the Eiffel Tower is 300m, but the antenna mast on the top of the tower increases its height reaching at 325m The lowest temperature in March 2015 in Paris was 8oC on March 1, 2015 and the highest temperature in July 2015 was 26 oC and measured on July 18, 2015 Calculate how many centimeters the tower has grown from March to July 2015, assuming the tower is 325 meters tall on March To this exercise, the students majoring in physics relate and mobilize to the following contents - Every solids have changes in length because of the change of temperature, which is described by the following formula l = l0 + l0 ( t − t0 ) , in which l0: The length of the object at time of temperature t0; l: The length of the object at time of temperature t; and : Long expansion coefficient - Students look up the board to find out the length of expansion due to the heat of iron: 11.10-6 From this the answer can be given However, students show their ability at different levels through solving the problem Solution 1: The height of the tower in July is l = 325 + 325.11.10−6 ( 26 − ) 325,064(m) The height difference between March and July is 325,064–325= 0,064 (m) = 6,4 (cm) Solution 2: For better students, they have mastered a better formula to calculate the difference: l = l0 ( t2 − t1 ) , thus, the difference in height of the tower between March and July is calculated: 325.11.10-6(26-8) 0,064(m) = 6,4(cm) 1.3.5 The fifth elemental competency: Competence in mathematical modeling practical problems This competence can be showed by the following activities Firstly, It can build mathematical models for practical issues Secondly, it can orient and solve model problems and master practical problems 1.3.6 The sixth elemental competency: critical and creative ability This ability can be displayed via some compatible activities Firstly, it can collect results of solving problems with right situations and right limits Secondly, it can make comments on problem-solving results, compare problem-solving methods to choose the best solution Besides, it requires students to adapt and critique in assessing the process of problem-solving, contemplate solutions, detecte and correct mistakes; and solve the problem from a new perspective and expanding 1.4 Levels of improving the competency of exploring and obtaining knowledge in teaching maths in high school The competency of exploring and obtaining knowledge is classified according to various levels *) The first level: Students fulfill the basic requirements of discovering and acquiring knowledge when situations and issues are clearly raised by teachers *) The second level: Students have a clear awareness of situations and problems that are raised by teachers, and they know how to complete improving the knowledge grasping actively and effectively Teachers act as acatalyst to accelerate students' discovery and acquisition of knowledge *) The third level: Students actively identify unclear issues through activities of discovering, predicting conditions and commenting on approaches to firmly, flexibly and critically Moreover, by this way, they also know how to expand and master the subject Below is example showing the levels of improving the competency of exploring and obtaining knowledge Example 1.2 There is a park showed in Figure 1.1.a It is planned to put a lamppost to illuminate the park Find an option to locate the lamppost This problem is conducted in the following steps: +) Step 1: Measuring and simulating the park as Figure 1.1b 11 a) Students have dfficulties in learning the content of subjects: Because learners bring incomplete insights into the subject, they often bring the previous knowledge into the relevant content b) Some research findings on teachers: straight teachers will greatly influence students' style, ability, and learning outcomes c) The obstacles from the current textbook program: Author Dao Trong Thi wrote: “The program and the amount of knowledge shown in the textbooks is too much compared to the understaning of students to Besides, the content puts more emphasis on theory and imparting knowledge without paying much attention to train students' skills and personality” [195] 1.7.5.2 Analysis of quantitative results Most of surveyed participants (93.8%) find that it is necessary to design and organize activities for students to discover and gain knowledge Only 6.2% of them did not see the need to organize this activity In contrast, 100% of teachers and administrators agree on the necessity of organizing discovery activities and gaining knowledge, and most of teachers and managers are interested in the manifest activities of discovering and acquiring knowledge, but the capacity and organization of those activities are not equal Students are encouraged to practice activities that are part of the ability to discover and dominate knowledge, but these activities are not conducted regularly It is said that students’ improving the competency of exploring and obtaining knowledge is only at fair level Besides, there are some activities that shows their ability only at average level, accounting for nearly 50% This shows that the capacity to discover and dominate knowledge should be paid attention and continue research Most of students want to be trained in compatible activities to improve the knowledge – grasping competence, but the training of those activities in learning is chaired and controlled by teachers has not been focused properly Teachers and students also think that about 25% -30% of students often face difficulties and passively when actively implementing compatible activities of discovering and acquiring knowledge in the classroom 1.7.6 Cause of the situation There are some main reasons Firstly, the teacher has not really considered improving the competency of exploring and obtaining knowledge in terms of competencies of component activities, and has not clearly identified the elemental competencies of this capacity, as well as typical manifestations of these competencies so that activities can be organized in accordance with the teaching content, which aims to formulate and train students' ability to discover and acquire knowledge Secondly, generally teachers have not identified measures and ways to implement measures to improve the competency of exploring and obtaining knowledge in association with the content of maths teaching Finally, teachers have difficulty identifying and exploring opportunities for students to be experienced, trained, and fostered improving the competency of exploring and obtaining knowledge in the teaching process 1.8 Conclusion of Chapter On the basis of research, analysis and practical experience, in Chapter 1, we draw some results as follows: Firstly, the thesis has codified the views of a number 12 of authors about receiving activities, make a point of of exploring and obtaining knowledge; as well as the status of exploiting activities of discovering and acquiring knowledge in teaching Maths in high schools today; Secondly, the thesis has given the viewpoint and the competency of exploring and obtaining knowledge in teaching mathematics for high school students; level of this ability in teaching mathematics among high school students; Thirdly, the thesis has presented the expression of the competency of exploring and obtaining knowledge through component competencies and component activities; Fourthly, the thesis has pointed out the urgency and reality of developing the of the competency of exploring and obtaining knowledge in teaching mathematics in particular, in educational innovation and practical needs in general At the same time, we gave a view on teaching mathematics to high school students specialized in developing the cmpetency of exploring and obtaining knowledge Chapter SOME PEDAGOGICAL MEASURES CONTRIBUTE TO FOSTERING THE COMPETENCY OF EXPLORING AND OBTAINING KNOWLEDGE FOR HIGH SCHOOL GIFTED IN MATHEMATIC, CHEMITRY AND BIOLOGY 2.1 Orientation of Construction 2.1.1 Ensuring the compatibility between content, goals with skills, knowledge standards of Mathematics curriculum 2.1.2 Thoroughly grasping the operational viewpoints in forming and developing competency of exploring and developing knowledge 2.1.3 Building a positive collaborative environment in studying, in which students are the center, always encouraged to experience - exchange - discuss explore - explore and gain knowledge 2.1.4 Focusing on helping students create relationships between Maths content, focusing on the relationship between Mathematics and the subjects of Physics, Chemistry, and Biology; contributing for dominating, applying with practice, discovering and developing new ideas in process of the teaching 2.2 Pedagogical measures 2.2.1 Measure 1: Creating opportunities for students to practise compatible activities with competency of exploring and developing knowledge components through installation of situation of discovery in teaching, finding and solving problem in teaching a) Scientific basis of measures Methods are forms and modes of operation of teachers and students in teaching environments preparing, aiming to achieve the purpose of teaching and developing individual competencies ([157]) *) Educators believe that teaching discovery has the following strengths ([83]): Firstly, As a method aimed at learners' activities, the activity is encouraged to consider that learning is their own work rather than teachers’ one, the level of learners' demands is also increased Second, A method to support the development of individual cognitive capacity and talent of learners 13 Thirdly, A method that allows learners to have time to acquire, update information and assess their true ability in the learning and research process + According to Jerome Bruner, teaching discovery consists of the following five principles: Problem solving; managering of student; Connection; analysising information analysis explanations; Failure and feedback *) With teaching method of discovery and solving of problem, it will develop students' creativity, competency of solving of, helping students acquire knowledge, skills, cognitive methods and self-development Meeting the requirements of modern practice, with experiential learning This measure will contribute to foster the overall of competency of exploring and developing knowledge b) Method of implementation of the measure + Solution 1: Organizing compatible activities in the direction of applying discovery teaching methods, fostering learners' interests and interests Through to Roger Bybee and his colleagues, we consensus with discovery teaching process consists of steps, called the 5E process: Engage; Explore; Explain; Elaborate; Evaluation + Solution 2: Organizing -compatible activities according to method of discovering and solving problem in order to increase the activeness, actively discovery c) Attention of using measures *) Applying teaching method of discovery requires the following conditions: almost of students must have enough knowledge and skills necessary to carry out activities organized by teachers; Teacher's participation in each activity should be necessary, not too little, not too much, causing students to understand exactly what they have to in each discovery activity To ensure this, teachers must understand their students' abilities and know their "nearest developmental area" *) Applying problem-solving and problem-solving learning, the following should be noted: + Giving students to discover and solve problems for a part of the learning content, teachers can help with different or lesser levels + The proportion of problems discovered and solved by the learners and the program depends on the characteristics of the subject, the object of the student and the specific situation + Teaching discovery and solving of problem can be applied in different stages of the teaching process: forming new knowledge, consolidating knowledge and skills, applying knowledge This method should be aimed at all students, not just for good students 2.2.2 Measure 2: Enhancing activities using visual means, teaching tools, activities relating to discover the nature of the problem so that students can improve their ability to observe, be proactive and interested in acquiring knowledge a) Scientific basis of measures The visual representations create vivid images, easy to impact on the human senses, thereby attracting attention, creating motivation for learners J.A Komensky (1592 - 1670) was the first educator setting up the principle of teaching systematically and scientifically Among the principles stated by the author, 14 visualization is ranked the firstly, J.A Komensky said: "There is no intellect in things that had not previously felt" The author said that in order to have solid knowledge, you must use visual methods Not only does visual media participates in the process of forming concepts, it but also support for teaching theorem to teach Math problems Visual media is a bridge and an intermediate step in the abstraction stage This measure will contribute to fostering elemental competencies 1, 2, and b) Method of implementation of the measure Teachers can perform follow these forms: + Organizing observation and experiment activities on models, drawings, , to help students draw attributes, typical signs of concepts, things and phenomena on the basis of that formed the symbol and came to dominate; + Training the ability to predict knowledge through specialization, generalization and similarization through the organization of open teaching phases; + Rationaly using of support software to enhance discovery, support for predicting and acquiring knowledge for learners Example 2.1 Having to hang a mirror with a height of 1m on a wall that how much the bottom edge of the flat so that a person with a height of 1.8 m can look in a mirror, knowing that person is 0.8m from the mirror? How will the result change when we change the distance between the person and the mirror? c) Attention of using measures From the obtained results, we find that in teaching mathematics at high schools, it is necessary to use visual means and activities to discover the nature of teaching, but to get the desired effect, we must be paid to the following principles: + Principle 1: Visual aids need to be built and applied in a spirit that must meet the purpose of teaching mathematics in high school + Principle 2: Visual aids in teaching must respect and inherit curricula and textbooks + Principle 3: The construction and using of visual aids must be based on the orientation of innovating teaching method, notably creating a positive and self-motivated working environment + Principle 4: Means of teaching must focus on students' self-discovery, independently explore problems, actively dominate and solve problems 2.2.3 Measure 3: Organizing teaching activities in the direction of integration and differentiation, focusing on mathematical modeling problems of practical issues and STEM-oriented learning topics a) Scientific basis of measures *) According to Do Ngoc Thong “In general education, integration and differentiation are two requirements that cannot be ignored and need to be done in a uniform and unified manner in all aspects: i) Desgigning program; ii) Compiling textbooks; iii) Organizing teaching and iv) Testing, evaluating ” Integration is one of the educational perspectives that have become a trend of teaching in high schools and in developing of curriculum in many countries around the world “STEM education is an interdisciplinary approach in the process of study, in the conceptual academic principles and it is integrated with real-world lessons, 15 where students apply the knowledge in the department study, technology, engineering and Mathematics into specific contexts, helping to connect schools, communities, workplaces and global organizations… to develop STEM capabilities and competitiveness in a new economy” + There are important characteristics of STEM education: i) Interdisciplinary approach; ii) Integrating with real-world lessons and Connecting from schools, communities to global organizations *) In terms of differentiation in teaching, teaching must ensure a differentiation for each student, for a whole group of participants The division of children into a group to ensure the mutual support between members of the group as well as assigning appropriate work for them, that is the guarantee of differentiated teaching, from the perspective of perceptions of Vygotsky about "nearest development zone", differentiated teaching to ensure the following requirements: The plan of teaching bases on the interests, readiness and outstanding intelligence of the learners; students are often guided and supported by making choices basing on their interests, thereby facilitating independent and cooperative development of skills; The assessment process is ongoing, helping teachers have the appropriate orientation and adjustment Students are assessed in a variety of ways and forms; The classification process bases on academic achievement and students *) Regarding the modeling problems, basing on Nguyen Danh Nam and other athors, we think that modeling is a situation, a problem with a model or a real context that can be used with mathematical knowledge to solve In other words, this is the situation, the problem containing elements in the real world, but they were simplified, specialized, concretized, adding appropriate conditions, assumptions, and limited factors of elements which are not necessary, allowing learners to access and solve by math as tools according to their intentions, however they still reflect a part of the situation in the world Formulating situations, the problem should be associated with a real, familiar, closed context with students This measure also helps students directly connect Math with practice through learning It is an opportunity for students to practise knowledges of theories, applying knowledge to solve practical problems This measure will contribute to fostering elemental competency 2, competen and b) Method of implementation of the measure *) The first solution: Organizing interactive activities with high experience like project-based teaching *) The second solution: Strengthening integrated and differentiated teaching, teaching according to STEM orientations to ensure specificity of the subject and ability of the learners, which can be conducted in the following directions: Organizing teaching mathematics and the meaning of Mathematics to study Physics, Chemistry and Biology well; Using tools that are subjects, contents of Physics, Chemistry and Biology subjects to teach Maths well; Seeing Mathematics, Physics, Chemistry, Biology is a tool for sovle problem well, this is the orientation of teaching STEM For example, with the solution of organizing teaching according to STEM orientations, considering Mathematics as a part of STEM, lessons can be designed as 16 follows: Setting up the objectives of; Identifying potential integration of organizations; Identifying cored and suggestive questions; Designing potential activities in subjects, mapping activities and setting allocation of time; The final step is to evaluate the integrated lesson and share the solution Figure 2.1 Students of high school for gifted of Vinh university in studying STEM class, March, 2019, The candidate participated in classroom research with projects "Measuring and monitoring quality of water" and "Sensors and data" *) The third solution: Organizing the construction of activities selecting and using situations, modeling issues *) The fourth solution: Organizing the activities creating opportunities for students to collect and exploit modeling problems containing inter-subject content Example 2.2 (Reproduction of amoeba metamorphosis): An amoeba can divides into amoebas after a second And every second, every child also divides into amps Counts to see how many amoebes after 30 seconds? After 30 seconds, the number of amps is: S = + + 22 + +230 - is the sum of an geometrich sequense with 31 terms, u1 = 1, q = 2, so: S = 231 − = 2.147.483.647 (amps) −1 For students gifted in Chemistry, we can include the following example Example 2.3 guessing the spatial structure of the methane CH4 Students with certain intuition predict that: there are C-H bonds created and to create the stability of the molecule, the structure needs to ensure symmetry, the distance between H and H is equal, the bonding angles H - C - H¬ are equal, therefore, CH4 must be composed of regular tetrahedra, where C is the center of the molecule Figure 2.2 Model of methane’s In fact that, experiments have shown that molecule the prediction is correct, but to get the results, chemists have gone through more complex theories, such as the VB theory of valence and the re-orbitalization AO (Figure 2.2) c) Attention of using measures *) In order to effectively ensure differentiation and integration in study, in our opinion, teachers need: to be flexible, dynamic, creative and constantly strive to improve their professional qualifications and competence; the requirements of renovating methods; understanding students clearly, the style of learners, Since, teachers designed the content suitable for each student and demonstrated the differentiation; It is necessary to have an overview of the subjects and each subject in order to adjust and integrate the content in design of lessons accordingly, contributing 17 to the format and develop of the capacity for students; focusing on both time and conditions of classroom *) In addition, in order to apply widely and effectively implement of differentiated teaching perspective, educators need to understand the nature of the problem, and at the same time, they must change their awareness in content development, the program as well as in teaching and testing, assessing the reality of the teaching process In order to organize a successful division of teaching, teachers need to create a democratic relationship between teachers and students, between students and students to help students be more ativitive and confident In particular, in differential teaching, students need to follow the process: Investigating and surveying students before teaching; Making of plans, preparing lessons from students' needs analysis; During the lesson, teachers must combine a variety of teaching methods, selecting the appropriate in teaching arrangements for the lesson objectives; Checking and assessing the progress of students during the process of studying *) Situations and modeling problems must be arranged from easyly to difficultly, from simply to complexly Students who themselves solve a problem have great psychological meaning On the contrary, the failure from the first problem easily makes of losing their morale, easily creates a negative mood for the organization of the next activity Therefore, while designing a system of model of situations and problems, teachers need to pay attention to the levels of modeling, Nguyen Danh Nam has divided into different levels and built the Compatible situations to foster modeling capabilities for students in teaching +) If mathematical situations are constructed, it is necessary to ensure: The situation is brought up closedly, linking with the real context Each situation may contain one or several related questions, but it need to ensure of various solutions; Increasing opened questions, promoting flexibility and creativity; closed situations, attracting students +) To collect mathematical modeling problems, it satisfies that the following requirements must be ensured: Learners must necessary have mathematical knowledge; Learners should know practical knowledge at their age appropriated level and experience, natural language skills, ability to switch to mathematical language or vice versa in general; Learners must recognize the hidden mathematical knowledge in real-life situations in general and in specific subjects Understanding how to link Math knowledge with knowledge in practice and other subjects, with personal experiences in real life +) In practical activities, it is necessary to ensure: the good teaching of prescribed practice lessons, looking for more practical opportunities from Maths topics In practice, it is able to organize practice in the classroom and outside the classroom To attract students of participate and meaningful exercises, the exercises should be associated with specific situations, specific phenomena in the real life 2.2.4 Measure 4: Organizing students acitivities of practical thinking and some other intellectual aim to practice their ability of discovery of the natural problems arcodding logically, as well as to expand problems a) Scientific basis of measures 18 According to Dang Vu Hoat, intellectual activity competence is expressed in the capacity to manipulate intellectual manipulations, especially thinking manipulation confronting of a problem, the most important thing is to predict how to approach the problem correctly and provide a simulation, the way of solving the problem This is important because it cultivates a sense of self-confidence as well as a sense of wellbeing, certainty in thinking and solving problem Generalization has an organic relationship with other thinking manipulations and is closely related to specialization and similazation G Polya asserts: "Specialization, specialization, and similarization often cooperate with each other in solving mathematical problems" and "generalizations, specializations and the like are combined together naturally in trying to solve the problem.” Not only are comparisons, analogies, generalizations, abstractions, etc a means to conduct cognitive activities and solve problems they but also provide methodological knowledge for students This measure focuses on fostering elemental competency 1, elemental competency and elemental competency b) Method of implementation of the measure We support to the opinion of the author Nguyen Thi My Hang, training of thinking manipulations through compatible activities can be conducted such as: + Practicing for students to analyze the contents, externalities of mathematical concepts, as well as the ability to apply those concepts to solving mathematical problems + Training for students to clarify the meaning of each element given in the hypothesis, thereby seeing the applicability of the theorem + Training for students to express definitions, theorems, and solving problems in different ways + Instructing students to find common signs, the nature of elements and relationships in a problem or a class of problems + Create opportunities for students to practice similarization skills in the process of solving math problems + Encouraging students to propose new problems that is based on exploiting the given problem + Developing some situations that contain solutions to problems with mistakes, guiding students to analyze to identify common mistakes, whereby finding appropriate teaching methods to overcome c) Attention of using measures The activities aimed at training the thinking and other intellectual activities presented above are mainly conducted through provocative questions of teachers' instructions and groping, exploring, creative experience of students Through activities, not only students solve of problems, they but also learn the ways of thinking, way of discovering solutions Therefore, the role of the teacher in the direction and control is very important, the question system and the anticipation of the discoveries and experiences of the students must be really good, the teachers need to direct the situations occuring in student activities, contributing to supporting students well in conducting intellectual activities and fostering thinking manipulations 19 2.2.5 Measure 5: Enhancing activities that create opportunities for students to argue, learn personally, activities to detect and correct mistakes, to gain knowledge and practice critical thinking a) Scientific basis of measures The school of psychology with typical representatives such as L X Vygotsky, A N Leonchiev, A.V Petrovski, the age of high school is a very important stage in development of thinking, especially theoretical thinking (due to the development of the brain and due to changes in the natural activity) study) Intellectual development includes changing of both quantitative and qualitative More activetily, they are more active and increase the ability to think independently and creatively with the subjects This depends on the attractiveness of the subject according to their perceptions High school students' thinking is more critical, rigorous, consistent, and more grounded than in their teens Although there is a period of strong and profound development in thinking, but because of that, they can not avoid wrong thinking This measure will make an important contribution in training students the elemental competencies of competency of exploring and obtaining knowledge in particular, and to competency of exploring and obtaining knowledge in general This measure focuses on fostering elemental competency 3, 4, and b) Method of implementation of the measure First way: training for students to ask and answer questions themselves; Second way: creating opportunities for students to find and analyze grounded arguments when solving problems; Third way: activities situations in designing collaborative learning for students to debate for finding out solutions, presenting and evaluating the solutions; Fourth way: exploiting and using reasonably exercises that have the advantage of refraining in critical activities in the teaching process; fifth way: Organizing teaching multiple choice exercises Exercises of this type enable students to reason in different ways to find the fastest way to choose the correct option, eliminating the interference plans However, the disadvantage of solving objective tests is difficult to assess the thinking process leading to the test results, so it is difficult to test thinking capacity (especially creative thinking) and detect and correct mistakes for students Therefore, objects of designing test assignments need to ask students to analyze the alternatives as a good opportunity for students to present their reasoning At the same time, the teacher also knows why students choose the answer c) Attention of using measures In order to effectively implement this measure, attention should be paid to the following: Firstly, teachers design situations that can lead to mistakes for students in challenging with them; Secondly, paying attention to the requirements: education, timeliness, accuracy in the process of detecting and correcting mistakes for students; Thirdly, it is necessary to create conditions for students to express mistakes through training the self-examination and self-assessment skills 2.2.6 Measure 6: Organizing forstudents activities of analyzing, identifying relationships, activities of transformation of language from different perspectives, focusing on interaction in communication activities, in order to enrich the 20 expression and increase of solving problem, contributing to practice critical thinking and creativity a) Scientific basis of measures: *) About language conversion activities, according to L.X Vygotsky [159], language is the process by which individuals use a certain language to communicate and to think The ability to identify and represent the same mathematical problem in different representations, and the flexibility to move from one representation to another, is crucial in learning math The author said that the meaning of the word is both a language and a thought, so it is a unit of linguistic thinking Mathematical language is a scientific language that requires shortness, accuracy and comprehension, students are often not good at expressing mathematical language in many aspects, such as not being able to grasp other forms of problems, can not convert to equivalence problems *) Activities aimed at cultivating critical and creative thinking, teaching in the direction of fostering competence requires teachers to orient and help students promote positive, proactive and creative skills, to verify information correctly and choose problem-solving methods getting high efficiency According to Toan Nguyen: “To be creative must have critical thinking, to have critical mind must have an independent spirit Indeed, creativity makes a new product better than the old one, critical thinking helps to evaluate the product, which comes from its independent thinking”; "Creative thinking develops from independent thinking, critical thinking" [141, p.59] Critical and creative thinking intertwined, penetrating each other, operating in a way of criticism - creativity - criticism - creativity - criticism in which the next level of creativity is higher Nguyen Ba Kim analyzed: "Flexibility, independence and criticism are necessary conditions of creative thinking" The creativity of Thinking is evident in the ability to create new things: discovering new problems, finding new directions, creating new results Creativity and criticism always go hand in hand, complementing each other This measure contributes to fostering elemental competencies 1, 3, 4, 5, and b) Method of implementation of the measure *) In order to practice language conversion activities, we use the following solutions: Encouraging and asking students to analyze exercises according to different aspects from which to express problems in various forms; Encouraging students to use the representations to illustrate the requirements of exercises; Organizing the conversion of languages within an discipline of Mathematics, as well as from different subjects, transforming from a subject, a subject to content related to the subject, the other subject and vice versa again Example 2.4 In teaching physics for students of the 10th division specialized system, it is possible to Figure 2.3 teach students the following exercise: Two homogeneous iron balls of the same weight are fastened on a homogeneous iron bar as shown in Figure (iron bars are negligible mass) Hanging vertically on a wire at the midpoint of AB, the iron is balanced horizontally as Figure A I B E A I C B 21 Contining to hang the same three marbles on A, B, C as Figuer 2.3 I is the point of satisfaction of equality, where, E is the midpoint of BC In order for the iron rod to stand still horizontally, it is at the point I Please explain the results above Can you generalize the results of the above problem? *) For the practice of critical and creative thinking, the following methods may be conducted: +) Conducting according to Ngo Bao Chau's point of view: “ Approach in a way not from definition, proof, and to exercises that we have to learn the opposite, from solving problems firstly, then we learn concepts that help us understand those issues Even that concept, we don't necessarily have to learn right away, we need to localize it into conceptual areas We have these black boxes, we manipulate and think about using those black boxes to solve our questions When we are proficient what those concepts reflect our thoughts, then we will find out the definitions, theorems, proofs of form and that are necessary, that is foundation for our knowledge to be more solid ” +) Organizing in the way of author Nguyen Canh Toan on the process of expanding a concept, a theorem, a problem to foster critical thinking: including steps +) Organizing activities, helping students see the problem from different perspectives, fostering critical thinking and creativity of learners c) Attention of using measures *) In defining terminology, transferring to equivalence problem, we need to anticipate difficulties, initial conditions of the original problem in order to rewrite to equivalent conditions of the following problem and students should be noted to avoid misunderstandings; Teaching students to exploit the above typical applications will enable students to explore and discover new knowledge related to the subject in other situations; increase the ability to associate, see the problem from many ways Thereby, improving the ability to change the language of the problem *) At the same time, this measure also contributes to educating dialectical thinking, creative perspective, educating the scientific worldview for students to look at things and phenomena in relation, different phenomena and in a state of constant change Therefore, it is important to pay attention to the following: Teachers, educators need to know how to look at the "particular" from different ways, because each way gives us a direction to find out the individual in the common, the common in the private; And, needing to guide students to persevere in the implementation, then success in learning Math will achieve high results 2.3 Conclusion of Chapter The main content of this chapter deals with pedagogical orientations and measures to foster the competency of exploring and obtaining knowledge for students in teaching Maths in high school Orientations are a necessary condition for developing the knowledge - grasping competence In the process of implementing the measures, we focus on teachers paying the experiential activities of students, activities of exploring, actively mastering knowledge At the same time, in some measures, we also focus on 22 situations, ways to create motivation, forms of leading students in the direction of experience, and positive activities to promote their ability The measures also focus on creating vivid, interesting images, associated with reality, enhancing student experience The coordinated combination of these measures will promote the development the competency of exploring and obtaining knowledge for high school students in general and high school gifted in Physic, Chemistry and Biology in other hand Chapter PEDAGOGICAL EXPERIMENT 3.1 Purpose of pedagogical experiment: Pedagogical experiments are conducted in order to test the feasibility and effectiveness of the pedagogical measures that have been proposed to foster competency of exploring and obtaining knowledge in teaching mathematic for students of Grade 10 and 11 3.2 The content of pedagogical experiment At the time of conducting the first round of experiments, Vinh University High School is using the 10th grade book of basic books edited by Tran Van Hao At the time of conducting the second round of experiments, Vinh University High School is using the 10th grade book of basic books edited by Tran Van Hao Two experiments were conducted in 2018 and 2019, namely: the first one took place from September to December 2018, and the second one took place from January to April 2019 The first: Experiments are conducted for students of grade 10, 26 periods of content Geometry 10 Session: Experiments are conducted in 36 periods of contents of Algebra and analysis 11 3.3 organizing experiment Organizing the experiment was conducte in two phases: * Phase I: Experimental class is class 10A4; teacher of experimental class: Dang Thi Viet Ha; The control class is grade 10 A5; The teacher of the control class is: Phan Xuan Vong Phase II: Experimental class is class 11A3; teacher of experimental class: Nguyen Cong Chuan; The control class is grade 11 A4; The teacher of the control class is: Phan Viet Bac 3.4 Qualitative assessment according to empirical process Teachers lecturing in expermental classes have the opinion that : It is not obstacles and feasible to apply the views and measures in the thesis obstacles, not feasible to apply the views and measures in the thesis Opinions, especial suggestions on how to organize that the role of teacher control is ensured, the central role, the student's selfexperience; and making of questions and the appropriative direction, moderate and diversified activities for students; not only does such the organization stimulates students' activeness, independence in exploration and discovery, it but also motivates them to acquire methodological knowledge in problem solving Many other observations have led to the result that students in the experimental class have developed and fostered 23 the elemental competencies of the competency of exploring and obtaining knowledge, the level of perfection is getting better at the end of the experiment, while in the control class, this compehtency of the students did not improve and it is more clearly to see that the increasing of the distance between the experiment and the control in order to the later of the experiment 3.5 Assessing by the tests 3.5.1 Content of the tests During each experiment, the students had been finished two tests: the between and the end of the experiment The first experiment, we conducted in Geometry, while in the second, we conducted in Algebra and Analysis 3.5.2 Qualitative assessment the tests 3.5.2.1 Assessing of the content of test Analysing above, it can be seen that the four tests above have clearly shown their intention: to survey the knowledge - grasping competence of students gifted in Physics, Chemistry and Biology In empirical classes, because of training of this competency through pedagogical measures, the quality of the tests is better 3.5.3 Assessing quantitativily through the tests’ result The test results of students of experimental classes and students of general classes are shown through the following statistics Diagrams: Diagram 3.1 The ratio of the results of the tests I and II of the first and second round of experiment Diagram 3.1 is show that: The average mark; The percentage of students who achieved good and excellent mark in the experimental class is higher than the control class The question is: Is the method of experimental class better than the method of control class, or is it just random? We propose the statistical hypothesis H0: "There is no difference between the two methods" and using Method U (See [73, p 58]) to refute H0” With significance level = 0.05, the critical value in all four tests are shown evidently: Because u U , then Hypothesis H0 is rejected So the method of teaching in the experimental class is better than the method of teaching in the control class 3.6 General conclusion of Chapter The experimental process with the results drawn after the experiment shows that: the purpose has been completed, the feasibility and effectiveness of the views have been confirmed Implementing pedagogical measures in teaching Maths to students specializing in Physics, Chemistry, and Biology, which the thesis proposes will contribute, is very well to foster the competency of exploring and obtaining knowledge for students of these specialized subjects and high school in general At the same time, the results of the thesis have contributed significantly in theory and practice to improve the effectiveness of teaching mathematics in high schools, specialized high schools, and especially the specialized system of Physics, Chemistry and biology 24 CONCLUSION There are some main findings of the paper It has systematized scientists’s views on activities, cognitive activities; giving defintion of the term improving the knowledge – grasping competence, which contribute on the identification of typical components of improving the knowledge – grasping competence It has introduced high school students the concept of improving the knowledge – grasping competence in teaching maths through six elemental competencies that has been described by compatible activities The paper also has studied the reality of improving the knowledge – grasping competence in teaching maths in gifted high schools in general high school studens majoring in physics, chemistry and biology in particularly It is essential to develop this competence by applying pedagogical measures and teaching topics It has introduced directions and developed six pedagogical measures to improve the knowledge – grasping competence in teaching maths for high school students majoring in physics, chemistry, and biology The paper has designed some topics that can be suitable for the main aim of the research It has organized pedagogical experiments to illustrate the feasibility and effectiveness of the proposed pedagogical measures LIST OF SCIENTIFIC RESEARCH OF AUTHOR RELATED TO THE THESIS Nguyen Tran Lam (2015), “Some measure contributing to training the problem identifying and solving skills in teaching Chaper III: The space coordinate method (12th grade Geometry), Jounal of Education Science – Ministry of Education and training, Volume 355, p 52 – 55 Nguyen Van Thuan - 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