MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF PEDAGOGY HO CHI MINH CITY HOA ANH TUONG Specialization: Theory and Methods of Teaching and Learning Mathematics Scientific Code: 62.14.01.11 SUMMARY OF DOCTORAL THESIS ON EDUCATIONAL SCIENCE HO CHI MINH CITY– 2014 THE THESIS COMPLETED IN: UNIVERSITY OF PEDAGOGY HO CHI MINH CITY Supervisor: Assoc Prof Dr Tran Vui Reviewer 1: Prof Dr Dao Tam Vinh University Reviewer 2: Assoc Prof Dr Nguyen Phu Loc Can Tho University Reviewer 3: Dr Le Thai Bao Thien Trung University of pedagogy Ho Chi Minh city The Thesis Evaluation University Committee: UNIVERSITY OF PEDAGOGY HO CHI MINH CITY Thesis can be found at: - General Science Library of Ho Chi Minh City - Library of University of Pedagogy Ho Chi Minh City THE PUBLISHED WORKS OF AUTHOR RELATED TO CONTENT OF THESIS Hoa Anh Tuong (2009), Lesson study-a view in researching mathematical education, Journal of Science and Education, Hue University’s College of Education, ISSN 1859-1612, No 04/2009, pp 105-112 Hoa Anh Tuong (2009), Research to make opportunity for students to communicate mathematics, Journal of Education, Ministry of education and training, ISSN 0866-7476, No 222 (period 2-9/2009), pp 50-52 Hoa Anh Tuong (2010), Mathematical creativity in teaching exercises, Journal of Saigon University, ISSN 1859-3208, No 04 (9/2010), pp 54-60 Hoa Anh Tuong (2010), Using lesson study in the lesson “Area of a Polygon”, Journal of Science, Ho Chi Minh city University of Education, ISSN 18593100, No 24 (12/2010), pp 133-140 Hoa Anh Tuong (2011), Thalès theorem-A research to improve the quality of teaching and learning, Journal of Science, Ho Chi Minh city University of Education, ISSN 1859-3100, No 27 (4/2011), pp 54-61 Hoa Anh Tuong (2011), Approach a problem by solving different ways, Teaching and learning today, Journal of Vietnam central learning promotion association, ISSN 1859-2694, No (2011), pp 59-60 Hoa Anh Tuong (2011), To look at the problem in different ways, Journal of Science, Saigon University, ISSN 1859-3208, No 07 (9/2011), pp 105-111 Hoa Anh Tuong (2011), Using the external visual representations in mathematics for teaching students grade 6, Journal of Science, Vinh University, ISSN 1859-2228, Vol 40, No.1A (2011), pp 56-65 Hoa Anh Tuong (2011), Using “Open–ended problem” stimulus student to communicate mathematics, Journal of Science, Ho Chi Minh city University of Education, ISSN 1859-3100, No 31 (10/2011), pp 121-124 10 Hoa Anh Tuong (2012), Approaching “Open–ended problem” helps students study geometry actively, Journal of Science, Vinh University, ISSN 18592228, Vol 41, No 1A (2012), pp 85-91 11 Hoa Anh Tuong (2013), Focus on innovative teaching method-a view of practical researcher, Journal of Science, Saigon University, ISSN 1859-3208, No 14 (6/2013), pp 81-87 12 Hoa Anh Tuong (2010), Theoretical basis of constructivism theory in mathematics teaching, Proceedings of the scientific conference of master students and PhD students in 2010, Ho Chi Minh city University of Education, pp 92-102 13 Hoa Anh Tuong (2010), Lesson study- Theoretical basis applying in mathematics teaching, Proceedings of the scientific conference of master students and PhD students in 2010, Ho Chi Minh city University of Education, pp 103-116 14 Hoa Anh Tuong (2012), The Use Of Visual Representation In Reasoning And Expanding Mathematics Problem: Lesson Study On The Area Polygon, 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 417-424 15 Hoa Anh Tuong (2013), Applying "open - ended task" to help secondary students to communicate mathematics, Proccedings of the 6th International Conference on Educational Research (ICER) 2013, ASEAN Education in the 21st century, February 23-24, 2013, Mahasarakham University, Cambodia, pp 394-405 16 Hoa Anh Tuong (2013), Solution to decrease distance between training teachers of education mathematics and teaching mathematics of new teachers in vietnamese secondary school, International Conference on Mathematical Research, Education and Application, December 21st-23rd, 2013, UEL, VNUHCMC 2013, pp.105 (abstract) 17 Hoa Anh Tuong (2014), Apply model of lesson study in teaching mathematics, Proceedings of the scientific conference on the teaching of natural sciences in 2014, An Giang province, pp 127-134 INTRODUCTION Definition of terms Mathematical communication is a way of sharing ideas and clarifying understanding Through communication, ideas become the objects of reflection, refinement, discussion, and amendment The communication process also helps to build meaning, permanence ideas and makes them public (Lim, 2008) Mathematical communication competence: including the disclosure is our own political opinions about the mathematics problems, understand people's ideas when they present the matter, express their own ideas crisply and clearly, use mathematical language, conventions and symbols (Pham Gia Đuc Pham Đuc Quang, 2002; Mónica Miyagui, 2007) Open-ended problems are often thought of as tasks for which more than a single correct solution is possible (Erkki, 1997) Foong (2002) describes the open-ended problems as “ill-structured” because they comprise missing datas or assumptions with no fixed procedures that guarantee a correct solution Lesson study is a professional development form in which research on teaching and learning in classroom is carried out systematically and collaboratively by a group of teachers in order to improve their teaching practices (James W.Stigler & nnk, 2009; Nguyen Thi Duyen, 2013) A group of teachers collaboratively designs the lesson plan, implements and observes the lesson in the classroom, discusses and reflects on the lesson which is taught, revises the lesson plan, and teaches the new version of the edited lesson plan A study lesson is a lesson that the lesson study group chooses to explore in the lesson study process Introduction Mathematical communication and lesson study have been much interested in many countries: • “Communication process helps students understand mathematics more deeply” (NCTM, 2007) • “Communication has been identified as one of the core competencies for students to develop” (Luis Radford, 2004) Chang (2008) stated “The first goal of mathematical communication is to understand the mathematical language” Emori (2008) stated “All the mathematical experiences are done through communication Mathematical communication is needed to develop mathematical thinking because thinking development is explained by the manner's language and ways of communication” • Lesson study helps teachers continuously innovate teaching and improve learning for students In lesson study, teachers play a central role in deciding what is new in teaching and learning and directly implement innovation in the real classroom Through lesson study, teachers accumulate real experience, and improve lesson study In this study, we tried to design lesson plan discussed colleagues by the process of lesson study in order to provide the opportunities for students to show, debate, deduce, and present the proof Since then, they need to communicate and evoke mathematical ideas in the process of constructing new knowledge We choose the research topic: "Using lesson study to develop mathematical communication competence for secondary school students." Purpose of the study • How to organize classroom to promote and develop students’ mathematical communication competencies • To research and design a number of lesson contents in mathematics grade to promote students to communicate mathematics • To look at the scale levels of mathematical communication competence are used in evaluating students through some of study lessons been studied experimentally Research Questions The first research question: How to use basic way of communicating mathematics effectively (mathematical representation, interpretation, argumentation, and presenting the proof) in mathematics classroom? The second research question: How to organize mathematics classroom that can promote and develop students’ mathematical communication competencies? The third research question: Which lesson contents in mathematics grade and how to design lesson plans create opportunities for students to promote mathematical communication process? • The fourth research question: How to evaluate the development of communicative competence of students through studied lessons? Research tasks • To find out the basic way of communicating mathematics is suitable for secondary school students • To find out the conditions or situations in the classroom can occur the basic way of communicating mathematics • To choose some of study lessons implemented by lesson study process can create conditions for students to show the basic way of communicating mathematics • To give the scale levels of mathematical communication competence The significance of the study The thesis will be meaningful for education by: • Surveying the basic way of communicating mathematics which is expressed by Vietnamese students in the classroom • Proposing forms of teaching methods to develop mathematical communication competence of students according to their mathematical ability; thereby, forming confidence for Vietnamese students in sharing, discussing with peers and teachers • Designing some lesson plans in mathematics grade has many opportunities to promote students to communicate • Proposing the scale levels of mathematical communication competence The layout of the thesis The thesis included chapters except for the introduction and conclusion remark Chapter Mathematical communication in classrooms Chapter Lesson study and open-ended problems Chapter Methods Chapter Developing mathematical communication competence through lesson study Chapter Conclusion and recommendation Chapter The results of the research questions Summary of introduction Chapter Mathematical communication in classrooms 1.1 Origin of mathematical communication “Mathematical communication is a kind of communication Greek origin of the word communication is related with community… Mathematical communication is the communication in mathematics” (Isoda, 2008) 1.2 Communication in mathematics classrooms Communication in mathematics classrooms is the interaction between students-teacher-students, through verbal communication and using everyday language 1.3 Other studies in mathematical communication We present some mathematical communication practices in some countries In thesis, we choose the meaning of communicate mathematical is the way students express their mathematical perspectives (Brenner, 1994) Mathematical communication has three distinct aspects: Communication about mathematics, communication in mathematics, communication with mathematics 1.4 The role of mathematical communication in classrooms Emori (2008), “Mathematical communication is a key idea which is important not only for the improvement to learn mathematics but also for the development of necessary skills in the development of sustainable society knowledge” 1.5 The scale levels of mathematical communication competence 1.5.1 The six proficiency levels in mathematics In six proficiency levels in mathematics, from the third level, it has proficiencies: Representation, interpretation, argumentation and reasoning 1.5.2 The basic way of communicating mathematics In this research, I choose the basic ways of communicating mathematics: Representation, interpretation, argumentation, presenting the proof because these basic ways are related to communicating mathematics 1.5.3 Standard about communicating mathematics 1.5.3.1 Four forms of communication in Mathematics classroom Oral communication; listerning; speaking communication; and writing 1.5.3.2 Standard about mathematical communication Organize and consolidate their mathematical thinking through communication Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Analyze and evaluate the mathematical thinking and strategies of others Use the language of mathematics to express mathematical ideas precisely (NCTM, 2007) 1.5.4 The scale levels of mathematical communication competence 1.5.4.1 The scale levels of mathematical communication competence Level zero No communication Level Expressing initial idea - Students describe and present methods or algorithms to solve the given problem (not to mention the method is right or wrong) - Students know how to use the mathematical concepts, terminologies, symbols and conventions formally Level Explaining - The students explain the method acceptably and present reasons why they choose this method - Students use the mathematical concepts, terminologies, symbols and conventions to support their ideas logically and efficiently Level Argumentating - Students argue the validity of a method or algorithm Students can use examples or counter-examples to test the validity of the method or algorithm - Students can argue which appropriate mathematical concepts, terminologies, symbols and conventions they should use Level Proving - Students use mathematical concepts, mathematical logic to prove the given results - Students use mathematical language to present mathematical results 1.5.4.2 Example of mathematical communication We illustrated a lesson on October 3, 2010 in class 6A3 (51 students) of Saigon Practical High school Friday, August 26th 2011 was Vi’s birthday a) After days was of her mother’s birthday What should be the day and date of her mother's birthday? Why? b) What should be the day after 52 days from the birthday of Vi? Why? c) November 20th 2011 was Vi father’s birthday What should be the day? Why? • Student showed the basic way of communicating mathematics as follows: Representation: Students could use calendar to find the date in a week from 26/8 to 2/9; Monday schedule of every month from 17/10 to 21/11 to find a solution They knew days respectively 1week, 30 days or 31 days respectively month Interpretation: Students tried to find a solution Depending on their abilities, they had different ideas The easiest way was writtern a specific schedule in a week If you change the assumptions of the problem, such as sentences b) changes 52 to 520 and change the sentence c) November 20, 2011 to December 27, 2014; this way will be not suitable Student should understand which solutions could still be used when the requirements have changed It means that students are aware of the rationality of expression Argumentation: Students know how to use the law of day cycle, the day will repeat From there, students know how to find remainder in the division by to find the day In addition, students remembered how many days have in August and September, then to perform operations unless suitable to find how many days have in the month Students recognized the problems may have link to each other Presenting the proof: Students themselves understood how to solve the problem by listening to peers who demonstrated the problem • Evaluate the scale levels of mathematical communication competence: Expressing initial idea: Students described a way to solve the problem by writing the calendar in the week from 26/8 to 2/9, Monday schedule in the month from 17/10 to 21/11 They applied the algorithm based on the remainder in the division by They used reasonable mathematical operations: addition, subtraction, division Explaining: Students recognized the validity of each solution Students realized algorithm to find the remainder in the division by more reasonable than other solution Argumenting: Students expressed logical reasoning, solution of each problem clearly Proving: Students used algorithms to find the remainder in division 7, the language of mathematics, logical reasoning in the presenting the proof 1.6 Summary of chapter Chapter LESSON STUDY AND OPEN-ENDED PROBLEMS 2.1 Lesson study 2.1.1 Origin of lesson study 11 To use flexible representations: represented by language, visual images and symbols To train the ability to use language for students To establish and develop the wisdom qualities This topic can integrate real life situations in lesson plans and is not heavy proving Mathematical knowledge is not too difficult for students to understand • Solving problems by using equations is a difficult form (Algebra 8) for lower secondary school students Through this theme, we want: To give students some analytical skills, write a solution, express, choose unkowns to solve problem simply and briefly To help students find the representing the correlations between quantities by method of establishing tables which has many benefits To guide the students detect problems Students have the opportunity to debate, explore, and give comments 3.7.3 An overview of the study lesson 3.8 Summary chapter Chapter DEVELOPING MATHEMATICAL COMMUNICATION COMPETENCE THROUGH LESSON STUDY In teamwork, the leader prepares ideas and sends email to other members before proceeding to discuss: • Set the example question: To teach this lesson for all students to be interested positively, how will teachers design a lesson plan? • Provide ideas to design lesson plans to promote mathematical communication competence for students After that, team members exchange ideas to reach agreement • After the experimental teaching, team leader exchanged data collection to colleagues, colleagues comment the effect of lesson plan and adjust to further promote mathematical communication competence for students • Focus on discussion: Whether the activities are suitable for students or not? How these need to adjust reasonably? What kind of questions or teachers’ suggestions posed to students are suitable? How teachers’ activities promote students to communicate effectively? 4.1 Study lesson The area of trapezoid 12 a) Designing the lesson plan Mr Hoa: Students are good at mathematics, they themselves can find a way to prove by It is difficult for other students to implement We should have a clearer suggestion Mr Tuan: For example, “We can divide the trapezoid into two triangular areas that can be found” Mr Long: Students learned about the formula for the area of a triangle, square, and rectangle in elementary school Mr Tuong: If teacher suggests: Connect the diagonal AC to divide the trapezoid ABCD into two triangles Based on the formula for calculating the area of a triangle, set the formula for calculating the area of a trapezoid This is forcing for students! Ms Phan: If the teacher doesn’t suggest, can the students themselves establish a formula for calculating? Mr.Thong, Ms Trinh: When students have experience in setting up formulas for calculating the area of a trapezoid, they can set the formula for calculating the area of a parallelogram Mr Si: Based on the property “If the area of a shape is divided into H1, H2,…,Hn without common points, the area S of H will be calculated S = S1 + S2 + + Sn Teachers guide students to divide a trapezoid into triangles, squares, or rectangles that students have already known how to calculate the area (maybe they have many ways) Activity 1: What are the areas of figures having in figure 4.1 Why? (unit is one small square) Note: students are not allowed to use the formula to compute the area of trapezoid and parallelogram A B Figure 4.2 Trapezoid ABCD h D a C Figure 4.3 Parallelogram ABCD Activity 2: Give trapezoid ABCD with two base sides are a, b and altitude h as figure 4.2 Find different ways to set up the formula to calculate the area the trapezoid ABCD? Activity 3: Give parallelogram ABCD with base side a and altitude h as figure 4.3 Find different ways to set up the formula to calculate the area the parallelogram ABCD? A G B H E F 13 D K I C Activity 4: In figure 4.4, give trapezoid ABCD with mid-segment EF and rectangle GHIK Find different ways to compare the area of trapezoid ABCD with rectangle GHIK Figure 4.4 Trapezoid and rectangle have the same area Figure 4.5 Land of three family Activity 5: A piece of rectangular land ABCD belong to three families: An (trapezoid ABHG), Ba (trapezoid HGFE), and Ca (trapezoid FECD) as shown in figure 4.5 One day, three families discussed how the lines GH and FE can be changed to divide the rectangle ABCD into pieces of rectangular land that have the same area as the area of trapeziums at first Find way to help them b) Implementing and observing the lesson in the classroom When students are working, teachers monitor, observe and record the activities of students c) Discussing and reflecting on the lesson Activity 1: Visual figures support students in the exploitation to find different reasonable solutions Figure 4.7 Rearrange the figure into the Figure 4.6 Divided the figure into triangles and rectangles polygon has to know the area Activity 2: Students demonstrate the capability and know how to use activity in the general case Figure 4.8 Divide the trapezoid into two triangles Figure 4.9 Divide the trapezoid into two triangles and a rectangle 14 Activity 3: Students find different ways and express their deduction in every proof Students use mathematical language, mathematical conventions and mathematical symbols in describing the proof In addition, students know the parallelogram is a special trapezoid with two equal bases and transform the parallelogram into rectangles that had the same area Activity 4: The figure is available, students demonstrate more rapidly Activity 5: Students applied activity into activity In addition, from mistakes of other students, through interaction between teachers and students, they know how to solve the problem Teacher: “If we draw HM is perpendicular to AD at M How will we chance HM to KL that is parallel to HM (figure 4.12) so rectangle BKLA and trapezoid BHGA have equal areas?” Student: “The area of BHGA minuses the area of triangle IKH and adds the area of triangle IKH then equal the area of BKLA” Then, student deduces if the area of BKLA and the area of BHGA have equal areas, KL pass through midpoint I of GH segment because triangles IKH and IGL are equal and have equal areas Figure 4.12 Land of three family An, Ba, Ca reorganization d) Edited lesson plan • Find different ways to calculate the area of the figures in figure 4.1 • According to you, the following representations may help you have more ways to find the area of the figures in figure 4.1? Why? • Figure 4.17 Orientation finding the area of the figure “The following representation can F help you transform the trapezoid into g a triangle had the same area? Why?” ur i e 4.18 Trapezoidal and triangular have the same area 15 4.2 Study lesson Practice Area of a polygon 4.3 Study lesson Practice Area of a polygon 4.4 Study lesson Solving problems by using equations The lesson plan Argue about four available solutions in problem Problem 1: The distance between An’s house and school is 1200 m, The distance between Binh’s house and school is 1650 m Velocity of An is equal to Binh Time for Binh go to school is more than An minutes Calculate the velocity of An In your opinion, which solution is right or wrong? If solution is wrong which step is wrong? Why? In your opinion, which solution you should choose? Why? To solve this problem well, what is your experience? Solution 1: Solution 2: Denote velocity of An is x Denote velocity of An is x (km/h) Because velocity of An is equal to (x>0) Binh so velocity of Binh is x Because velocity of An is equal to 1200 Binh so velocity of Binh is x Time for An go to school is x (km/h) Time for Binh go to school is 1650 x Time for An go to school is 1200 x Time for Binh go to school is more (hour) 1650 than An minutes so we have Time for Binh go to school is 1650 1200 x − =5 equation: x x (hour) 1650 − 1200 450 Time for Binh go to school is more ⇔ =5⇔ = ⇔ x = 90 x x than An minutes so we have 1650 1200 In conclusion, velocity of An is 90 − =5 equation: x x 1650 − 1200 450 ⇔ =5⇔ = ⇔ x = 90 x x In conclusion, velocity of An is 90 km/h Solution 3: Solution 4: 1200m= 1,2km; 1650m= 1,65km; Denote velocity of An is x (m/min) (x> 0) minutes = hour 12 Because velocity of An is equal to Denote velocity of An is x Binh so velocity of Binh is x 16 (km/h) (x > 0) (m/min) 1200 Because velocity of An is equal to Time for An go to school is x Binh so velocity of Binh is x (minute) (km/h) Time for An go to school is 1, x (hour) Time for Binh go to school is 1,65 x Time for Binh go to school is 1650 x (minute) Time for Binh go to school is more than An minutes so we have 1650 1200 (hour) − =5 equation: x x Time for Binh go to school is more than An minutes so we have ⇔ 1650 − 1200 = ⇔ 450 = ⇔ x = 90 x x 1,65 1, − = equation: x x 12 (condition satisfied) 1,65 − 1,2 0, 45 In conclusion, velocity of An is 90 ⇔ = ⇔ = x 12 x 12 m/min ⇔ x = 5,4 (condition satisfied) In conclusion, velocity of An is 5,4 km/h Problem 2: A train goes from A to B in 10h40' If the train decreases the speed 10km/h, it will come 2h8’ later than to B Calculate the distance AB and the speed of the train Firstly, please select an unknown represented one quantity Secondly, tabulate to represent quantities and establish equations Finally, students write detail of the solution 4.5 Summary chapter Chapter THE RESULTS OF THE RESEARCH QUESTIONS 5.1 The results of the first research question a) Teachers make the mathematical representation suitably to help student solve open-ended problems and facilitate opportunities for students to communicate mathematically Students used visual representations to communicate with peers when they study by working group to form and consolidate new mathematical knowledge b) When students are asked to explain the understanding of mathematics as well as the results of their work with others, they can have self-regulation and develop mathematical knowledge certainly When students are asked to 17 explain their proof in the class, they argue which results need to be used to solve problem c) Students convince others by giving exact or creative solutions 5.2 The results of the second research question 5.2.1 Students’ communicational abilities in mathematical classroom Students are not good at mathematics, they don’t have any idea Students are good at mathematics, they actively build and present their ideas in class 5.2.2 Learning environment survey Students will learn more effectively when they discuss in groups or need the teachers’ help in time In addition, lesson content has visual representations and to be suitable for students promote them study positively When students study mathematics, students look for connectional problems, apply old knowledge to solve new problems, and find different ways to solve Learning integrated to solve the real-life situations would actually help students loving mathematics Students learn mathematics positively depending on making friendly learning environment and the lesson content 5.2.3 Organize class to promote mathematical communication a) The situations containing the conflict between old and new knowledge were really impacted on students’ cognition, urged them to perceive the benefits of learning mathematics from mathematical communication b) Collaborative learning environment help students more confidently present and post their opinions c) The open-ended problems encourage students communicate mathematically because they have different ways to solve problem 5.3 The results of the third research question 5.3.1 The role of lesson study Using lesson study to organize classroom communicate mathematically is expressed through the practicality of lesson study 5.3.2 Design some lesson plans The lesson plans put the strongest into the mathematical thinking and multiple representaions Teacher advantages the opportunity for students to 18 consolidate, inculcate contents, and base on old content to have new knowledge 5.3.3 A number of lesson contents in mathematics grade have many opportunities to promote students to communicate mathematics Lesson plan integrates open-ended problems and mathematical representation to real-life situations with the aim of: Students mobilize and apply the old knowledge to solve problem; Through specific case, students can predict the outcome in the general case; Develop the student’s capabilities such as inference, reasoning, explaining the nature of the problem 5.4 The results of the fourth research question 5.4.1 Evaluate the basic way of communicating mathematics Representation Students know how to use algebraic notation reasonably for unknown represented one quantity to present the proof simply and briefly Students use mathematical conventions to give the condition and the unit for unknowns which are illustrated through solving problem by using equation Students apply mathematical concepts appropriately Students understand the meaning of algebraic notation, and express in speech by different ways Students exploit visual figures that are reasonable and effective Interpretation Students know how to explain the reasoning, realize their mistake when peers present the proof Students find many different solutions and will realize the affect of each solution Students know how to explain the validity of each solution Students know how to make argument step by step of each solution Argumentation Students argue which knowledge should be used to solve problems Students argue the validity of each solution Presenting the proof Students present the proof by writing on the paper clearly and accurately When students say, there are some errors 19 5.4.2 Evaluate the scale levels of mathematical communication competence Study lesson The area of trapezoid Level and level 2: Students know how to demonstrate and give different ways to solve the problem Students have a long or shorter or brief proof Students use algebraic notation to support their mathematical ideas Students present their solution by speaking fluently, clearly, rather accurately Level 3: Students communicate reflectively and explain why they choose solution in each activity Students express a logical inference when they activities To solve new problems, students transform solved problems to familiar problem Level 4: Students use mathematical language, mathematical conventions, mathematical symbols and logical reasoning in presenting the proof Study lesson: Solving problems by using equations Problem 1: Through the reading of the solution is available, students understand the content and express their opinion which solution is right or wrong and analyze the error of wrong solutions (level and 2) Student comments should choose the best solution to apply solving the actual problem (level 2) Students themselves draw experiences when they solve this problem: depending on asking of the problem, we selected an unknown represented one quantity and units of this quantity appropriately (level and 3) Problem 2: Students themselves selected an unknown represented one quantity and units of this quantity appropriately (level 1) Students can communicate with peer through setting up tables represented the quantities (11 groups setting up right table and group setting up wrong table) Since the withdrawal of experience in problem 1, students carefully change time unit, create right condition for unknown represented one quantity Students actively learn, develop thinking depending on their cognitive capacity Students transfer a realistic situation to set up table represent and give simple or complex equations 20 From reading the problem carefully, students understand problems, connect the quantities to unkown represented one quantity and set to tables represent quantities (level 2) Through dialogue between teachers and students: Students confidently speak and write detailed answers At the same time, students have the opportunity to regulate the written and expressed ability (level 1, 2, 3) - - 5.5 Summary chapter Chapter CONCLUSION AND RECOMMENDATION 6.1 Conclusion 6.1.1 The conclusion to the first research question Students used multiple representations flexibly The basic way of communicating mathematics: interpretation, argumentation increasingly promoted when students are asked to solve open-ended problems, the intellectually conflicted situations stimulate awareness of students 6.1.2 The conclusion to the second research question Need to build mathematical learning environment cooperation in pairs or small group, situations that integrate to open-ended problems, realistic situations, visual representation Teachers organize language activities for students 21 Teachers create an enviroment for students not only to have solution, argue with teachers, but also discuss, debate with classmates to find out the answer and solve the posed problem by their opinion 6.1.3 The conclusion to the third research question To promote students to communicate mathematically, study lesson should be utilized: Attractive, interesting content, actual visual figure about the daily life rather than pure mathematics Students that read content can deduce immediately, not take much time for them to think Contents are not too complex, combine the use of different representation inferences: symbols, tables, diagrams Contents that are not only computing but also focusing on students’ thinking to find different ways to solve the problem and have unique, creative solutions 6.1.4 The conclusion to the fourth research question Evaluate mathematical communication process of student through: Students can express how to solve problems and refer reasoning about solution or basis that causes them to think how to solve it Students select and use appropriately mathematical representations Students express reasonable inferences in finding results Students explain the rationale for each solution Students use mathematical concepts, conventions, mathematical language in presenting the proof 6.1.5 The conclusion to the study lesson Study lessons of the research topic are different from lessons according to the current teaching methods in Vietnam as follows: In each lesson, lesson plans focus on students-centered: • Promoting the students’ ability to look figure carefully as well as understand the language, and use and link mathematical representations • Each action and each hint of teachers have a non-imposition and suggestive nature, provoke the learning ability of students • Students communicate reflectively, teacher’s oral foster students to express their thoughts and solutions - 22 • Students express their mathematical views, explaining how they solve the problems is feasible and reasonable, and present the proof clearly, precisely through using mathematical symbols, mathematical language • Teachers discuss, observe classroom to help them evaluate the effectiveness of the lesson plans; then, group of teachers adjust, revising the lesson plan to increase the ability to communicate of students Each lesson has the collaboration of teachers: • Teachers speak up their views in designing lesson plans • Teachers predict any difficulties that students may encounter and propose oriention to help students solve problems in time • Teachers agree steps in each activity towards creating the opportunity for students to argue If students don’t have any opinion, teachers will help them but the teachers’ help still promote them to express mathematical respective • In the opening activity: Teachers discuss carefully because it is the platform activity that support their students to solve the problem in the following activities Teachers orient students to have different ways to solve so as to evoke capacity to learn mathematics of students 6.2 Applying • The thesis is documented for practitioners, fellows and teachers In addition, the thesis helps secondary mathematics teachers earn experience about: point of view, teaching ideas, classroom context, the results of student, mistakes, the creativity of students, using the mathematical representation and mathematical communication competence of students • We can expand research to develop pupils' thinking In secondary schools, the visual figure is limited in textbooks, while the mathematical representation contributes students to learn mathematics We can deeply research on the using of mathematical representation in a way to develop mathematical communication competence and mathematical thinking for students 6.3 Proposal Some strategies that use lesson study to develop communicative competence of students: Select the lesson content in the program: not too focus on knowledge and skills, content is not too difficult and fits to the majority of students, 23 integrates with real life situations The amount of knowledge in each lesson much be appropriate Strengthen the cooperation, discussing, sharing experience of teachers about content of each lesson During group meetings, teachers raise questioning, guide students, and solve educational situations to help students develop the capacity of study and communication Provide model training of lesson study for teachers Reducing the number of teaching period for teachers when they participate in this model Learning environment: Teachers create opportunities for students to have small-group discussion, sharing, exchange and reflect, debate selfconfidently CONCLUSION REMARK Through research process, the thesis were obtained the following results: To unravel theory about lesson study through practicing lesson plan in which applies mathematics classroom have mathematical communication and the basic way of mathematical communication To suggest orientations for building the learning environment, how to organize classroom to promote and develop mathematical communication for students To suggest lesson content and design some lesson plans in mathematics grade to create many opportunities to promote students to communicate To propose and use the scale levels of mathematical communication competence to evaluate students in experimental lessons Result of the thesis can conclude: Students will learn more effectively when students try to find different ways to solve problem and need the teachers’ help in time When students are solving problem, students find different ways to solve new problems When students not understand anything in mathematics, they are always looking for more information to clarify the problem, and to understand new concepts in mathematics; they try to link what they know with them Students themselves learn to solve problems The mathematical representations support student to communicate mathematically Using harmonious mathematical representation supports 24 student to construct new knowledge Visual mathematical representation create mathematics environment efficiently Using different mathematical representations help student to access nature of the problem, to find out how to solve the problem Organizing classroom promoting mathematical communication should have a combination of the following factors: The situations containing the conflict between old and new knowledge, active cooperation between the members in classroom and how to design lessons During small group discussion, they exchange and express ideas by writing them down on paper and talking in speech When they express those ideas that they use specific signs such as diagrams, drawings, letters, symbols, icons , this means they use the mathematical representation Mathematics lessons have visual representations and consistent to students will promote them actively participate in the lessons Lesson plans integrate to solve real life situations to help students feel excitement and like to study mathematics We should design lesson plans with open-ended problems or integrate lessons to real-life situations and use the visual representation, multiple representations and mathematical thinking Meanwhile, students confidently express their opinions on raised problems through communication Students solve the open-ended tasks and use multiple representations to promote student to communicate mathematically Students know how to justify, argue, and resolve their problems to construct mathematical knowledge themselves Through above activities: teachers not only taught the content but also hold students activities, and evaluated students’ learning Teachers create an environment in mathematics classroom for all students to be interested in positive, selfdiscipline working, and to communicate reflectively to have new knowledge sustainably When students positively solve mathematical situations designed in the thesis, we found that we can develop students’ ability such as deduction, exploring, observation, description, analysis, comparison, explain, generalization Students express the basic way of communicating mathematics: Representation, interpretation, argumentation, presenting the proof In summary, students show levels of mathematical communication 25 competence from low level to higher level: level (expressing initial idea), level (explaining), level (argumenting), level (proving) The success of the thesis is to design lesson plans in mathematics grade by applying process of lesson study which is suitable for students' cognitive psychology, to create the learning environment "student-centered" and contribute to innovate in mathematics education in lower secondary schools towards the development of mathematical communication competence for students ... to school is x (km/h) Time for Binh go to school is 1650 x Time for An go to school is 1200 x Time for Binh go to school is more (hour) 1650 than An minutes so we have Time for Binh go to school... Time for An go to school is x Binh so velocity of Binh is x (minute) (km/h) Time for An go to school is 1, x (hour) Time for Binh go to school is 1,65 x Time for Binh go to school is 1650 x (minute)... distance between An’s house and school is 1200 m, The distance between Binh’s house and school is 1650 m Velocity of An is equal to Binh Time for Binh go to school is more than An minutes Calculate