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MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY NGUYEN DANG MINH PHUC INTEGRATING DYNAMIC MANIPULATIVE MODELS WITH ELECTRONIC MATHEMATICS TEACHING AND LEARNING ENVIRONMENT FOR ENHANCING THE STUDENTS’ ABILITY OF EXPLORING NEW KNOWLEDGE Major: Theory and Methods of Teaching Mathematics Code number: 62.14.01.11 SUMMARY OF THE DOCTORAL THESIS OF EDUCATION SCIENCE NGHE AN, 2013 ii The work was done at Vinh University The supervisor: Assoc Prof Dr Tran Vui Reviewer 1: Assoc Prof Dr Tran Kieu Reviewer 2: Assoc Prof Dr Trinh Thanh Hai Reviewer 3: Dr Tran Trung The thesis will be defensed in front of the Board of University level in Vin University, On .hours .date .month .year……… This thesis can be found at: - National Library of Vietnam, 31 Trang Thi street, Hanoi - Information Center – Library Nguyen Thuc Hao, Vinh University INTRODUCTION The mathematical objects shown on the blackboard or on paper are static, the characteristics and their relationship often described by the language or symbolic representations However, in a dynamic geometry environment, the object will show the characteristic behavior and can become the raw material used to "experiment" The idea for students to perform experiments as an empirically they often in the other sciences subjects of mathematics educators have become ever more viable in the dynamic geometry environments They can conduct the experiment for suggesting hypotheses, verifying results, detecting invariants, finding relationships to constructing new knowledge With this study, we hope to survey the status of students learning mathematics that forms the foundation for using software in teaching, research the constructing of electronics teaching and learning environments, integrating the dynamic mathematics models to the environment to help improve students' ability to discover new knowledge CHAPTER 1: INTRODUCTION STUDY BACKGROUND 1.1 Introduction With the student center of the teaching process, the autonomy of students in discovering knowledge was impressed In a positive learning environment, they have many more conditions in the exchanges, study with classmates through learning group or through interaction The dynamic manipulative models designed on the software can help students to discover knowledge through operations on that model with initial guidance Students will reduce the dependency on teachers in the reception of new knowledge Instead, base on to the active and their initiative actions, students will discover new knowledge with teachers as mentors The development of a positive learning environment depends on many factors such as the lesson contents, the existing infrastructures, teachers’ capacity, the ability to adapt to the environment of the students Teachers actually are researchers based on reality to build a suitable environment, using appropriate teaching equipment and integrating process of equipment into that environment for meeting effective teaching and learning 1.2 Research needs Dynamic geometry software, with its original strength, which can preserve the invariance of the geometrics images or keep the rational role relationships of objects With the interaction between students and computer models, students can discover, explore new knowledge for themselves With learning environments that use teaching facilities in the application of information technology, students have more opportunities to explore mathematical knowledge The problem is the need of how to create learning environments, design and build dynamic mathematics models as well as the integrating models into electronic learning environment to be effective in teaching and learning 1.3 The study The study needs to find a way to integrate the dynamic mathematics models to this learning environment to build e-learning environments to enhance mathematical thinking and experimentation We chose the research topic: "Integrating dynamic manipulative models with electronic mathematics teaching and learning for enhancing the students’ ability to explore new knowledge" 1.4 Research purposes The purpose of the thesis is to:  Study the effectiveness of dynamic manipulative models in supporting to enhance the students’ ability to explore new knowledge  Research and develop electronic teaching and learning in supporting to enhance the students’ ability to explore new knowledge  Develope the students’ ability to explore new knowledge through abduction and induction when conducting the investigation on electronic dynamic manipulative models  Research on the experiment mathematics on dynamic manipulative models in supporting students enhancing their ability of exploring new knowledge 1.5 Research question With the goal of integrating the model with environmental manipulation in electronic teaching to enhance the ability to discover new knowledge of students, this study proposed scientific hypothesis as follows: If the model integrated with environmental manipulation in electronic teaching a scientific basis will enhance the ability to discover new knowledge through experimentation students math Research Question 1: performing math operations in the mathematical model of the impact of electronic support improving the ability to discover new knowledge of how the students? Research Question 2: Building the e-learning environment, such as how to effectively support students in improving the ability to discover new mathematical knowledge? Research Question 3: Develop the ability to discover new knowledge through the student's reasoning and inductive extrapolation models on the electronic manipulation like? Research Question 4: Empirical estimates of the impact model sports electronic support students enhance explore how new knowledge? 1.6 The meaning of the study The findings of the study will help teachers create e-learning environment at the high school level math, including the integration of mathematical models to support dynamic manipulations improve students' ability to care break new knowledge, thereby fostering self-exploration capabilities, learning, enhance creativity in problem solving and decision making CHAPTER 2: LITERAL REVIEW 2.1 Background history 2.1.1 The development of the learning environment In recent years, parallel to the traditional learning environment, have emerged and developed new learning environment First of all, there is the appearance of the teaching facilities of information technology applications such as overhead, projector, designed on the model of the software used mixed with auxiliary equipment such as tables, handout episode With the desire to help students more easily in the knowledge discovery, these new devices are developed and constantly upgraded Number of schools equipped with these devices and more A number of studies have been done with this new learning environment as research to create electronic teaching equipment, namely models on software design activities, to support student learning Math 2.1.2 The shift in mathematics education In the 1970s, the interest in mathematics education mathematics Tan (New Mathematics), which emphasizes the development and introduction of new content as the algebraic structure, and transformation matrix In the 1980s, back to the basic theme (Back- to- Basic) to be concerned, at which math skills are taught as a core content of school mathematics So the math content related to the development of skills for students to be included in many of the math textbook Deductive inference from that is emphasized in the classroom Students acquire the knowledge, methods, and forms to practice math skills to apply them in homework However, in the 1990s it was realized the problem is that students need to learn and need to learn the most when tectonic theory commonly accepted in the computing educators worldwide 2.1.3 Dynamic Geometry soft wares and Applications Dynamic geometry environments are becoming popular in schools There are many different arguments about the effectiveness of dynamic geometry software in mathematical reasoning of students However, the dynamic geometry software has proved useful in the development of their reasoning The learning environment needs a new course teaching the theory underlying sense reasoning 2.1.4 The teaching theories that affect mathematics education reform 2.1.4.1 Activity theory With a focus on operations, activity theory emphasizes the basics of teaching methods are "exploiting the potential of each activity content as the basis for the organization of the teaching process achieve targets set" Operational perspective in teaching methods, which can be expressed in the mainstream of thought: Give the students perform and practice the activity component is compatible with the content and teaching objectives, suggested activities base for learning activities; led students create new knowledge, especially knowledge as the means and methods of operating results, distribution levels and activity as a basis for teaching process control 2.1.4.2 Situations theory The basic premise of the theory of situations of Brousseau's knowledge, or use the built-in situations are determined by the suppression of this situation Therefore, it is thought that by creating artificially suppressed, the teacher has the ability to stimulate students to build a certain type of math knowledge The interest in the meaning of knowledge on the subject has made theoretical situation more humane 2.1.4.3 Constructivism theory Theory tectonic theory is somewhat similar to the situation because it is particularly interested in how people learn Basically, learning theory associated with the interaction of two factors: the diagram of learner knowledge and new knowledge 2.2 Theoretical Framework 2.2.1 Basic constructivism The view is created for that, the development of mathematical ideas are explained through interactive culture of human society, in which special attention is devoted to examining how the performing multiple systems, symbols and tools provide an opportunity for the formation of meaning 2.2.2 Constructivism in education Tectonic theory, as a theory in psychology, and cognitive talk about how people learn , it does not provide the specific model of teaching , and also suggest what should not be in programs Tectonic theory , here are the key ideas to help people grasp the significance of learning and thus more applicable to general education and mathematics education in particular has been formed 2.2.3 Constructivism perspective in Teaching and Learning mathematics Tectonic theory focuses on the role of internal cognitive processes and "install data" in the first private individual students in their own learning math ¬ reachable collaborative learning organizations to create opportunities for students to exchange and discuss approaches to understand their problems According to this view, there are many approaches to improving math education: finding different ways to attract individual student participation, develop informationrich environment to study math, preparing multiple threads accounting or related problems to help students to empirical evidence 2.2.4 Constructivism theory for e-learning Tectonic theory is a theory of the superiority of teaching used in the current educational reform This theory encourages students to construct their own knowledge based on personal experiments and directly applicable to their environment The learning of each individual student at the center of the learning process Theory tectonic theory is considered ideal for electronic learning (e -learning ) for the following reasons:  tectonic theory views the student is the center of the learning process  tectonic theory that knowledge is built and applied consistent with the empirical personal  tectonic theory views the student as entities rather than passive activities to fill information CHAPTER 3: METHODOLOGY AND RESEARCH PROCESS 3.1 Design the research process To answer the research questions outlined in Chapter 1, the study was conducted according to the following steps:  Research results and researchers have done studies to determine the strengths and the support of mathematical models sporting impact in improving the ability to discover new mathematical knowledge  Research studies , articles , results from previous studies had to examine the current learning environment through the investigation process and from which to build the mathematics learning environment that integrates active manipulation of the mathematical model , which defines the roles of teachers , students , the teaching model support , assessment methods and related issues  Research Design and integration of mathematical models sporting impact on the mathematics learning environment , especially e-learning environment used to build mathematical models on the dynamic geometry software in order to develop the ability to discover new knowledge by students  Research the available research results about reasoning , especially reasoning and inductive extrapolation from which mathematical models designed to manipulate support students in developing reasoning to extrapolate , extrapolation of the with inductive as well as conclusions on the role of software in the creation of a supportive environment for students to create extrapolation  Research the available research results, models designed to manipulate the ability to perform research experiments mathematics through manipulation of the model in improving the ability to explore is new students 3.2 Research Subjects Audience research is the process of interaction with the dynamic mathematical model manipulation, interaction with other students or student - teacher in the process of implementing the learning task Subjects of investigation and learning environment survey included 278 students from the class of Middle School, including two in the city of Hue and schools in neighboring districts Students will be surveyed about the learning environment and the personal opinion of them For the results of research on inductive inference and extrapolation , we experimentally on students from three classes 11A2 , 11B1 and 11B2 of the Hai Ba Trung Street , Hue The experimental results by the empirical estimates will be conducted in two classes of schools Phan Boi Chau , Nghe An 3.3 Research tool Research tools include the mathematical models designed on The Geometer 's Sketchpad software , tools, survey data analysis , lesson plan , handout 3.4 Methods of data collection To answer the research questions outlined in Chapter 1, we carried out the data collection methods for the following topics:  Collect data from the available studies on the ability to perform math and show the performances of dynamic geometry software as well as information collected through the experimental teaching in order to capture the effect of integration models to manipulate the environment in order to improve mathematics teaching ability to discover new knowledge by students  Collect data from sources such as books, newspapers , yearbooks , articles , materials science download via the internet on mathematics learning environment positive ; gather information about the teaching environment is locally through the survey as well as the existing research on positive teaching environment  Collect data from studies in extrapolating inferences as well as abroad At the time of the research data on deductive reasoning, inductive were also collected to assess suitable for the discovery of new knowledge through extrapolation on the dynamic model manipulation The data obtained through experimentation will be collected for analysis and evaluation  Collect data from the empirical study of mathematics in the development of mathematics, especially experimental studies using mathematical software 3.5 Method of data analysis  From the data gathered through the study of mathematics performances , we can now perform the operations on dynamic geometry software designed by the positive model From there we analyze mathematical models in supported operations enhance the discovery of new knowledge of students  From the data obtained through the research findings have , we evaluated the role of integrating the perspectives of students in teaching mathematics With the data collected through the survey process for students , we conducted statistical data to evaluate existing learning environment , the student's perspective , the ability to perform survey work on manipulation of the model Since then, we evaluate the ability to integrate the operations model for building dynamic e-learning environment  From the data collected through the research had to deduce inferences , inductive and extrapolation , we analyze the differences between the types of inference , focusing on extrapolation inference , especially extrapolate the impact on sport models The mathematical model has to be positive we analyze the possibility of developing applications for the extrapolation inference for students as well as design new models The data from the experiments will be analyzed to look for quantitative extrapolation of students doing surveys on dynamic models  From the data collected by the empirical study of mathematics , we analyze the empirical estimates of the historical development of mathematics as well as through experimental computer algebra software such as Maple , Mathematica  From these data we suggest that the empirical math students through the design pattern on dynamic geometry software 3.6 Scope of the study  The content of the experimental lesson: Some search problems to develop rules inductive reasoning and deductive problem developed extrapolation  Students participated in the experimental lesson: There are 174 students in relation to empirical thesis Including 110 students in the class 11A2, 11B1 and 11B2 of High School Hai Ba Trung Street, Hue City, 64 students in the class 11A1 and 11A2 specialized school of Phan Boi Chau, Nghe An  Objects Survey: Students at schools in the city of Hue, Hai Ba Trung High School (83 children), Dang Tran Con (73 children); students in neighboring districts: Ha Trung high school (52 children), To Huu (63 children) CHAPTER 4: INTERGRATING DYNAMIC MANIPULATIVE MODEL WITH ELECTRONIC TEACHING AND LEARNING MATHEMATICS ENVIRONMENT 4.1 The study results 4.1.1 The results for the research question 4.1.1.1 Mathematical representation There are many different definitions about performances in mathematics education Most mathematics education researchers to distinguish between internal and external 11 4.1.2.2 Surveying the learning environment We analysed the survey results for the 271 students have used computers from two high schools in the city centre of Hue (group A, including 156 children) and two schools in neighbouring districts (group B, 115 children ) on May 03, 2009 4.1.2.3 The feedback for the development of e-teaching and learning environment We conducted the survey through questionnaires for students of four secondary schools in four schools of Thua Thien Hue provice and feedback of students following deserve attention • Number of students exposed to computer • How often you use basic computer programs • The level of proficiency in perform basic operations on the computer • Level students / teachers of students using computers, projectors, lights in school 4.1.2.4 Some survey results Survey results showed that the average student in group A had time to use the computer almost reached to years , group B to cross the threshold from to years The standard deviation of the two groups were similar and low Survey how often they use basic computer programs showed average both groups A and B level between and in Group A despite frequent communication with the computer more The standard deviation of the two groups are almost the same and are quite high because there are many more activities surveyed and a rating Survey proficiency in implementing basic operations on the computer showed that in group A had an average of between levels and Overall, the children in this group can perform good actions either alone or have the help of others The students in group B average remains close to 2, the more you need the support of others than group A Comparison of standard deviations in the two groups was found in Group A have a more uniform level 4.1.2.5 Electronic teaching and learning mathematics The role of students and teachers in the teaching environment is described electronically as follows:  Students are actively promoting their abilities  The teacher acts as a mediator to guide their ideas to achieve the purpose of the lesson 12 4.1.3 Results for the research question 4.1.3.1 Type of inferences 4.1.3.2 Plausible inference By Polya, we protect their mathematical knowledge by inference proven but we support his hypothesis with plausible inferences 4.1.3.3 Inductive inference Mathematics, in a way that is suitable experimental material for the study of inductive inference Induction usually begins with observation and survey Inductive hypothesis is born as a result of observations, and has been verified by separate examples Subsequently, finding more cases separately is necessary to reinforce the theory Of course, if a case is found, the hypothesis contradicts that theory completely eliminated In contrast, the initial hypothesis is strengthened and more rational 4.1.3.4 Abductive inference In general, abdution is the process of inference to the best hypothesized to explain the observed results A process according to extrapolate inferences J Josephson and S Josephson (1996) is shown in the following steps: An event (events, results ) S is observed; Appears hypothese G explanation for S; No other hypothesis explains well for S as G Then G is the best explanation for S 4.1.3.5 The popularity of the abductive inference Despite its uncertainty, is deduced extrapolating an essential part of everyday life of the people When scientists form a hypothesis to explain the data they collected, they really are extrapolated inference In everyday life, extrapolating inferences presence almost everywhere For example, when people generate hypotheses to explain the behavior of others, explain the facts, phenomena … 1.3.6 Basic forms of abductive inference We describe the form of extrapolation in mathematics and illustrated through four examples These examples can be used to develop inferences extrapolated to students in the next section a Selective abduction: Choose the number of cases available a case can justify the conclusion b Creative abduction: When the case is available unexplained, need to find another case to justify the conclusions 13 c Observe abduction: Make observations during extrapolated to the case might justify conclusions have been d Manipulative abduction: Use to manipulate objects in the inference process to find the proper explanation The concept of manipulative abduction to cover a large part of the scientific findings where the role of the operations center and the results can sometimes be located in hidden form: activities can provide information allows researchers to solve the problem by implementing a suitable extrapolation process to build or select hypothesis 4.1.3.7 Some models for developing inductive inference Model Students observe the triangle pattern sequence is designed on GSP Model Calculate the first n terms of a sequence of consecutive odd numbers Model Calculate the nth term of a sequence of dots form a square ladder Model The model divided the circle by lines Model The model divided by the supply circle 4.1.3.8 Some models for developing abductive inference Model Constructing a triangle ABC, AB took over arbitrary point M and k = AM ratio measurements Taking B as center, point C of the order according to score AB points N k Similarly, taking C as the center, the point P from point A to follow the ratio k When M changes, the two triangles ABC and MNP have certain common features? Model For several axes including axes parallel to each other Each point x in the upper shaft is connected to a point f(x) at the bottom of the shaft in a straight line (x and f(x) can move on the number line by dragging the knob at point x and f (x)) Look for the relationship between two quantities x and f(x) 4.1.3.9 Evaluate some experimental results We chose classes 11 including 11A2, 11B1 and 11B2 for pedagogical experimentation We choose models to evaluate experimental results pedagogy, which helped develop two models extrapolate and infer the rest for development of inductive inference The experimental results are statistically rather than by numbers, we analyzed the thinking process of the children, the manipulation operations on the pattern a The model of building stairs 14 All students in the experimental lessons have to answer in the number of squares to be used in step 10 However, not all students make the right argument b The model of apple garden From the observation and manipulation to find the relationship between the windbreak and n, the number of children analyzed Windbreak each side of the garden to come to a conclusion Using analysis on the number n general, they can infer the number of tree windbreaks on the value n Final conclusions, n2 > 8n will be incorrect in any case extend the garden, however, the student's understanding is that they think will increase with the square shape faster than most forms c The model of two squares Teachers conduct experiments up two squares ABCD and EFGH a side, placed so that the top of the center coincides with ABCD E, F peaks move Next, teachers measured EMCN quadrilateral area is the intersection between the two squares When changing the position F, the value quadrilateral area EMCN not change By observing and manipulating models, to prove two triangles are equal, students can use the rotation angle of 900 E Center When it turned into a N M C was transformed into B With the use of rotations to prove successful, students use proven methods follow two equal triangles angle-side-angle case to create a way for their own interpretations From the observation area of the intersection of two squares is constant, a number of square EFGH to bring you a special position: EF//AB, now part of communication is also a square Since then they provided the other cases of this particular case d The model of total distance When extrapolating the number m is the length of the equilateral triangle, students have different approaches to explain this There you find the relationship between the distances of x, y, z corresponding to each high street in an equilateral triangle When determining the path of his explanations, students often go to the same persistence The following explanation of a student rather cumbersome and also errors, but showed her perseverance to perform the PE and PF distance of distance PQ, with GQ = AH 15 4.1.4 Results for the research question 4.1.4.1 Experiment mathematics Experimentation is an activity or operation is performed under defined conditions to detect, verify, and illustrate a theory, hypothesis or event 4.1.4.2 Some of experiment mathematics models Requirements of a good support model for conducting empirical estimates:  The parameters, the initial conditions can be changed  The model objects in relation to each other mathematically rigorous  Demonstrate the intermediate process in motion and change  Empirical Results only appear after experimentation  The experimental results can be observed and analyzed easily The model introduced here is our design on The Geometer's Sketchpad software (GSP5) in calculus in high school theme Model Fixed point of a function containing parameters Model The graph of the functions f(x), f ' ( x) and f '' ( x) Model The graph of the derivative function Model The graph of the antiderivative function 4.1.4.3 The experiment mathematics role of dynamic manipulative models We emphasize the role of experimental models of mobile computing: a Illustrations intuition and mathematics content understanding b Detection of events, rules and relationships c Urbanization to express the data structures or rules d Check closely, confirm or refute the hypothesis e Suggested approach for proving inference f Reduce the cumbersome calculations by hand g Verify the results 4.1.4.4 Evaluate some experiment teaching results We evaluate the use of dynamic manipulation, collaboration between students and the construction of knowledge about the very little quantity in two computing tasks: limitation of sequences of numbers and the derivative concept 16 CHAPTER 5: CONCLUSIONS, INTERPRETATIONS AND APPLICATIONS 5.1 Conclusions and interpretation 5.1.1 The conclusion to the research question 5.1.1.1 The teaching approaches by dynamic multiple representations We should not assume that people will perceive the same mathematical knowledge from each other only as a performer Performing multiple opportunities for the learners to study options for working performers, performers manipulate to create the relevant variants Performing multiple active instances is not a mathematical knowledge in different forms, but each also has its representation in the mathematical relationship closely Each change in representation may entail changes in the other representation to help students see the relationship between the performers as well as the opportunity to discover their laws, accounting invariant study, hypothesize, test hypotheses 5.1.1.2 The role of mathematical representations  The performances provide students with effective thinking tools  Performing visual for students providing a cost effective learning environment  The harmonious combination between performances help teachers better support students create new knowledge  Use various representations to help students reach the essence of the problem, then finding out how to solve the problem  Information technology support for the design of better performing multiples 5.1.2 Interpretation for the research question 5.1.2.1 The dynamic manipulations on representations The impact on the operations performed under the support of the dynamic geometry software has become more interesting to students than ever before You drag the points, lines; rotate objects as if you work with them so directly Indeed, such a move for the two ends of the line segment (the segment AB as point B), the computer will perform high-level tasks as follows: (1) new updated coordinates for the peak B on the screen, (2) delete the image point B in the same location, (3) develop a B in the new coordinates, (4) delete the line segment connecting points A and B ex, and finally (5) construction period straight AB to point B in the new location 17 5.1.2.2 Relation between the representations The performances keep the various aspects of mathematical knowledge With each performance, an aspect of knowledge that is expressed in focus For example, the concept of the slope of the tangent to express aspects of image clarity and "slope" as expressed through visual representation and the intuitive There is a need for students to see math knowledge under various performances 5.1.2.3 The environment for mathematics explorations The dynamic mathematical model manipulation become an essential component in exploring mathematical environment The final performances are vividly expressed with mathematical closely linked For each object that students are interested, they can perform the operation to capture the object properties, said its relationship with other objects Environment mathematical discoveries based on the dynamic geometry software is optimized for this task 5.1.2.4 Mathematical representations, behaviour perspective and constructivism perspective Study the representations will serve as a bridge between the theoretical behavior (emphasizing the outside) and tectonics theory (emphasizing the inside) The relationship between the performances were particularly interested in and we can best electronic computing environment thanks to the support of dynamic geometry software Through interaction with the environment to perform math on electronic payment systems represented within the student's development 5.1.3 Conclusions for the research question 5.1.3.1 Evaluation of feedback results The familiarity with computers will help students focus on mathematical tasks performed However, in the survey of the level of use of computers, lights, projectors showed only occasional teacher or almost no opportunity for students to carry out the survey estimates This may be due to the following reasons:  Lack of computers for tasks they perform  The lesson plan is still in the form of performance, knowledge accounting information  Teachers not master the technology  The school shall have the use of information technology requires more preparation time 18 In summary, although the teachers have found the benefits of implementing school students towards the implementation of the survey estimates in e-learning environment but also the implementation of difficult and subjective to objective 5.1.3.2 Building electronic teaching and learning mathematics environments For teachers, the dynamic model of electronic manipulation has been designed and is ready to be used immediately for the school The problem is that the math teachers need to build e-learning environments in accordance with classroom towards enhancing positive and proactive student, the model integrates the electronic manipulation of a component core in that environment to support children math 5.1.3.3 Install the electronic teaching and learning mathematics environments The school depending on its conditions, usually equipped with computers for classrooms in different forms, from simply having a computer in a classroom equipped with modern The software is designed geometry suitable for the different types of it Teaching plan, of course, also changed to match the existing conditions Class with one computer In this case, each group will have the opportunity to use computers in a short time during school hours A single computer without a projector or large screen will be limited in use as a presentation tool A class has many students will have difficulty tracking on a small computer screen A computer and a projector The dynamic geometry software is designed to work well with the projector You or a student can manipulate the model to survey the class to ask questions like: "Next we'll try to do?", "Should I build one straight line ?" , "Object should I move ?" or" If this object moves, what will happen? With a projector, original and your students can prepare for the presentation, or students can introduce their search on the projector The class has a few computers If you can divide the class into small groups of to students so that each group has a computer, you will be a plan to teach through the computer survey A modern computer lab Experimental use of the GSP teachers in the classroom suggests that, even if enough computers for all students individual work, they should also work together in pairs Students will learn best when they communicate with each other what they are learning, and students can work together to coordinate and support the idea of learning together 19 5.1.4 Interpretation for the research question 5.1.4.1 Integrate the students attitude on teaching Although the approach to incorporate the views of students on teaching has been widely discussed, people are still concerned about the appropriateness and effectiveness of the approach The study also showed that students' perceptions of the learning environment has a great impact on the effectiveness of their learning rather than inherent characteristics of the environment itself Teachers often face difficulties when teaching students about perceptions of the learning environment are often not consistent For example, students may not like the learning environment causing decreased interest in learning the children, but the teacher does not see that and think that the current environment is good for children and may increase the excitement of learning This context requires educators and teachers need to design learning environments to better suit the student's point of view so as to achieve higher academic performance 5.1.4.2 Electronic teaching and learning mathematics The deployment of e-learning environment can not take place in a short time It should start by examining students' readiness, preparation of teachers, facilities Next is the installation environment according to the existing conditions The implementation of teaching in e-learning environment is essential, however, not forced Reality shows are not always taught math in e-learning environment is more successful than traditional learning Not content in textbooks can also be designed to manipulate models supporting students create new knowledge mathematics 5.1.5 Conclusions for the research question 5.1.5.1 The relationship among the inferences Extrapolation inference is a form of reasoning to explain than it is to predict because the results can not be directly known Extrapolating similar to induction in that both are related to the findings However, while inductive discovery rules, then extrapolating trends discovered new facts Induction will help test a hypothesis extrapolated through experimental and increase the level of success in the test, ie increase the confidence level of the hypothesis 5.1.5.2 Combining inference with dynamic visual representations A combination of three types of inference with the visual representation is shown in the following diagram: 20 Figure Combining types of inference with dynamic visual representations 5.1.6 Interpretation for the research question 5.1.6.1 Observations and manipulations on dynamic visual representations The following note is of concern when designing the model: These stakeholders should be designed together with the same color The object should be colored up more separate Consider fixed between the spot and fade marks Considering the number of objects on the image to appear Survey before any case can the model 5.1.6.2 Manipulative abductive inference The experimental results also showed that students improved their operations on the model From the nature of the manipulation of trial and error, students gradually more careful in the manipulation and spend a certain amount of time to predict the outcome before performing operations In dynamic geometry environment, many theories given extrapolation and manipulation will help eliminate false hypotheses, reinforcing the hypothesis reliable In many cases, students can hardly hypothesize extrapolating if not done any further action 5.1.7 Conclusion for the research question 5.1.7.1 Explore new knowledge through experiment mathematics Technology has a huge role involves the experimental use of mathematics and mathematics education Experimental School math when it is part of the concern related to math education experiments on the specialized mathematical software With optimized for interactive dynamic geometry software, students should be encouraged to conduct surveys, experiments on mathematical models in order to achieve a deeper understanding of mathematics 21 5.1.7.2 Experiment mathematics and manipulative abductive inference Electronic environment mathematical models have integrated operations support for our students to conduct experiments using the accounting impact sports These actions help students formed the extrapolation inference operations During the survey on the model, the above reasoning can be increased reliability to become the best hypothesis That theory is further strengthened through the dynamic manipulations This process continues in operation solving mathematical tasks students through the software 5.1.8 Interpretation for the research question 5.1.8.1 The divergence in the investigations Clearly, the students in the class will use the various operations performed on the same in accordance with the direction of their thinking This stimulates creativity itself and not be influenced by trends available The different reactions of the audience impact upon students to help guide them to the next operation and the process repeated until the students get what she wants 5.1.8.2 Collaboration in the experiment mathematics environments Even in a class each student has a computer, but the cooperation in the empirical model estimates are needed and should be encouraged Collaboration in e-learning environments, like in a traditional environment Students exchange ideas with each other, and cooperate to perform the tasks The task assigned to members can interchangeably, creating uniform experiences between members as well as create opportunities for all members perform all the different stages of the task mathematics Clearly, the empirical model, each student has its own approach and acknowledge, represent issues through the prism of the children 5.1.8.3 Experiment with and without dynamic manipulative models The experiments on the software creates different experience for students For example, the construction of a line from a point outside the contact circle is fundamentally different from a point on the circle So as you drag point P until it lies on the circle, the rendering process is not for us anymore because tangent point with P tangent to build can not be identified anymore 5.2 Application 5.2.1 Application for teachers and students The model is designed to support teachers can use in the lesson plan to help students achieve higher efficiency in mathematics The model also helps teachers use to teach a particular lesson, save time building models Indeed, the design of a 22 computer model is not simple, each model must often have design ideas, solve many modeling problem, which combines the tools of dynamic geometry software which is reached purpose 5.2.2 Application for pre-service teachers Part of the thesis is used for modules related to the use of information technology and communication in teaching mathematics Students are acquainted with the design model manipulation on dynamic geometry software The traditional method of assessment should be integrated in a systematic way with the products evaluated in practice The combination of assessment and evaluation according to the product set is necessary to improve the quality of the evaluation process of student learning 5.2.3 Application for further studies The findings in this thesis creates direction for further research issues are addressed 5.2.3.1 Research on mathematical representations The mathematics education researchers interested in the students' performance outside but the results are still limited The use of math when performing students perform academic tasks should be studied further, many more pupils Performing outside and show the world the tendency of students Foreseeing this, teachers can help students optimize the performances to support the tasks they complete their studies In addition, in performances, which are not visible, tangible progress when the students perform accounting tasks is a topic for further study interference between school education and neuroscience, even although outside the main performers are clearly shown by the performers inside 5.2.3.2 Integrating the attitude of students The integration of the students' views on teaching mathematics is not mathematics educators interested Typically, the research findings or proposed measures to help students understand this knowledge, acquire other methods However, these measures only to answer the question "How to teach?" While answering the question, "How to learn?" Re-oriented to teaching 5.2.3.3 Experiment mathematics The results of the thesis research in empirical estimates are still many limitations, such as with different conditions, whether the environment can make effective empirical estimates? The effects of experimental mathematics to mathematical tasks require only paper and pen to use? 23 CONCLUSIONS AND RECOMMENDATIONS OF THE THESIS Through researching process, thesis were obtained the following results: The dynamic model that we designed in accordance with accounting software programs Secondary education and practice in the classroom has really good support for pupils proceed to the direct manipulation of mathematical objects to survey create new knowledge The teaching environment with integrated electronic payment model in our experiments have supported students to manipulate, observe invariance, which predicted hypothesis, detection rules for examination breaking new mathematical knowledge The teaching environment that we electronically building contains the mathematical facts surprising when students observe and manipulate, since they have the opportunity to perform inference and inductive extrapolation a positive way, through proactive model to explore new knowledge Computing environment has allowed experiments students conduct the survey on the model appropriately according to their level of understanding of mathematics so that they themselves create new knowledge in mathematics for themselves The study results allow us to conclude that: Students will have the advantage opportunities to construct mathematics knowledge for themselves if such knowledge is expressed in different forms of representations through dynamic manipulative model designed on the mathematics software It is need to build electronic teaching and learning mathematics environment integrated dynamic manipulative models to support students to manipulate objects for observing the mathematical invariants to construct mathematical knowledge Proficiency in basic tasks on a computer is necessary for students to perform mathematics tasks The visual representations containing mathematical challenges that effectively support students developing abductive and inductive reasoning in the investigation process, solve math problems in order to come to a reasonable solution Experiment school mathematics is a division of interest and research practice of mathematics education Students should be encouraged to conduct experiments on the dynamic manipulative model to reach a deep mathematics understanding 24 LIST OF PUBLICATIONS I Journal papers Nguyen Dang Minh Phuc (2008), Design some dynamic manipulative mathematics models in internet environment for supporting high school students in constructing mathematics knowledge, Journal of Science and Education, College of Education, Hue University, ISSN 1859-1612, No 02(06) 2008, pp 113-121 Nguyen Dang Minh Phuc (2010), The meaningful feedback of high school students towards mathematics e-learning enviroment, Journal of Education, Ministry of Education and Training, ISSN 0866-7476, No 239, Vol (6/2010), pp 42-44 Nguyen Dang Minh Phuc (2010), Designing teaching equipments using multiple representations to support high school students in exploring knowledge of derivative, Journal of Educational Equipment, Ministry of Education and Training, ISSN 1859 – 0810, No 59 (Jul – 2010), pp 21-22 & 41 Nguyen Dang Minh Phuc (2010), Develop abductive inference through electronic dynamic manipulative mathematics models, Journal of Science, Vinh University, ISSN 1859-2228, Vol 39, 2A, Aug.-2010, pp 51-59 Nguyen Dang Minh Phuc (2011), The role of mathematical experiments of dynamic geometry softwares, Journal of Science, Hanoi University of Education, ISSN 0868 – 3719, Vol 56, No 5, pp 101-108 Nguyen Dang Minh Phuc (2012), Performance assessment based on the eportfolios for the mathematics pedagogical students, Journal of Science, Hanoi University of Education, ISSN 0868 – 3719 (Accepted for publication) II Conferences papers Nguyen Dang Minh Phuc (2010), The students’ abductive inference when conducting observations and manipulations on dynamic visual representations, Proceedings of Science conference of PhD Students 2010, Vinh University, pp 98103 Nguyen Dang Minh Phuc (2011), The role of dynamic visual representation on supporting student exploring limitation of functions, Proceedings of National Conference about School Mathematics Education, Ministry of Education and Traning, Vietnam Publishing House, Mar 2011, pp 494-499 Nguyen Dang Minh Phuc (2011), Design dynamic mathematics models in Etextbooks to improve students’ abductive inferences, Proceedings of APECUbon Ratchathani International Symposium 2011, Innovation on Problem Solving-Based Mathematics Textbooks and E-textbooks, Ubon Ratchathani University, Thailand, pp 117-125 25 Nguyen Dang Minh Phuc (2011), Manipulative Abductive Inference via Experimenting on Dynamic mathematics models, Proceedings of the 4th International Conference on Science and Mathematics Education (CoSMEd), “Transforming School Science and Mathematics Education in the 21st Century”, Penang, Malaysia, pp 103 Nguyen Dang Minh Phuc, Pham Sy Nam (2012), Experiment School Mathematics in Constructing knowledge of Infinitesimal small quantities, Proccedings of the 5th International Conference on Educational Research (ICER) 2012, Challenging Education for Future Change, September 8-9, 2012, Khon Kaen University, Thailand, pp 309-319 III Books 1.Tran Vui (editor), Le Quang Hung, Nguyen Dang Minh Phuc (2007), Exploring Algebra & Analysis 11 with The Geometer’s Sketchpad Education Publishing House, Hanoi, Vietnam, 2007 (joint work with Tran Vui & Le Quang Hung) 2.Tran Vui (editor), Le Quang Hung, Nguyen Dang Minh Phuc (2009), Designing of teaching and learning mathematics models in secondary school with The Geometer’s Sketchpad, Vietnam Education Publishing House, 2009 (joint work with Tran Vui & Le Quang Hung) 3.Tran Vui (editor), Le Quang Hung, Nguyen Dang Minh Phuc (2009), Exploring Analysis 12 with The Geometer’s Sketchpad, Vietnam Education Publishing House, 2009 IV Research projects Participant of science and technology research project at main Ministry level: “Integration of active models in Lesson Study with a focus on mathematical thinking to develop mathematics teachers’ profession”, Code: B2008-ĐHH 0341 TĐ (2008 – 2010) Principal Investigator of science and technology research project “Teaching and learning mathematics through the internet with the support of dynamic models”, Thua Thien Hue, Department of Education and Training (2008 – 2010) Principal Investigator of science and technology research project at University level: “Developing students’ abductive and inductive argumentation by using dynamic manipulations in teaching mathematics”, Vinh University (2009 – 2010) ... supervisor: Assoc Prof Dr Tran Vui Reviewer 1: Assoc Prof Dr Tran Kieu Reviewer 2: Assoc Prof Dr Trinh Thanh Hai Reviewer 3: Dr Tran Trung The thesis will be defensed in front of the Board of University