Luận án Dạy học một số nguyên lí của toán rời rạc trong chương trình bồi dưỡng học sinh khá và giỏi ở trường trung học phổ thông

d and excellent students in high school in Vietnam. + Researching the system of contents that are necessary and possible to teach some of the principles of Discrete Mathematics in high school. + Proposing a number of teaching measures applied some principles of discrete mathematics for fostering program of good and excellent students in high school + Pedagogical experimentation to assess the feasibility and effectiveness of the research topic. 4. Research Methodology + Methods of theoretical research: Learning the history of research and theoretical issues related to the topic directing the research. Researching contents of Discrete Mathematics, contents of Mathematics in high school and teaching methods of Mathematics. + Methods of survey and observation: Collecting and analyzing data through survey, observations of teaching process of Discrete Mathematics topic in high school. + Methods of Case study. + Methods of Pedagogical experimentation; Deploying pedagogical experimentation, teaching this topic based on the results of the topic in gifted schools to test the feasibility and effectiveness of the topic. 3 5. Scientific hypothesis If teaching principles of Discrete Mathematics in fostering program of good and excellent students in high school under contents and measures proposed in the thesis, the quality of teaching and learning this topic in high school will be raised. 6. The issues given to defend + The demand and the necessity to put more contents of some principles of discrete mathematics for fostering program of good and excellent students in high school. + The contents and methods of teaching some principles of Discrete Mathematics for good and excellent students in high school are proposed in the thesis with scientific and practical sense. + The measures proposed in the thesis are feasible and effective. 7. The contributions of the thesis + The thesis has proven necessity and possibility of teaching some principles of Discrete Mathematics in fostering program of good and excellent students in high school today. + The thesis has proposed contents and methods of teaching some principles of Discrete Mathematics for good and excellent students in high school. + Experiments have confirmed the feasibility and effectiveness of the solutions that the thesis has proposed. 8. The structure of the thesis In addition to the introduction, conclusion, references and appendixes, the thesis includes four chapters: Chapter 1: Theoretical and practical foundation. Chapter 2: Objectives and contents of teaching the principles of Discrete Mathematics in fostering program of good and excellent students in high school. Chapter 3: Some methods of teaching the principles of Discrete Mathematics in fostering program of good and excellent students in high school. Chapter 4: Pedagogical Experimentation. 4 Chapter 1 THEORETICAL AND PRACTICAL BASIS 1.1. Overview of Research Issue 1.1.1. These studies on putting the Discrete Mathematics into Mathematics program in high schools in some countries in the world. In 1989, the National Council of Teachers of Mathematics (NCTM) of America announced Curriculum and evaluation standards for Mathematics. This document recognizes the importance of the topic Discrete Mathematics in high school program. This is an important milestone for the encouragement of putting Discrete Mathematics into elementary schools and secondary schools in the United States. After this document was published, numerous studies on Discrete Mathematics have confirmed the importance of teaching Discrete Mathematics and described the content of the Discrete Mathematics in high schools. In addition to, some programs have been built in preparation for the teachers in teaching Discrete Mathematics and attract them to combine Discrete Mathematics in the classroom. In 2000, NCTM released the revision of Curriculum and evaluation standards for Mathematics into Principles and standards for school mathematics [PSSM], in which there is no separate Discrete Mathematics criterion as in the previous version but the topics of Discrete Mathematics are distributed on the standards, from kindergarten to grade 12. However, many researchers are working to integrated Discrete Mathematics into the curriculum, school textbooks. The view of many authors is that Discrete Mathematics is not just a collection of interesting and new math topics. More importantly, the Discrete Mathematics is considered as a mean which provides teachers with new ways of thinking about math topics and new strategies to engage their students to learn math. 1.1.2. Some research works mentioning the principles in Discrete Mathematics a, In foreign countries b, In Vietnam Basing on statistics of principles mentioned in various documents, we find 6 principles which are mentioned most: The sum rule, The Product Rule, The Box Principle, The Inclusion – Exclusion Principle, The Induction Principle and The Invariance Principle. This is one of the foundations for us to choose the principle to shift in the next chapter. 1.2. Discrete Mathematics and its role in mathematics and in practice. 1.2.1. The formation and development history of Discrete Mathematics. 5 "Discrete Math or Discrete Mathematics is the common name of many branches of mathematics which have the object of study is discrete set; these branches were gathered from the appearance of computer science as the mathematical basis of computer science. It also called the mathematics for computers. There are some factors which are often mentioned in discrete mathematics: combinatorial theory, graph theory, complexity theory, Boolean algebra. Rosenstein, Franzblau and Roberts (1997) confirmed that: Over the years, Discrete Mathematics has formed and developed rapidly. Discrete Mathematics becomes an important field of mathematics. Increasingly, Discrete Mathematics is used in many branches of work. Discrete Mathematics is the language of the great science units. 1.2.2. The role of Discrete Mathematics in Mathematics programs in high schools Through these documents, we see the role of Discrete Mathematics in Mathematics programs in high schools presents in the following basic points: - Discrete Mathematics can be taught to all students at all levels. - Discrete Mathematics encourages a discovery approach in teaching. - Discrete Mathematics can be applied to everyday situations. - Discrete Mathematics helps teachers to have a new look compared to the traditional mathematics. - Discrete Mathematics provides the mathematical problems which are quite challenging but accessible for the students who love math. - Discrete Mathematics is a great tool to develop thinking and mathematics-solving skills. 1.2.3. The role of Discrete Mathematics’ principles in practice Discrete Mathematics in general and Principles in Discrete Mathematics in particular have contributed to create many new scientific achievements. These achievements have high applicability in the areas of life such as telecommunication, transportation, industrial production and distribution of energy 1.4. The current status of teaching Discrete Mathematics in high schools in Vietnam. 1.4.1. Methods and ways for investigation of the current status. a, Discrete Mathematics in Mathematics program of Vietnam - In high school curriculums and textbooks. - In Mathematics textbooks and other references. b, Conducting a survey through the expert opinions 6 We have conducted three surveys through using opinion sheet for collecting information. - The 1st survey on August 2012, in the professional training period for the key Mathematics teachers all over the country. The object of survey is 70 teachers from Specialized High Schools and mathematics specialists of Department of Education and Training of the provinces in the country. - The 2nd survey on December 2013 in Hai Phong. We survey the opinions of 40 key Mathematics teachers from schools, Department of Education and Training of 14 Northern provinces. - The 3rd survey at Hung Vuong Summer Camp of Specialized High Schools in the midland and the northern mountainous region held in Quang Ninh Province. The object of survey is Teachers and Students who are good at Math of Viet Bac Upland High school and 16 Specialized High Schools in the midland and the northern mountainous region. The results are as follows: * 100% of surveyed teachers agreed with the two following contents: + The content of Discrete Mathematics in current Mathematical textbooks and references specialized in Mathematics are not sufficient to use as documentation to foster good students. + It’s necessary to put some principles of Discrete Mathematics into fostering program of good and excellent students in high school. * Over 90% of teachers agreed to teach the topic of Discrete Mathematics for good and excellent students in high school according to the following sequence: + Class 10: Teaching the basic contents: The sum rule, The Product Rules, Combination, Arrangement, Permutations, Newton's binomial, The Pigeonhole Principle, The Inclusion – Exclusion Principle, The Induction Principle. + Class 11: Basic teaching of principles: The Extremal Principle, The Invariance Principle. Advanced teaching of learned Discrete Mathematics’ topics + Class 12: Advanced teaching of Discrete Mathematics’ topics. * The teaching hour for principles of Discrete recommended by teachers is about 15-50 hours. * The survey results show that teachers and students are facing with many difficulties in teaching the Discrete Mathematics in high schools and wish to have measures to overcome these difficulties. The survey results 7 also orient us to propose measures for teaching the principles of Discrete Mathematics in the Chapter 3 of the thesis. c, Statistics of test results having the Discrete Mathematics content of International Math Olympiad Team (IOM) of Vietnam and some countries in the world. Statistics to compare the level of students in our IOM Team in the field of Discrete Mathematics to students of advanced countries and some countries in the world. 1.4.2. Evaluation of the survey’s results of the current status of teaching Discrete Mathematics in high schools. Through the results of survey, we can confirm that: - The content of Discrete Mathematics in high school textbooks and reference materials in Vietnamese in our country is not sufficient to meet the needs of fostering good and excellent students. - The level of our students in recent years in the field of Discrete Mathematics compared to students in advanced countries in the world and some countries in the region is limited. - It is necessary to put more content of Discrete Mathematics into fostering program of good and excellent students in high school. The first step of this work is to put the content of Discrete Mathematics’ principles into the program. Need to propose the content and methods of teaching these principles for good and excellent students in high school. Chapter 2 TEACHING OBJECTIVES AND CONTENTS OF THE PRINCIPLES OF DISCRETE MATHEMATICS IN FOSTERING PROGRAM OF GOOD AND EXCELLENT STUDENTS IN HIGH SCHOOL 2.1. Teaching objectives for the principles of Discrete Mathematics in fostering program of good and excellent students in high school 2.1.1. The basic features of high school good and excellent students 2.1.2. Teaching objectives for the principles of Discrete Mathematics in fostering program of good and excellent students in high school + Teaching mathematical reasoning and proof techniques for students in the process of principles teaching. Develop creative thinking, logical thinking for students. + Forming some capacities for students such as: problem-solving ability, creative ability, self-learning ability, communication ability, cooperating ability, calculation capacity, the capacity to use language... 8 + Supplementing mathematics contents associated with the practice in Mathematics program for high school good and excellent students. + Stimulating the student’s passion in mathematical studies through interesting topics of Discrete Mathematics. + Training and Developing the Discrete Mathematics knowledge for good and excellent students of high school in Vietnam. Equipping with the knowledge for students in the future to prepare capacity of accessing to the modern science and technology in the world. 2.2. Shift in pedagogy 2.2.1. Concept of shift in pedagogy In education, knowledge shift is seen as a philosophical cornerstone of the teacher. Shift phenomenon is probably the most important, but little known in the process of teaching - learning. According to accepted views, knowledge shift is considered as the application of a solution to a situation unknown prior to that time. Shift is based on generalization skills and abilities of abstraction. In psychology, shift is defined as behavior in which an affection for a person, an object is spread to others. According to Prof. Nguyen Ba Kim, Methods of teaching Mathematics, page 201: On the knowledge components, in theory of teaching, Yves Chevallard first analyzed the overall process of the transformation of scientific knowledge into teaching knowledge and called pedagogical shift (Chevallard 1985 and Verret 1975). In this process knowledge is considered in terms of three levels: scientific knowledge, program-based knowledge and teaching knowledge. Scientific knowledge: The knowledge found by researchers. After non-chemical circumstances, timeless goods, non-personalized, scientists announced under a generalized form as possible, according to the current rules expressed in the scientific community. Program-based knowledge: is scientific knowledge after the screening, and determining the required level and manner of expression consistent with the objectives and conditions of the society to ensure the interoperability of the learning system with its environment. Teaching knowledge: At the level of class, we talk about teaching knowledge. To achieve teaching objectives, the teacher must organize the teaching knowledge specified in the program, textbooks and transform them into teaching knowledge in accordance with their pedagogical ability, the constraints of class, student performance and other learning conditions. 9 According to didactic, program-based Knowledge of Mathematics is also known as Knowledge necessary to teach. Teaching Knowledge is also known as Knowledge taught. Shift in pedagogy or pedagogical transformation (transposition didactique) is a process with two stages: transformation from the scientific knowledge into program-based knowledge and from program-based knowledge into teaching knowledge. The main stages of the pedagogical transformation process are: In this thesis, we have greater emphasized on stage 2. 2.2.2. The need for pedagogical shift from scientific knowledge into teaching knowledge in teaching math in high school The Era is getting developed, to keep up with the development of science and technology, the knowledge of Mathematics for high school students must be changed: removed the old and backward sections, adding new, necessary sections in accordance with the requirements of new real life. Therefore, we have to select some contents of scientific knowledge consistent with high school students, then design the ways of content organizing and teaching for high school students. This shift is a crucial principle, has occurred and will occur in high school education activities 2.3. The teaching content of some principles of discrete mathematics for fostering program of good and excellent students in high school Orientation of content building for the curriculum of some Discrete Mathematics principles for fostering program of good and excellent students in high school: (1) Content proposed must includes the basic theoretical issues, questions and exercises with the different levels of difficulty and complexity. The system must include basic mathematic forms, are classified from easy to difficult, from simple to complex to be suitable for good and excellent students in high school. (2) Systems of theory and exercises must develop mathematical thinking, form the capacity of the students and is the basis for implementation of teaching methods mentioned in chapter 3. (3) The exercises proposed must have guidance or detailed explanation. Polymath knowledge (institution creating knowledge) Knowledge to teach (transforming institution) Transfer transformation Knowledge taught (teaching institution) Learning transformation 10 (4) Content must be selected in accordance with the objects of high school’s good and excellent students. From the above directions, we propose to teach the principles of discrete mathematics for good and excellent students: The sum rule and The Product Rules, The Box Principle hay The Pigeonhole Principle, The Induction Principle of Mathematics, The Inclusion – Exclusion Principle, The Invariance Principle. 2.3.1. The sum rule and The Product Rules 2.3.1.1. The sum rule and The Product Rules a. The sum rule b. The Product Rules 2.3.1.2. Applied exercises 2.3.2. The Box Principle or The Pigeonhole Principle 2.3.2.1. The Box Principle or The Pigeonhole Principle 2.3.2.2. Applied Exercise: 2.3.3. The Inclusion – Exclusion Principle 2.3.3.1. The Inclusion – Exclusion Principle 2.3.3.2. Applied Exercises 2.3.4. The Induction Principle of Mathematics 2.3.4.1. The Induction Principle of Mathematics 2.3.4.2. Applied Exercises 2.3.5. The Invariance Principle 2.3.5. 1. The Invariance Principle 2.3.5.2. Applied exercise 2.4. Some proposals for teaching organization of some principles of discrete mathematics for fostering program of good and excellent students in high school Content proposed above could be the basis for educators, professionals and people who make the Mathematics program for good and excellent Students to perform stage 1 of the process of pedagogical shift. It was moved from scientific knowledge into knowledge of program for six principles outlined in the thesis. With the idea of focusing on the second phase of pedagogical shift, we propose teaching methods for the principles of Discrete Mathematics in Chapter 3. Chapter 3 SOME MEASURES OF TEACHING PRINCIPLES OF DISCRETE MATHEMATICS IN FOSTERING PROGRAM OF GOOD AND EXCELLENT STUDENTS IN HIGH SCHOOL Orientation of developing measures: 11 1) Measures which are consistent with educational objectives and current trend in innovative instructional methods can be executed in actual conditions of process teaching discrete mathematics to good and excellent students in high school. 2) Those measures must heighten students’ role in self-building knowledge based on their previous knowledge and experience. The measures help students to form some capacities such as: problem-solving ability, creative ability, self-learning ability, communication ability, cooperating ability, calculation capacity, the capacity to use language... 3) Measures should assist good students in high school to get interest in study, thereby stimulating their curiosity, as well as developing their creative minds, critical thinking, and logical thinking for good and excellent students in high school. 3.1. Measure 1: Evoking motivation and creating excitement for students through the use of visual appliances, information technology products and problems with practical contents. a, Purposes, significances Using the visual appliances or some educational software in discrete mathematics lessons will create excitement for students. Besides, teachers can use it to organize activities for students so that students may understand problems better. Therefore, they can form methods of answering mathematical problems. Teaching software is tool for teachers and students designing games and lectures, thereby, students may self- consolidate their knowledge spontaneously. Discrete mathematics is the math of life. It is easy for teacher to find mathematical problems with practical contents to create motivation for students. b, Scientific Basis Visual problems have been played an important role in teaching. One of the problems bringing efficiency in teaching is how to select and use visual elements in education. Visualization in teaching activities means the concept denoting nature of cognitive activities, in which information are obtained from external things and phenomena by direct perception of human sensory organs. c, Implementation methods Teachers should actively use visual appliances, making full use the support of the teaching software in order to reduce abstraction of discrete mathematics problems. Thus, students may easily understand raised problems in each specific case. Then, students can understand problems in the general case. 12 Technique 1: Evoking motivation and creating excitement for students by using some visual appliances in teaching principles of discrete mathematics. To support our teaching, we have used sets of magnets attached to magnetic board during teaching problems on counting or on games related to stones. Instead of stones in mathematical problems, teachers can use beans which are available in their family. Similarly, in mathematical problems on chess board, we can use the side board with cross-shaped cells. Or simple strings can also be used in model construction to generate excitement for the children in all lessons on the Injection principle and the Bijection principle. The general principle of teaching discrete mathematics by visual appliances is: use the right purpose, right time, right place, the right level and intensity; ensure consistency between the specific and the abstraction. Visualization is a basis to predict and discover. Students should know abstract thinking immediately after using visual appliances. Technique 2: Evoking motivation and creating excitement for students by using some IT software during teaching principles of discrete mathematics. In addition to use IT software in support of teaching, teachers should encourage and guide students to use above software to design games and group reports on topics of discrete mathematics. Through the design process, students are able to collect a system of exercises and to self-reinforce their knowledge. They will be more confident and excite when teachers use their products during the lesson to reinforce knowledge of other classmates. Example 3.2: Using Adobe Presenter software to design e-learning lessons through the game "Adventures of Mario in combinatory country". (There are discs enclosed with thesis). Lectures integrate instructional programs under branching program and straight line program. The idea of the game is created by a group of 3 students specializing in Mathematics 10 K22 in Thai Nguyen High School for gifted students. Then, we have applied, developed and designed E - learning lectures. Under the guidance of Prof. Dr. Nguyen Ba Kim, this lecture has been shortlisted in the finalists of "Designing E - learning lesson” in national contest, academic year 2011 - 2012 and received gifts from organizing committee. Purpose of the game is to review and reinforce some Combination knowledge. The main idea of the game is that a pupil plays a role of Mario to wander the Combination country. Destination is Combinatory Mount with a victory flag plugged on its slope. Mario must go through 3 or 4 13 zones: the island of two basic counting rules (The sum rule, the product rule) (Zone 1), Permutation – Arrangement – Combination Gulf (zone 2), Land of danger (zone 3), Newton’s binomial forest (zone 4). In each zone, player encounters mathematical problems which must be answered before moving to the next. The player starts at zone 1 and moves to zone 2. If player can answer a difficult problem in the end of zone 2, he is offered a shortcut to zone 4. If this problem cannot be answered, the player will fall in zone 3 and move to zone 4. In the final stage, player moves from zone 4 to Combination Mount. The image of Mario plugged victory flag in the mount slope but not its peak implies: There is more various, interest and difficult Combination knowledge in particular and discrete mathematics knowledge in general waiting for student to discover. We have designed various types of exercises: player must fill the answers in the blank, choosing one in given answers or connecting options. When player has given out the answer, evaluation and specific solution of the problem will be shown. Most of mathematical problems are designed in a straight-line program. Specifically, special problems are designed in branching program. We have scheduled the two most common errors when students solve this problem. The, students must choose one of three given answers (only one is correct). In fact, this is three paths. If students give the correct answer, the will receive a compliment and two solutions of the problem, then moving on. If students give the wrong answer, they will get the hint which explains the reason for wrong answers. In this case, students are asked to click on a link to a special problem to do it again. Technique 3: Evoking motivation and creating excitement for students by problems with practical contents during teaching principles of discrete mathematics. Lenin pointed out the general path of human perception is: "From vivid intuition to abstract thinking and from abstract thinking to practice - which is the dialectical path of perception of truth and perception of objective reality". Stemming from practical problems, knowledge on discrete mathematics is developed; then obtained knowledge is further developed and applied in solving more difficult problems. By applying this path, we wrote the article "Around Euler’s problem in dividing candy" which was published in Journal of Mathematics and Youth on 10/2012. This article has been reported in Hanoi Mathematics Seminar held in Thai Nguyen on 11/2012 and in training course for approximately 60 teachers of 14 Mathematics of Thai Nguyen province on 8/2013. Through the report, we also presented above views and received the support of professionals and colleagues. However, the way starting from practice and creating the most appropriate attention for students is derived from mathematical problems associated with events happening in the classroom. In the course of practicing and creating new mathematical problems, a group of students specializing in Mathematics K 25 wrote a series of problems which mention memories of the class, daily work taking place in classroom. 3.2. Measure 2: flexible application of some active teaching methods for Discrete Mathematics principles teaching. a. Objectives Besides the objective of attracting students in learning, this measure has been developed mainly: to promote students’ self-discipline, passion and self-achievement under the instruction of teachers. b. Scientific basis Teaching methods are teachers’ activities and behaviors to hold activities and interaction between the teacher and students to achieve teaching purpose, according to Prof. Dr. Nguyen Ba Kim. In reality, there is no so-called effective teaching method for all teaching objectives and contents. The combination of different methods and teaching forms in teaching process, therefore, is an important direction to promote students’ enthusiasm and to improve education quality. c. Applying methods The principle “students will complete tasks assigned under the teacher’s organization and guidance on their own” must be implemented by the teachers when applying the teaching methods. We proposed some teaching methods for teaching Discrete Mathematics principles: Learning games, group teaching, problem solving, self-study. Technique 1: The application of teaching methods in official class hours Example 3.4: (Holding activities for students through the use of visual items to find tactics to win the Nim game – a Chinese traditional game) Step 1: The teacher raises specific problems: “There are 3 piles of pebble including a pile of 2 pebbles, a pile of 3 stones and the other of 4 pebbles. Two students in turn take away a pile of pebbles or some pebbles in one random pile. The student who gets the last pebble shall be the winner. 15 Students will in pairs play the game in class held by the teacher. The two pairs use magnets mounted on the classroom board. Students are required to record their status played (see illustration) This is the situation of raising problem finding tactics to win the game needed to be solved. When two students play together in the game, the winner can be identified after a finite number of steps. Students believe that they will find out rules to win due to limited numbers of pebbles. It is, however, uneasy to find such rules. Step 2: The method for counting sum of nonnegative integer of Nim being similar to that of numbers presented in binary form will be guided by the teacher. Then, apply to count the Nim sum of numbers in each recorded status (see illustration). Step 3: Looking at the change in the Nim sum to find the rule of victory. Since the final status when there is no pebble to take having sum of Nim equal to 0, students will find the tactics: the person who always changes the different Nim sum into that of 0 after his/her turn will be the winner. In the given problem, students can find the specific rule without difficulty on account of the limited number of pebbles. Students figure out that where the two players have already known the rule, the victory will depend on the beginning status. The second player will win the game if the initial state has a total Nim of 0. Where the total Nim is different from 0, the other will be the winner. Example 3.5: The use of scaffolding technique in macro level designed teaching to help student grasp the problem of candy division of Euler. Problem: There are n identical candies divided for m babies (m, n are positive integers) How many ways to divide the number of candies? This is an interesting mathematic problem providing many applications in combinatorial problem solving. To handle this problem, students are assumed to already know some basic knowledge as below: - The concept of k combination of n particles. 16 - The concept of a binary sequence. - The number of binary sequences of n in length, each sequence with k particles equal to 1 is: knC A small experience should be provided to students by the teachers before raising the mathematic problem is: A problem with two main subjects can be taken back to the binary sequence one for an easier resolution. Before solving the problem, the teacher can conduct some following preparations for the micro scaffolding method: Option 1: Scaffolding technique for students who got the idea of changing the problem concerned into a binary sequence-related one. ? What is the assumption of the problem? ! There are n identical candies divided to m babies ? What are the problem requirements? ! To count how many ways to divide candies that satisfies initial requirements of the problem ? In your opinion, how many main subjects are there in the problem? ! There are two main subjects that are the babies and candies. ? Is there any idea come up with out of this question? ! I will change the problem into binary sequence-related one? ? So what subject shall be considered 0 and what is 1? ! I think that of two main subjects, one subject is regarded as 0 and the other is 1. For example, I regard each baby as 0 and each candy as 1. ? How to have candies divided corresponding to a random sequence of binary? ! I have to put the numbers 0 and 1 in a row ? So what is your next coming idea? ! If the baby is given the number of 0 candy, I will write only: 0 In case he/she is given a number of k candies, I will write: 0 sè 1 11 ...1 k I will continuously write from the first baby to the m one to form a binary sequence having m of 0 and n of 1 ? Can you make it clear by giving an example? ! For instance, if there are 7 identical candies divided among 3 babies and the first baby got 3 candies, the second one received no candy and the last one got 4 candies, I can write down 0111001111. ? Well done! So do you know how to solve the initial mathematic problem? 17 ! Each way of candy division will be corresponding to a binary sequence with length of (m+n); in which it exists m of 0, n of 1 and one particle of 0 will always come at the beginning of the sequence. Hence the results needed are”: ? If we refer xi as the number of candy given to the baby number i, what is the result? ! We have equation: x1 + x2 +... + xm = n (1). ? Each natural solution of the linear equation (1) is a set of (x1, x2,...,xm) that satisfies (1) and Can you see any association between that solution and the above dividing method? ! Each solution will be corresponding to a dividing method and vice versa. ? So what conclusion can you draw? ! The number of natural solutions of equation 1 is 1 1 m n mC    ? You should remember the method to solve problems for finding natural solutions of equation form 1with xi being blocked. Therefore, applying to solve actual mathematic problems. Option 2: (To students having the idea of changing the problem into the one of finding natural solutions of a linear equation) ? What is the mathematic problem assumption? ! There are n identical candies needed to divide among babies. ? What are the problem requirements? ! To count how many dividing methods that satisfy problem requirements. ? What is your idea? ! Refer xi as the number of candies given to baby number i, 1,i m . The number of dividing method will be equal to the natural solutions of the equation: x1 + x2 +... + xm= n (1). But I am stuck here. ? We try to solve the following sub-problem: Give a grid of squares. The nodes are numbered from 0 to (m-1) in the direction from left to right and from 0 to n vertically from bottom to top. How many different paths are there from node (0, 0) to node (m-1, n) if only allowed to go on the edge of the square in the direction from left to right or from bottom to top. 18 1 1 1 1 0 0 0 0 0n m-1 (m-1,n) (0,0) The teacher guides students to solve the sub-problem. ? What are the features of each route which meets the requirements of the mathematic problem? ! Merely allow moving on edges of the box from left to right or upward. ? It means that there are 2 core factors: Following the horizontal line or following upward. So can we make it as the binary problem? ! Yes. I will encode each upward segment by the number 0, each horizontal segment will be coded by the number 1 (each segment will have the length equivalent to the length of the box). ? Therefore each segment satisfies requirements equivalent to the binary sequence. What are the characteristics of each binary sequence? .. ? Can you try to consider again the assumption? What are the more characteristics of each route? ! Originating from node (0, 0), end at node (m-1, n) being only allowed to go horizontally from left to right or upward. ? By that characteristic, do you have any inference about the number of horizontal segments and upward segments in each route? ! There have only (m-1) horizontal segments and n upward segments. Thus the mentioned above binary sequence has the length of (m + n - 1) in which there have only (m-1) element of 0. Therefore you will have answer for the supplementary problem. ? Let look at the image and regard each vertical line passing through the respective nodes (0,0), (1,0), ... (m - 1, 0) are m babies, what factors are the candies that the babies can be received? Can you look at the image which we have above? 19 ! The first baby can be considered the first vertical line, by the image, our route passing through one segment of this vertical line. The second baby can be considered the second vertical line; our route does not pass through any segment of this vertical line. The total candies are rightly equivalent to the number of segments in each route. You have already understood. In the above circumstance, the first baby is received 1 candy, the second baby is received none. ? Do you have answer for the mathematic problem of dividing candies? ! Considering that each route meet the requirements, call xi is the number of segments in the vertical line through the node (i-1, 0), 1,i m , you will have the equation (1) mentioned above. Each route satisfies equivalent to the natural solution of (1). Each solution is equivalent to one way of dividing candies. There have 1 1 m n mC    routes meeting the requirement, so there have 1 1 m n mC    ways of dividing n similar candies to m babies. Techniques 2: Enhance self-studying through the material of Discrete Mathematics Self-studying is a suitable method to good and very good students. Teacher should step by step guide the method of self- studying to students. Start with the way of self-noting in the lessons, the way of making plan for Discrete Mathematics topic for themselves and searching for materials on the Internet. 3.3. Measure 3. Training students to solve the problems applying the principles of Discrete Mathematics under the levels: smoothness – flexibility – creativeness. a, Purpose and Meaning General education in our country is changing from approaching education contents to learner’s capacity. This means we take care of students, who applied something instead of learnt something. To ensure that, it is necessary to carry out successfully the movement from teaching methods under “one-way transmittance” to the way of studying and applying knowledge, practicing skills and forming capacity and quality. b, Scientific basis According to Professor Bui Van Nghi (2009), the capacity is subjective or natural ability and condition to do something. It is qualities of psychology and physiology that make people to fulfill something with high quality. Capacity to apply the principles in solving the problems includes basic skills: identifying the principles need to apply in solving the 20 problems, analyzing to find out the answer of the problems, creating from old problem to new one. c, Implementation methods Technique 1: Training students to recognize the principles, which need to apply, thenceforth, finding out the methods to solve the problems. Teachers direct students to understand the contents of each principle and kinds of problem. Then, teachers guide students to identify the principles, which will be applied through exercises system. Technique 2: Step by step develop the capacity in solving the discrete mathematics for students from expansion, generalization to major exercises. Beginning from depend on specific cases to extend gradually the problems by adding or cutting the datum in the assignments. Technique 3: Finding out the way to solve the problems in general thanks to review the specific cases. In November 2013, a Mathematic seminar held in Thai Binh Province by gifted schools in the Coastal and North Delta areas. The subject of the seminar is how to teach the Discrete Mathematics for students of high school. Conclusion of the seminar affirmed that: the methods of approaching many subjects of the Discrete Mathematics are little cases in fact and then becoming to a general problem. This way is applied by Teacher Nguyen The Sinh (Nguyen Trai Gifted School-Hai Duong) in the article: “Starting from little cases”. It is published in the Proceeding of Workshop and classified as excellent. In this article, the writer gave many examples and reasons for application of this opinion in solving the Discrete Mathematics. 3.4. Measure 4: Combining the teacher’s assessment with self- assessment of students in teaching the principles of Discrete Mathematics to get more effectiveness in teaching and studying. a, Purpose and Meaning Test and assessment are integrated in using during teaching process to adjust teacher’s teaching methods and student’s learning methods aim to support for teaching and studying to come into effective. b, Scientific base: Assessment is comments and judgments the result or advancement. The Decree of 8th Central Conference, XI session on basic and comprehensive innovation of education and training stated: “basic innovation of examination form and methods, test and assessment of education and training result need to as step by step as the advanced standards, which is 21 trusted and recognized by society and educational community in the world. Combining and using the assessment result during learning period and at the end of semester and school year; assessment of teachers as students’ assessment; assessment of school with assessments of families and society. c, Implementation methods: Technique 1: Giving the comments and compliments, pointing the mistakes through reviewing the written test and right after the students solved the examples and class exercises in the lesson. Specifically, we executed the following works: - Marked the errors that students had met. Analyzed the reasons and measures for students to fix those mistakes. - Repaired the exercises in front of the class carefully. Teachers collected the common and rare mistakes, the good way to recommend for students. Technique 2: Using the period assessments, making diverse of the exam form (motivation and opportunities for students). We used the assessing tools as below:  Tests under the program distribution of the Ministry of Education and Training  Observed form the positive attitude of students in class.  Learning result or records with coefficient of 2. Technique 3: Students self-assess We recommended two assessment forms. They are self-assessment and peer assessment. 3.5. Measure 5: Overcoming and repairing the mistakes in applying the principles of Discrete Mathematics for good students in High School. a, Purposes and Meaning: Discovering their own mistakes or someone else will help students to understand the knowledge more certainly. b. Scientific base: According to the opinion of Brousseau: “Mistake is not only consequence of unknown, unsure, accidental reasons of people, who is following the empiricism and behaviorism, but also maybe consequence of existing knowledge. The knowledge used to be useful for study before but not true or not suitable to learn new one any longer”. c, Implementation Methods: 22 Technique 1: Directing students to grasp the concepts and contents of principles and forming logic skill through the problems. Technique 2: Organizing activities in mistaken situations aim to help students to detect the mistakes and figure out how to fix it. During the period of teaching the principles, external forces – impact of teachers and studying environment will improve and resonate with internal forces – self-study capacity of students to complete the outlined objective by using the recommended measures. Ideology changing from “external” into “internal” of pedagogical movement in phase 2 is carried out in here. Chapter 4 PEDAGOGICAL EXPERIMENT 4.1. Purpose, Request, Contents of Pedagogical Experiment 4.1.1. Purpose Pedagogical Experiment aims to review the feasibility of applying the recommended measures in teaching some principles of discrete mathematics for good and excellent students students in High School. At the same time review the efforts of teaching under such measures to study result of students. 4.1.2. Request Pedagogical experiment has to undertake the objectivity of experiments. It has to be suitable with students and close to real status of teaching. 4.1.3 Contents of pedagogical experiment 4.1.3.1. Content 1: Experiment to teach chapter II. Combination and Probability, Section A. Combination in Advanced Algebra and Calculus 11 Program, including 7 periods. 4.1.3.2. Content 2: Experimenting at some specialized classes on Dirichlet Principles Topics for students specializing on math 10 with 6 periods. 4.1.3.3. Content 3: Experiment depends on research about 5 students of math class course 25 at Thai Nguyen Gifted High School 4. 2. Time, procedure and methods to assess the pedagogical experiment 4.2.1. Time for pedagogical experiment From January, 2010 to August, 2014 4.2.2. Experiment objects 23 a, Experiment objects of the content 1 are two chemistry-specialized, biology-specialized classes of 11 grade - course 23 of Thai Nguyen Gifted High School. b, Experiment object of content 2 are two math classes of grade 10 c, Experiment object of content 3: Researching 5 students of math- specialized classes-course 25 of Thai Nguyen Gifted High School. 4.2.3. Progress of Organizing the Experiment 4.2.4. Assessment methods of the Pedagogical Experiment’s result. 4.3. Progress of Pedagogical Experiment 4.3.1. Content 1 of Pedagogical Experiment 4.3.2. Content 2 of Pedagogical Experiment 4.3.3. Content 3 of Pedagogical Experiment After experiment, we draw comments from basis of results as follows: - Suggested contents at section 2.3 are suitable with good students in High school. - The use of recommended teaching methods in previous chapter is possible and effective at the beginning that contributed to the improvement of the teaching and learning quality of Discrete Mathematics in High School. - Many students of experiment classes get the way to self-study the Discrete Mathematics and work in groups and create new problems, new games. Those help students to learn Discrete Mathematics. Thus, the thesis can be considered to pass the purpose of research. 24 CONCLUSION The thesis collected the following results: 1. Demonstrated that the demand and necessity to further add the contents of some principles of discrete mathematics for fostering program of good and excellent students in high school 2. Recommended the purpose and contents in teaching the principles of discrete mathematics for fostering program of good and excellent students in high school 3. Recommended 5 measures often used in teaching the principles above. 4. Organized the pedagogical experiment to illustrate the feasibility and effectiveness of pedagogical measures above. The result of experiment showed that the measures recommended in the thesis are feasible and obtained initial good results. From the main results above, we can conclude that: Scientific theory of the thesis is acceptable. The researching purpose of topic is completed. The contributions of the thesis can be deployed and applied in practice of teaching Discrete Mathematics in High School for good and excellent students. The research works of the author involving the thesis published 1. The scientific articles 1. Nguyen Thi Ngoc Anh (2013), Teaching the Box Principle for good students in high school, Scientific Journal – HNUE (vol. 58), p. 28 – 35. 2. Nguyen Thi Ngoc Anh (2014), Applying teaching theories on Vygotsky's zone of proximal development in teaching discrete mathematics for good students in high school, Scientific Journal – HNUE (vol. 59, No. 2A), p. 136 – 144. 3. Nguyen Thi Ngoc Anh (2014), Guiding good and excellent students in high school to apply bijection method to solve some of counting problems, Journal of Science and Technology - Thai Nguyen University (vol. 128, No. 14), p. 127 – 131. 4. Nguyen Thi Ngoc Anh (2014), Suggesting motivation, creating excitement for students by using visual appliances, information technology products, and problems from the practice of teaching discrete mathematics in high school, Journal of Education (vol. 1, No. 345), p. 48 – 50. 5. Nguyen Thi Ngoc Anh (2014), Situation of teaching Discrete Mathematics in high schools in Vietnam, Journal of Education (vol. 1, No. 345), p. 52 – 55. 2. Reports at scientific conferences, training courses for high school teachers 1. Nguyen Thi Ngoc Anh (2012), Around Euler's candy division problem, report in the Scientific Conference by the Hanoi Mathematical Society held in Thai Nguyen in November, 2012. 2. Nguyen Thi Ngoc Anh (2014), Guiding good students in high school to apply bijection method to solve some of counting problems, report in the training course of Mathematics teachers of Thai Nguyen Province in August, 2013. 3. Nguyen Thi Ngoc Anh (2014), Teaching the box principle for students specializing in Mathematics, report in the training course of teachers specializing mathematics held by Institute of Advanced Mathematics in September, 2014.